Cornerstones
Emmanuele DiBenedetto
Ugo Gianazza
Partial
Differential
Equations
Third Edition
Cornerstones
Series Editor
Steven G. Krantz, Washington University, St. Louis, MO, USA
Editorial Board Member
Robert Lazarsfeld, Stony Brook University, Stony Brook, NY, USA
Peter Petersen, University of California, Los Angeles, CA, USA
Alan Tucker, Stony Brook University, Stony Brook, NY, USA
Scott Wolpert, University of Maryland, College Park, MD, USA
Cornerstones comprises textbooks that focus on what students need to know and
what faculty should teach regarding various selected topics in pure and applied
mathematics and related subjects. Aimed at aspiring young mathematicians at the
advanced undergraduate to the second-year graduate level, books that appear in this
series are intended to serve as the definitive advanced texts for the next generation
of mathematicians. By enlisting only expert mathematicians and leading researchers
in each field who are top-notch expositors with established track records,
Cornerstones volumes are models of clarity that provide authoritative modern
treatments of the essential subjects of pure and applied mathematics while capturing
the beauty and excitement of mathematics for the reader.
Emmanuele DiBenedetto • Ugo Gianazza
Partial Differential Equations
Third Edition
Emmanuele DiBenedetto (Deceased)
Ugo Gianazza
Department of Mathematics
University of Pavia
Pavia, Italy
ISSN 2197-182X
ISSN 2197-1838 (electronic)
Cornerstones
ISBN 978-3-031-46618-2 (eBook)
ISBN 978-3-031-46617-5
https://doi.org/10.1007/978-3-031-46618-2
© Springer Nature Switzerland AG 1995, 2010, 2023
This work is subject to copyright. All rights are reserved by the Publisher, whether the whole or part of
the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation,
broadcasting, reproduction on microfilms or in any other physical way, and transmission or information
storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology
now known or hereafter developed.
The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication
does not imply, even in the absence of a specific statement, that such names are exempt from the relevant
protective laws and regulations and therefore free for general use.
The publisher, the authors, and the editors are safe to assume that the advice and information in this book
are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the
editors give a warranty, expressed or implied, with respect to the material contained herein or for any
errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional
claims in published maps and institutional affiliations.
This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered
company Springer Nature Switzerland AG
The registered company address is: Gewerbestrasse 11, 6330 Cham, Switzerland
Paper in this product is recyclable.
Contents
0
1
PRELIMINARIES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
Green’s Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Differential Operators and Adjoints . . . . . . . . . . . . . . . . . .
2
The Continuity Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
The Heat Equation and the Laplace Equation . . . . . . . . . . . . . . .
3.1
Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
A Model for the Vibrating String . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Small Vibrations of a Membrane . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Transmission of Sound Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
The Navier–Stokes System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
The Euler Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
Isentropic Potential Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9.1
Steady Potential Isentropic Flows . . . . . . . . . . . . . . . . . . .
10 Partial Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1
2
3
5
5
6
8
11
13
13
14
15
15
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3c The Heat Equation and the Laplace Equation . . . . . . . . . . . . . . .
3.1c Basic Physical Assumptions . . . . . . . . . . . . . . . . . . . . . . . .
3.2c The Diffusion Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.3c Justifying the Postulates (3.3c)–(3.4c) . . . . . . . . . . . . . . . .
3.4c More on the Postulates (3.3c)–(3.4c) . . . . . . . . . . . . . . . . .
16
16
16
17
18
19
QUASI-LINEAR EQUATIONS AND ANALYTIC DATA .
1
Quasi-Linear Second-Order Equations in Two Variables . . . . . .
2
Characteristics and Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Coefficients Independent of ux and uy . . . . . . . . . . . . . . . .
3
Quasi-Linear Second-Order Equations . . . . . . . . . . . . . . . . . . . . . .
3.1
Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Variable Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Quasi-Linear Equations of Order m ≥ 1 . . . . . . . . . . . . . . . . . . . .
4.1
Characteristic Surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
21
21
23
24
25
27
27
28
29
V
VI
Contents
5
Analytic Data and the Cauchy–Kowalewski Theorem . . . . . . . .
5.1
Reduction to Normal Form ([32]) . . . . . . . . . . . . . . . . . . . .
Proof of the Cauchy–Kowalewski Theorem . . . . . . . . . . . . . . . . . .
6.1
Estimating the Derivatives of u at the Origin . . . . . . . . .
Auxiliary Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Auxiliary Estimations at the Origin . . . . . . . . . . . . . . . . . . . . . . . .
Proof of the Cauchy–Kowalewski Theorem (Concluded) . . . . . .
9.1
Proof of Lemma 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Holmgren’s Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
Proof of the Holmgren Uniqueness Theorem . . . . . . . . . . . . . . . .
11.1 Proof of Lemma 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
30
30
32
33
33
35
37
37
38
40
42
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1c Quasi-Linear Second-Order Equations in Two Variables . . . . . .
5c Analytic Data and the Cauchy–Kowalewski Theorem . . . . . . . .
6c Proof of the Cauchy–Kowalewski Theorem . . . . . . . . . . . . . . . . . .
8c The Generalized Leibniz Rule . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
9c Proof of the Cauchy–Kowalewski Theorem Concluded . . . . . . . .
43
43
44
45
45
45
THE LAPLACE EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.1
The Dirichlet and Neumann Problems . . . . . . . . . . . . . . .
1.2
The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Well-Posedness and a Counterexample of Hadamard . . .
1.4
Radial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
The Green and Stokes Identities . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
The Stokes Identities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Green’s Function and the Dirichlet Problem for a Ball . . . . . . . .
Green’s Function for a Ball . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
4
Sub-Harmonic Functions and the Mean Value Property . . . . . . .
The Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
4.2
Structure of Sub-Harmonic Functions . . . . . . . . . . . . . . . .
5
Estimating Harmonic Functions and Their Derivatives . . . . . . .
5.1
The Harnack Inequality and the Liouville Theorem . . . .
Analyticity of Harmonic Functions . . . . . . . . . . . . . . . . . . .
5.2
6
The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7
About the Exterior Sphere Condition . . . . . . . . . . . . . . . . . . . . . .
7.1
The Case N = 2 and ∂E Piecewise Smooth . . . . . . . . . . .
7.2
A Counterexample of Lebesgue for N = 3 ([163]) . . . . . .
8
The Poisson Integral for the Half Space . . . . . . . . . . . . . . . . . . . .
9
Schauder Estimates of Newtonian Potentials . . . . . . . . . . . . . . . .
10 Potential Estimates in Lp (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
11.1 Local Weak Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
12 Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
47
47
48
49
49
50
51
51
53
55
57
59
60
61
62
63
65
68
69
69
70
72
75
78
79
80
6
7
8
9
10
11
2
Contents
12.1
12.2
12.3
12.4
On the Notion of Green’s Function . . . . . . . . . . . . . . . . . .
Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Case f ∈ Co∞ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The Case f ∈ C η (Ē) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1c Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1c Newtonian Potentials on Ellipsoids . . . . . . . . . . . . . . . . . .
1.2c Invariance Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2c The Green and Stokes Identities . . . . . . . . . . . . . . . . . . . . . . . . . . .
3c Green’s Function and the Dirichlet Problem for the Ball . . . . . .
3.1c Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4c Sub-Harmonic Functions and the Mean Value Property . . . . . . .
4.1c Reflection and Harmonic Extension . . . . . . . . . . . . . . . . . .
4.2c The Weak Maximum Principle . . . . . . . . . . . . . . . . . . . . . .
4.3c Sub-Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
5c Estimating Harmonic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1c Harnack-Type Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2c Ill Posed Problems. An Example of Hadamard . . . . . . . .
5.3c Removable Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . .
7c About the Exterior Sphere Condition . . . . . . . . . . . . . . . . . . . . . .
8c Problems in Unbounded Domains . . . . . . . . . . . . . . . . . . . . . . . . .
8.1c The Dirichlet Problem Exterior to a Ball . . . . . . . . . . . . .
9c Schauder Estimates up to the Boundary ([222, 223]) . . . . . . . . .
10c Potential Estimates in Lp (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.1c Integrability of Riesz Potentials . . . . . . . . . . . . . . . . . . . . .
10.2c Second Derivatives of Potentials . . . . . . . . . . . . . . . . . . . . .
3
VII
80
81
82
83
83
83
83
84
84
85
85
86
87
87
88
89
90
90
91
92
93
93
94
95
95
96
BOUNDARY VALUE PROBLEMS BY DOUBLE
LAYER POTENTIALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
1
The Double-Layer Potential . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2
On the Integral Defining the Double-Layer Potential . . . . . . . . . 99
The Jump Condition of W (∂E, xo ; v) Across ∂E . . . . . . . . . . . . . 101
3
4
More on the Jump Condition Across ∂E . . . . . . . . . . . . . . . . . . . . 103
5
The Dirichlet Problem by Integral Equations ([192]) . . . . . . . . . 104
6
The Neumann Problem by Integral Equations ([192]) . . . . . . . . 105
The Green’s Function for the Neumann Problem . . . . . . . . . . . . 107
7
7.1
Finding G(·; ·) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
8
Eigenvalue Problems for the Laplacean . . . . . . . . . . . . . . . . . . . . . 109
8.1
Compact Kernels Generated by Green’s Function . . . . . . 110
9
Compactness of AF in Lp (E) for 1 ≤ p ≤ ∞ . . . . . . . . . . . . . . . . 110
10 Compactness of AΦ in Lp (E) for 1 ≤ p < ∞ . . . . . . . . . . . . . . . . 112
11 Compactness of AΦ in L∞ (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
2c On the Integral Defining the Double-Layer Potential . . . . . . . . . 114
VIII
Contents
5c
6c
7c
8c
4
The Dirichlet Problem by Integral Equations . . . . . . . . . . . . . . . . 115
The Neumann Problem by Integral Equations . . . . . . . . . . . . . . . 116
The Green’s Function for the Neumann Problem . . . . . . . . . . . . 116
7.1c Constructing G(·; ·) for a Ball in R2 and R3 . . . . . . . . . . . 116
Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117
INTEGRAL EQUATIONS AND EIGENVALUE
PROBLEMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1
Kernels in L2 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119
1.1
Examples of Kernels in L2 (E) . . . . . . . . . . . . . . . . . . . . . . . 120
2
Integral Equations in L2 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
2.1
Existence of Solutions for Small |λ| . . . . . . . . . . . . . . . . . . 121
3
Separable Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
3.1
Solving the Homogeneous Equations . . . . . . . . . . . . . . . . . 123
3.2
Solving the Inhomogeneous Equation . . . . . . . . . . . . . . . . 123
Small Perturbations of Separable Kernels . . . . . . . . . . . . . . . . . . . 124
4
4.1
Existence and Uniqueness of Solutions . . . . . . . . . . . . . . . 125
5
Almost Separable Kernels and Compactness . . . . . . . . . . . . . . . . 126
5.1
Solving Integral Equations for Almost Separable
Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.2
Potential Kernels Are Almost Separable . . . . . . . . . . . . . . 127
6
Applications to the Neumann Problem . . . . . . . . . . . . . . . . . . . . . 128
7
The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
8
Finding a First Eigenvalue and Its Eigenfunctions . . . . . . . . . . . 131
9
The Sequence of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
9.1
An Alternative Construction Procedure of the
Sequence of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
10 Questions of Completeness and the Hilbert–Schmidt Theorem . 134
10.1 The Case of K(x; ·) ∈ L2 (E) Uniformly in x . . . . . . . . . . 135
11 The Eigenvalue Problem for the Laplacean . . . . . . . . . . . . . . . . . . 136
11.1 An Expansion of the Green’s Function . . . . . . . . . . . . . . . 137
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2c Integral Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.1c Integral Equations of the First Kind . . . . . . . . . . . . . . . . . 138
2.2c Abel Equations ([2, 3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
2.3c Solving Abel Integral Equations . . . . . . . . . . . . . . . . . . . . . 139
2.4c The Cycloid ([3]) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 140
2.5c Volterra Integral Equations ([266, 267]) . . . . . . . . . . . . . . 140
3c Separable Kernels . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.1c Hammerstein Integral Equations ([114]) . . . . . . . . . . . . . . 141
6c Applications to the Neumann Problem . . . . . . . . . . . . . . . . . . . . . 142
9c The Sequence of Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10c Questions of Completeness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 142
10.1c Periodic Functions in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
Contents
IX
10.2c The Poisson Equation with Periodic Boundary
Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144
11c The Eigenvalue Problem for the Laplacean . . . . . . . . . . . . . . . . . . 144
5
THE HEAT EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1
Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1.1
The Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 145
1.2
The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
1.3
The Characteristic Cauchy Problem . . . . . . . . . . . . . . . . . 146
2
The Cauchy Problem by Similarity Solutions . . . . . . . . . . . . . . . . 146
2.1
The Backward Cauchy Problem . . . . . . . . . . . . . . . . . . . . . 150
3
The Maximum Principle and Uniqueness (Bounded Domains) . 150
3.1
A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.2
Ill Posed Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 151
3.3
Uniqueness (Bounded Domains) . . . . . . . . . . . . . . . . . . . . . 151
The Maximum Principle in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
4
4.1
A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 154
4.2
About the Growth Conditions (4.3) and (4.4) . . . . . . . . . 154
5
Uniqueness of Solutions to the Cauchy Problem . . . . . . . . . . . . . 155
5.1
A Counterexample of Tychonov ([263]) . . . . . . . . . . . . . . . 155
6
Initial Data in L1loc (RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 157
6.1
Initial Data in the Sense of L1loc (RN ) . . . . . . . . . . . . . . . . 158
7
Remarks on the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.1
About Regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
7.2
Instability of the Backward Problem . . . . . . . . . . . . . . . . . 160
8
Estimates Near t = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
9
The Inhomogeneous Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . 162
10 Problems in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 163
10.1 The Strong Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
10.2 The Weak Solution and Energy Inequalities . . . . . . . . . . . 166
11 Energy and Logarithmic Convexity . . . . . . . . . . . . . . . . . . . . . . . . 167
11.1 Uniqueness for Some Ill Posed Problems . . . . . . . . . . . . . . 168
12 Local Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168
12.1 Variable Cylinders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
12.2 The Case |α| = 0 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
13 The Harnack Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 173
13.1 Compactly Supported Sub-Solutions . . . . . . . . . . . . . . . . . 174
13.2 Proof of Theorem 13.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
14 Positive Solutions in ST . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 177
14.1 Non-Negative Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
2c Similarity Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 181
2.1c The Heat Kernel Has Unit Mass . . . . . . . . . . . . . . . . . . . . . 181
2.2c The Porous Medium Equation . . . . . . . . . . . . . . . . . . . . . . 182
X
Contents
2.3c The p-Laplacean Equation . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2.4c The Error Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184
2.5c The Appell Transformation ([10]) . . . . . . . . . . . . . . . . . . . . 184
2.6c The Heat Kernel by Fourier Transform . . . . . . . . . . . . . . . 185
2.7c Rapidly Decreasing Functions . . . . . . . . . . . . . . . . . . . . . . . 185
2.8c The Fourier Transform of the Heat Kernel . . . . . . . . . . . . 186
2.9c The Inversion Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 186
3c The Maximum Principle in Bounded Domains . . . . . . . . . . . . . . . 187
3.1c The Blow-Up Phenomenon for Super-Linear Equations . 188
3.2c The Maximum Principle for General Parabolic
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
4c The Maximum Principle in RN . . . . . . . . . . . . . . . . . . . . . . . . . . . . 190
4.1c Counterexamples of the Tychonov Type . . . . . . . . . . . . . . 191
7c Remarks on the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . 191
12c On the Local Behavior of Solutions . . . . . . . . . . . . . . . . . . . . . . . . 192
6
THE WAVE EQUATION . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
1
The One-Dimensional Wave Equation . . . . . . . . . . . . . . . . . . . . . . 195
1.1
A Property of Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . 196
2
The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 198
3
4
A Boundary Value Problem (Vibrating String) . . . . . . . . . . . . . . 200
4.1
Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
4.2
Odd Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202
4.3
Energy and Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
4.4
Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . 203
5
The Initial Value Problem in N Dimensions . . . . . . . . . . . . . . . . . 203
5.1
Spherical Means . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
The Darboux Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 204
5.2
An Equivalent Formulation of the Cauchy Problem . . . . 205
5.3
6
The Cauchy Problem in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 205
7
The Cauchy Problem in R2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208
8
The Inhomogeneous Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . 210
9
The Cauchy Problem for Inhomogeneous Surfaces . . . . . . . . . . . 211
Reduction to Homogeneous Data on t = Φ . . . . . . . . . . . . 212
9.1
9.2
The Problem with Homogeneous Data . . . . . . . . . . . . . . . 212
10 Solutions in Half Space. The Reflection Technique . . . . . . . . . . . 213
10.1 An Auxiliary Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 214
10.2 Homogeneous Data on the Hyperplane x3 = 0 . . . . . . . . . 214
11 A Boundary Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
12 Hyperbolic Equations in Two Variables . . . . . . . . . . . . . . . . . . . . 217
13 The Characteristic Goursat Problem . . . . . . . . . . . . . . . . . . . . . . . 217
13.1 Proof of Theorem 13.1: Existence . . . . . . . . . . . . . . . . . . . . 217
13.2 Proof of Theorem 13.1: Uniqueness . . . . . . . . . . . . . . . . . . 219
13.3 Goursat Problems in Rectangles . . . . . . . . . . . . . . . . . . . . . 220
Contents
14
15
XI
The Noncharacteristic Cauchy Problem and the Riemann
Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
Symmetry of the Riemann Function . . . . . . . . . . . . . . . . . . . . . . . 222
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
2c The d’Alembert Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
3c Inhomogeneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
3.1c The Duhamel Principle ([61]) . . . . . . . . . . . . . . . . . . . . . . . 223
4c Solutions for the Vibrating String . . . . . . . . . . . . . . . . . . . . . . . . . 224
6c Cauchy Problems in R3 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.1c Asymptotic Behavior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.2c Radial Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 227
6.3c Solving the Cauchy Problem by Fourier Transform . . . . 229
7c Cauchy Problems in R2 and the Method of Descent . . . . . . . . . . 230
7.1c The Cauchy Problem for N = 4, 5 . . . . . . . . . . . . . . . . . . . 231
8c Inhomogeneous Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . . . . . 231
8.1c The Wave Equation for the N and (N + 1)-Laplacean . . 231
8.2c Miscellaneous Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
10c The Reflection Technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
11c Problems in Bounded Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.1c Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
11.2c Separation of Variables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238
12c Hyperbolic Equations in Two Variables . . . . . . . . . . . . . . . . . . . . 239
12.1c The General Telegraph Equation . . . . . . . . . . . . . . . . . . . . 239
14c Goursat Problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
14.1c The Riemann Function and the Fundamental Solution
of the Heat Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
7
QUASI-LINEAR EQUATIONS OF FIRST ORDER . . . . . . . 241
1
Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
2
The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 242
2.1
The Case of Two Independent Variables . . . . . . . . . . . . . . 242
2.2
The Case of N Independent Variables . . . . . . . . . . . . . . . . 243
3
Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 243
3.1
Constant Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
3.2
Solutions in Implicit Form . . . . . . . . . . . . . . . . . . . . . . . . . . 245
4
Equations in Divergence Form and Weak Solutions . . . . . . . . . . 246
4.1
Surfaces of Discontinuity . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
4.2
The Shock Line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 247
5
The Initial Value Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248
5.1
Conservation Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
6
Conservation Laws in One Space Dimension . . . . . . . . . . . . . . . . 249
6.1
Weak Solutions and Shocks . . . . . . . . . . . . . . . . . . . . . . . . . 251
6.2
Lack of Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 252
7
Hopf Solution of The Burgers Equation . . . . . . . . . . . . . . . . . . . . 253
XII
Contents
8
9
10
11
12
13
14
15
Weak Solutions to (6.4) When a(·) is Strictly Increasing . . . . . . 254
8.1
Lax Variational Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . 255
Constructing Variational Solutions I . . . . . . . . . . . . . . . . . . . . . . . 256
9.1
Proof of Lemma 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
Constructing Variational Solutions II . . . . . . . . . . . . . . . . . . . . . . 258
The Theorems of Existence and Stability . . . . . . . . . . . . . . . . . . . 261
11.1 Existence of Variational Solutions . . . . . . . . . . . . . . . . . . . 261
11.2 Stability of Variational Solutions . . . . . . . . . . . . . . . . . . . . 261
Proof of Theorem 11.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 262
12.1 The Representation Formula (11.4) . . . . . . . . . . . . . . . . . . 262
1
12.2 Initial Datum in the Sense of Lloc
(R) . . . . . . . . . . . . . . . . 263
12.3 Weak Forms of the PDE . . . . . . . . . . . . . . . . . . . . . . . . . . . 264
The Entropy Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
13.1 Entropy Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 265
13.2 Variational Solutions of (6.4) Are Entropy Solutions . . . 266
13.3 Remarks on the Shock and the Entropy Conditions . . . . 267
The Kruzhkov Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 269
14.1 Proof of the Uniqueness Theorem I . . . . . . . . . . . . . . . . . . 269
14.2 Proof of the Uniqueness Theorem II . . . . . . . . . . . . . . . . . 271
14.3 Stability in L1 (RN ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 272
The Maximum Principle for Entropy Solutions . . . . . . . . . . . . . . 272
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
3c Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274
6c Explicit Solutions to the Burgers Equation . . . . . . . . . . . . . . . . . 275
6.2c Invariance of Burgers Equations by Some
Transformation of Variables . . . . . . . . . . . . . . . . . . . . . . . . 276
6.3c The Generalized Riemann Problem . . . . . . . . . . . . . . . . . . 276
13c The Entropy Condition . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 277
14c The Kruzhkov Uniqueness Theorem . . . . . . . . . . . . . . . . . . . . . . . . 279
8
NONLINEAR EQUATIONS OF FIRST ORDER . . . . . . . . . . 281
Integral Surfaces and Monge’s Cones . . . . . . . . . . . . . . . . . . . . . . . 281
1
1.1
Constructing Monge’s Cones . . . . . . . . . . . . . . . . . . . . . . . . 282
The Symmetric Equation of Monge’s Cones . . . . . . . . . . . 282
1.2
2
Characteristic Curves and Characteristic Strips . . . . . . . . . . . . . 283
2.1
Characteristic Strips . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 284
3
The Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 285
3.1
Identifying the Initial Data p(0, s) . . . . . . . . . . . . . . . . . . . 285
3.2
Constructing the Characteristic Strips . . . . . . . . . . . . . . . 286
4
Solving the Cauchy Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 286
4.1
Verifying (4.3) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 287
4.2
A Quasi-Linear Example in R2 . . . . . . . . . . . . . . . . . . . . . . 288
5
The Cauchy Problem for the Equation of Geometrical Optics . 289
5.1
Wave Fronts, Light Rays, Local Solutions and Caustics . 290
Contents
6
7
8
9
10
11
12
13
14
15
16
17
18
9
XIII
The Initial Value Problem for Hamilton–Jacobi Equations . . . . 290
The Cauchy Problem in Terms of the Lagrangian . . . . . . . . . . . . 292
The Hopf Variational Solution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 293
8.1
The First Hopf Variational Formula . . . . . . . . . . . . . . . . . 294
8.2
The Second Hopf Variational Formula . . . . . . . . . . . . . . . . 294
Semigroup Property of Hopf Variational Solutions . . . . . . . . . . . 295
Regularity of Hopf Variational Solutions . . . . . . . . . . . . . . . . . . . . 296
Hopf Variational Solutions (8.3) Are Weak Solutions of the
Cauchy Problem (6.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 297
Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
12.1 Example I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 299
12.2 Example II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
12.3 Example III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 301
More on Uniqueness and Stability . . . . . . . . . . . . . . . . . . . . . . . . . 303
14.1 Stability in Lp (RN ) for All p ≥ 1 . . . . . . . . . . . . . . . . . . . . 303
14.2 Comparison Principle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 304
Semi-Concave Solutions of the Cauchy Problem . . . . . . . . . . . . . 304
15.1 Uniqueness of Semi-Concave Solutions . . . . . . . . . . . . . . . 304
A Weak Notion of Semi-Concavity . . . . . . . . . . . . . . . . . . . . . . . . . 305
Semi-Concavity of Hopf Variational Solutions . . . . . . . . . . . . . . . 306
17.1 Weak Semi-Concavity of Hopf Variational Solutions
Induced by the Initial Datum uo . . . . . . . . . . . . . . . . . . . . 306
17.2 Strictly Convex Hamiltonian . . . . . . . . . . . . . . . . . . . . . . . . 307
Uniqueness of Weakly Semi-Concave Variational Hopf
Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 309
LINEAR ELLIPTIC EQUATIONS WITH
MEASURABLE COEFFICIENTS . . . . . . . . . . . . . . . . . . . . . . . . . 313
1
Weak Formulations and Weak Derivatives . . . . . . . . . . . . . . . . . . 313
1.1
Weak Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
2
Embeddings of W 1,p (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 315
2.1
Compact Embeddings of W 1,p (E) . . . . . . . . . . . . . . . . . . . 316
3
Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E) . . . . . . . . . 316
3.1
Some Consequences of the Multiplicative Embedding
Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 317
4
The Homogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . 318
5
Solving the Homogeneous Dirichlet Problem (4.1) by the
Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 318
6
Solving the Homogeneous Dirichlet Problem (4.1) by
Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
6.1
The Case N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 320
6.2
Gâteaux Derivative and The Euler Equation of J(·) . . . . 321
7
Solving the Homogeneous Dirichlet Problem (4.1) by
Galerkin Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 321
XIV
Contents
7.1
8
9
10
11
12
13
14
15
16
17
18
On the Selection of an Orthonormal System in
Wo1,2 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 322
7.2
Conditions on f and f for the Solvability of the
Dirichlet Problem (4.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
Traces on ∂E of Functions in W 1,p (E) . . . . . . . . . . . . . . . . . . . . . 323
8.1
The Segment Property . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 323
8.2
Defining Traces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 324
8.3
Characterizing the Traces on ∂E of Functions in
W 1,p (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 325
The Inhomogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . 325
The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 326
10.1 A Variant of (10.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 328
Constructing The Eigenvalues of (11.1) . . . . . . . . . . . . . . . . . . . . . 329
The Sequence of Eigenvalues and Eigenfunctions . . . . . . . . . . . . 331
A Priori L∞ (E) Estimates for Solutions of the Dirichlet
Problem (9.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 333
Proof of Propositions 14.1–14.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 334
15.1 An Auxiliary Lemma on Fast Geometric Convergence . . 335
15.2 Proof of Proposition 14.1 for N > 2 . . . . . . . . . . . . . . . . . . 335
15.3 Proof of Proposition 14.1 for N = 2 . . . . . . . . . . . . . . . . . . 336
A Priori L∞ (E) Estimates for Solutions of the Neumann
Problem (10.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
Proof of Propositions 16.1–16.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
17.1 Proof of Proposition 16.1 for N > 2 . . . . . . . . . . . . . . . . . . 340
17.2 Proof of Proposition 16.1 for N = 2 . . . . . . . . . . . . . . . . . . 341
Miscellaneous Remarks on Further Regularity . . . . . . . . . . . . . . . 341
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
1c Weak Formulations and Weak Derivatives . . . . . . . . . . . . . . . . . . 342
1.1c The Chain Rule in W 1,p (E) . . . . . . . . . . . . . . . . . . . . . . . . 342
2c Embeddings of W 1,p (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 343
2.1c Proof of (2.4) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 344
2.2c Compact Embeddings of W 1,p (E) . . . . . . . . . . . . . . . . . . . 344
3c Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E) . . . . . . . . . 345
3.1c Proof of Theorem 3.1 for 1 ≤ p < N . . . . . . . . . . . . . . . . . 345
3.2c Proof of Theorem 3.1 for p ≥ N > 1 . . . . . . . . . . . . . . . . . 347
3.3c Proof of Theorem 3.2 for 1 ≤ p < N and E Convex . . . . 348
5c Solving the Homogeneous Dirichlet Problem (4.1) by the
Riesz Representation Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . 349
6c Solving the Homogeneous Dirichlet Problem (4.1) by
Variational Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 350
6.1c More General Variational Problems . . . . . . . . . . . . . . . . . . 350
6.8c Gâteaux Derivatives, Euler Equations and
Quasi-Linear Elliptic Equations . . . . . . . . . . . . . . . . . . . . . 352
Contents
8c
9c
10c
11c
12c
13c
14c
15c
XV
Traces on ∂E of Functions in W 1,p (E) . . . . . . . . . . . . . . . . . . . . . 353
8.1c Extending Functions in W 1,p (E) . . . . . . . . . . . . . . . . . . . . 353
8.2c The Trace Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
8.3c Characterizing the Traces on ∂E of Functions in
W 1,p (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 355
The Inhomogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . 357
9.1c The Lebesgue Spike . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 357
9.2c Variational Integrals and Quasi-Linear Equations . . . . . . 357
The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 358
The Eigenvalue Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
Constructing the Eigenvalues . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
The Sequence of Eigenvalues and Eigenfunctions . . . . . . . . . . . . 359
A Priori L∞ (E) Estimates for Solutions of the Dirichlet
Problem (9.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 359
A Priori L∞ (E) Estimates for Solutions of the Neumann
Problem (10.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c) . . . . . 360
10 DEGIORGI CLASSES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
1
Quasi-Linear Equations and DeGiorgi Classes . . . . . . . . . . . . . . . 363
1.1
DeGiorgi Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 365
2
Local Boundedness of Functions in the DeGiorgi Classes . . . . . . 366
2.1
Proof of Theorem 2.1 for 1 < p < N . . . . . . . . . . . . . . . . . 367
2.2
Proof of Theorem 2.1 for p = N . . . . . . . . . . . . . . . . . . . . . 368
3
Hölder Continuity of Functions in the DG Classes . . . . . . . . . . . 369
3.1
On the Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 370
4
Estimating the Values of u by the Measure of the Set Where
u Is Either Near µ+ or Near µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . 370
5
Reducing the Measure of the Set Where u is Either Near µ+
or Near µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 371
5.1
The Discrete Isoperimetric Inequality . . . . . . . . . . . . . . . . 372
5.2
Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 373
6
Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 374
7
Boundary DeGiorgi Classes: Dirichlet Data . . . . . . . . . . . . . . . . . 375
7.1
Continuity up to ∂E of Functions in the Boundary
DG Classes (Dirichlet Data) . . . . . . . . . . . . . . . . . . . . . . . . 376
8
Boundary DeGiorgi Classes: Neumann Data . . . . . . . . . . . . . . . . 377
8.1
Continuity up to ∂E of Functions in the Boundary
DG Classes (Neumann Data) . . . . . . . . . . . . . . . . . . . . . . . 379
9
The Harnack Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 380
9.1
Proof of Theorem 9.1. Preliminaries . . . . . . . . . . . . . . . . . 380
9.2
Proof of Theorem 9.1. Expansion of Positivity . . . . . . . . . 381
9.3
Proof of Theorem 9.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 381
10 Harnack Inequality and Hölder Continuity . . . . . . . . . . . . . . . . . . 383
XVI
Contents
11
12
Local Clustering of the Positivity Set of Functions in
W 1,1 (E) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 384
A Proof of the Harnack Inequality Independent of Hölder
Continuity . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
11 LINEAR PARABOLIC EQUATIONS IN DIVERGENCE
FORM WITH MEASURABLE COEFFICIENTS . . . . . . . . . . 389
Parabolic Spaces and Embeddings . . . . . . . . . . . . . . . . . . . . . . . . . 389
1
1.1
Steklov Averages . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
2
Weak Formulations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 393
3
The Homogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . 394
4
The Energy Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
5
Existence of Solutions of the Homogeneous Cauchy–Dirichlet
Problem (3.1) by Galerkin Approximations . . . . . . . . . . . . . . . . . 397
6
Uniqueness of Solutions of the Homogeneous Cauchy–
Dirichlet Problem (3.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 403
7
8
9
10
11
12
13
14
15
def
Traces of Functions on Σ = ∂E × (0, T ] . . . . . . . . . . . . . . . . . . . 403
The Inhomogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . 405
The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 406
9.1
The Energy Inequality for the Neumann Problem . . . . . . 408
9.2
A Variant of Problems (3.1) and (9.1) . . . . . . . . . . . . . . . . 410
A Priori L∞ (ET ) Estimates for Solutions of the
Cauchy–Dirichlet Problem (8.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . 410
Proof of Propositions 10.1–10.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 412
A Priori L∞ (ET ) Estimates for Solutions of the Neumann
Problem (9.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 416
Proof of Propositions 12.1–12.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 418
Miscellaneous Remarks on Further Regularity . . . . . . . . . . . . . . . 422
Gaussian Bounds on the Fundamental Solution . . . . . . . . . . . . . . 423
15.1 The Gaussian Upper Bound . . . . . . . . . . . . . . . . . . . . . . . . 425
15.2 The Gaussian Lower Bound . . . . . . . . . . . . . . . . . . . . . . . . . 436
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 441
3c The Homogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . . . 441
5c Existence of Solutions of the Homogeneous Dirichlet Problem
(3.1) by Galerkin Approximations . . . . . . . . . . . . . . . . . . . . . . . . . 442
def
Traces of Functions on Σ = ∂E × (0, T ] . . . . . . . . . . . . . . . . . . . 445
The Inhomogeneous Dirichlet Problem . . . . . . . . . . . . . . . . . . . . . 445
8.1c Parabolic Quasi-Minima . . . . . . . . . . . . . . . . . . . . . . . . . . . . 446
9c The Neumann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 447
10c A Priori L∞ (ET ) Estimates for Solutions of the Dirichlet
Problem (8.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 448
12c A Priori L∞ (ET ) Estimates for Solutions of the Neumann
Problem (9.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 449
15c Gaussian Bounds on the Fundamental Solution . . . . . . . . . . . . . . 449
7c
8c
Contents
XVII
12 PARABOLIC DEGIORGI CLASSES . . . . . . . . . . . . . . . . . . . . . . 451
1
Quasi-Linear Equations and DeGiorgi Classes . . . . . . . . . . . . . . . 451
1.1
Parabolic DeGiorgi Classes . . . . . . . . . . . . . . . . . . . . . . . . . 456
2
Local Boundedness of Functions in the PDG Classes . . . . . . . . . 456
3
Hölder Continuity of Functions in the PDG Classes . . . . . . . . . . 459
3.1
On the Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . 460
4
Estimating the Values of u by the Measure of the Set Where
u is Either Near µ+ or Near µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . 461
5
Reducing the Measure of the Set Where u is Either Near µ+
or Near µ− . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
5.1
Proof of Proposition 5.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 464
6
Propagating in Time the Measure-Theoretical Information . . . . 467
6.1
Proof of Proposition 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 468
7
Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 470
8
Boundary Parabolic DeGiorgi Classes: Dirichlet Data . . . . . . . . 472
8.1
Lateral Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 474
8.2
Initial Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 475
8.3
Definition of Boundary Parabolic DeGiorgi Classes . . . . 476
8.4
Continuity up to ∂p ET of Functions in the Boundary
PDG Classes (Dirichlet Data) . . . . . . . . . . . . . . . . . . . . . . . 476
9
Boundary Parabolic DeGiorgi Classes: Neumann Data . . . . . . . 479
9.1
Lateral Boundary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 481
9.2
Definition of Boundary Parabolic DeGiorgi Classes . . . . 484
9.3
Continuity up to ST of Functions in the Boundary
PDG Classes (Neumann Data) . . . . . . . . . . . . . . . . . . . . . . 484
10 The Harnack Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 485
10.1 Proof of Theorem 10.1. Preliminaries . . . . . . . . . . . . . . . . 486
10.2 Proof of Theorem 10.1. Expansion of Positivity . . . . . . . . 487
10.3 Proof of Theorem 10.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 488
10.4 The Mean Value Harnack Inequality . . . . . . . . . . . . . . . . . 492
11 The Harnack Inequality Implies the Hölder Continuity . . . . . . . 494
12 A Consequence of the Harnack Inequality . . . . . . . . . . . . . . . . . . . 495
13 A More Straightforward Proof of the Hölder Continuity . . . . . . 498
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 499
2c Local Boundedness of Functions in the PDG Classes . . . . . . . . . 499
3c Hölder Continuity of Solutions of Linear Parabolic Equations . 500
6c Propagating in Time the Measure-Theoretical Information . . . . 501
6.1c Proof of Proposition 6.1c . . . . . . . . . . . . . . . . . . . . . . . . . . . 501
7c Proof of Theorem 3.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 505
11c The Harnack Inequality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 507
XVIII Contents
13 PARABOLIC EQUATIONS IN NONDIVERGENCE
FORM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
1
Introductory Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
1.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 509
1.2
The Pucci Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 513
1.3
The Bellman–Dirichlet Equation . . . . . . . . . . . . . . . . . . . . 516
1.4
Remarks on the Concept of Ellipticity . . . . . . . . . . . . . . . 517
1.5
Equations of Mini-Max Type . . . . . . . . . . . . . . . . . . . . . . . 518
2
Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
2.1
Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 519
2.2
Quasi-Linear Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . 523
3
The Aleksandrov Maximum Principle . . . . . . . . . . . . . . . . . . . . . . 529
3.1
Basic Geometric Notions . . . . . . . . . . . . . . . . . . . . . . . . . . . 529
3.2
Increasing Concave Hull of u . . . . . . . . . . . . . . . . . . . . . . . . 531
3.3
Auxiliary Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 537
3.4
Embedding by Normal Mapping . . . . . . . . . . . . . . . . . . . . . 539
3.5
Estimates of the Supremum of a Function . . . . . . . . . . . . 544
3.6
Maximum Principle for Nonlinear Operators . . . . . . . . . . 551
4
Local Estimates and the Harnack Inequality . . . . . . . . . . . . . . . . 552
4.1
A Local Maximum Principle . . . . . . . . . . . . . . . . . . . . . . . . 553
4.2
A Covering Lemma . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 558
4.3
Two Technical Lemmas . . . . . . . . . . . . . . . . . . . . . . . . . . . . 562
4.4
The Harnack Inequality for Linear Equations . . . . . . . . . 569
4.5
The Harnack Inequality for Quasi-Linear Equations . . . . 576
4.6
Local Hölder Continuity of Solutions . . . . . . . . . . . . . . . . . 580
4.7
Hölder Continuity of Solutions of Quasi-Linear
Equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 583
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
1c Introductory Material . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
1.1c Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 584
1.3c The Bellman–Dirichlet Equation . . . . . . . . . . . . . . . . . . . . 585
3c The Aleksandrov Maximum Principle . . . . . . . . . . . . . . . . . . . . . . 589
3.5c Estimates of the Supremum of a Function . . . . . . . . . . . . 589
14 NAVIER–STOKES EQUATIONS . . . . . . . . . . . . . . . . . . . . . . . . . 591
1
Navier–Stokes Equations in Dimensionless Form . . . . . . . . . . . . . 591
2
Steady-State Flow with Homogeneous Boundary Data . . . . . . . . 593
2.1
Uniqueness of Solutions to (2.1) . . . . . . . . . . . . . . . . . . . . . 594
3
Existence of Solutions to (2.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 595
4
Nonhomogeneous Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . 597
4.1
Uniqueness of Solutions to (4.1) . . . . . . . . . . . . . . . . . . . . . 599
4.2
Existence of Solutions to (4.1) . . . . . . . . . . . . . . . . . . . . . . 599
5
Recovering the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 600
6
Steady-State Flows in Unbounded Domains . . . . . . . . . . . . . . . . . 601
Contents
7
8
9
10
11
12
13
14
15
16
17
18
19
20
XIX
6.1
Assumptions on a and f . . . . . . . . . . . . . . . . . . . . . . . . . . . . 602
6.2
Toward a Notion of a Solution to (6.1) . . . . . . . . . . . . . . . 603
Existence of Solutions to (6.1) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 603
7.1
Approximating Solutions and A Priori Estimates . . . . . . 603
7.2
The Limiting Process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
Time-Dependent Navier–Stokes Equations in Bounded
Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 606
The Galerkin Approximations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 609
Selecting Subsequences Strongly Convergent in L2 (ET ; R3 ) . . . 611
The Limiting Process and Proof of Theorem 8.1 . . . . . . . . . . . . . 613
Higher Integrability and Some Consequences . . . . . . . . . . . . . . . . 615
12.1 The Lp,q (ET ; RN ) Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 615
12.2 The Case N = 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 617
Energy Identity for the Homogeneous Boundary Value
Problem with Higher Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 617
Stability and Uniqueness for the Homogeneous Boundary
Value Problem with Higher Integrability . . . . . . . . . . . . . . . . . . . . 620
Local Regularity of Solutions with Higher Integrability . . . . . . . 622
Proof of Theorem 15.1 – Introductory Results . . . . . . . . . . . . . . . 625
Proof of Theorem 15.1 Continued . . . . . . . . . . . . . . . . . . . . . . . . . . 627
Proof of Theorem 15.1 Concluded . . . . . . . . . . . . . . . . . . . . . . . . . 631
Regularity of the Initial-Boundary Value Problem . . . . . . . . . . . 632
Recovering the Pressure in the Time-Dependent Equations . . . 633
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 635
1c Navier–Stokes Equations in Dimensionless Form . . . . . . . . . . . . . 635
4c Nonhomogeneous Boundary Data . . . . . . . . . . . . . . . . . . . . . . . . . . 636
4.1c Solving (4.1) by Galerkin Approximations . . . . . . . . . . . . 637
1
4.2c Extending Fields a ∈ W 2 ,2 (∂E; R3 ), Satisfying (4.2)
into Solenoidal Fields b ∈ W 1,2 (E; R3 ) . . . . . . . . . . . . . . . 638
4.3c Proof of Proposition 4.3c . . . . . . . . . . . . . . . . . . . . . . . . . . . 644
4.4c The Case of a General Domain E . . . . . . . . . . . . . . . . . . . 645
5c Recovering the Pressure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 646
5.1c Proof of Proposition 5.1 for u ∈ H ⊥ ∩ C ∞ (E; R3 ) . . . . . 646
5.2c Proof of Proposition 5.1 for u ∈ H ⊥ . . . . . . . . . . . . . . . . . 647
5.3c More General Versions of Proposition 5.1 . . . . . . . . . . . . . 649
8c Time-Dependent Navier–Stokes Equations in Bounded
Domains . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 649
10c Selecting Subsequences Strongly Convergent in L2 (ET ) . . . . . . . 649
10.1c Proof of Friedrichs’ Lemma . . . . . . . . . . . . . . . . . . . . . . . . . 650
10.2c Compact Embedding of W 1,p into Lq (Q) for
1 ≤ q < p∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 650
10.3c Solutions Global in Time . . . . . . . . . . . . . . . . . . . . . . . . . . . 651
11c The Limiting Process and Proof of Theorem 8.1 . . . . . . . . . . . . . 651
12c Higher Integrability and Some Consequences . . . . . . . . . . . . . . . . 652
XX
Contents
13c Energy Identity for the Homogeneous Boundary Value
Problem with Higher Integrability . . . . . . . . . . . . . . . . . . . . . . . . . 652
15c Local Regularity of Solutions with Higher Integrability . . . . . . . 653
16c Proof of Theorem 15.1 – Introductory Results . . . . . . . . . . . . . . . 653
20c Recovering the Pressure in the Time-Dependent Equations . . . 656
15 QUASI-LINEAR FIRST-ORDER SYSTEMS . . . . . . . . . . . . . . 657
Hyperbolic Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 657
1
2
Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 658
2.1
Incompressible Euler Equations . . . . . . . . . . . . . . . . . . . . . 659
2.2
Reacting Gas Flow in 1–Space Dimension . . . . . . . . . . . . 660
2.3
A Weakly Hyperbolic System Arising in
Magnetohydrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 661
3
Uniqueness of Smooth Solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . 663
Existence of Solutions: The Linear Theory . . . . . . . . . . . . . . . . . . 665
4
4.1
A Family of Approximating Problems . . . . . . . . . . . . . . . . 666
4.2
Estimate of Hi , i = 1, 2, 3 . . . . . . . . . . . . . . . . . . . . . . . . . . 668
4.3
Proof of Theorem 4.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 671
5
Existence of Solutions: The Nonlinear Theory . . . . . . . . . . . . . . . 672
6
Counterexamples to Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . 674
Back to Quasi-Linear First-Order Strictly Hyperbolic
7
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 679
7.1
A First Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 681
7.2
A Second Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
8
Lax Shock Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 682
9
Shocks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 684
9.1
An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 687
10 Centered Rarefaction Waves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 688
10.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 690
11 Contact Discontinuities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 691
11.1 An Example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
12 The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 693
13 Convex Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 696
13.1 Examples of Entropies for 2 × 2 Systems . . . . . . . . . . . . . 703
14 The Glimm Existence Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 704
15 Some Final Comments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
Problems and Complements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 706
2c Some Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 707
5c Existence of Solutions: The Nonlinear Theory . . . . . . . . . . . . . . . 708
6c Proof of Theorem 6.1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 708
7c Back to Quasi-Linear First-Order Strictly Hyperbolic
Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 715
12c The Riemann Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 716
13c Convex Entropies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 720
Contents
XXI
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 723
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 737
Preface
Preface to the Third Edition
This is a revised and extended version of the 2010 second edition of the introduction to partial differential equations authored by Emmanuele DiBenedetto.
Even though the material is essentially the same for a large part, nevertheless
there is a relevant new portion, which covers different topics.
The guiding principle that led us during the preparation of this monograph
is twofold, and indeed it was already at the heart of the second edition.
On one hand, there is the close and existential connection between the
theory of PDEs and the modelling of physical phenomena: quite a number of
well-known physical facts are described in terms of partial differential equations; the nature of the phenomenon suggests what kind of results mathematicians should look for, and at the same time, it is a common fact that,
once established, analytical tools prove to be powerful even largely beyond
the problem they were originally developed for.
On the other hand, in the mathematical theory of PDEs there are topics,
which can by now be considered as classical, but at the same time serve as a
natural introduction to very active research fields. Therefore, it seems quite
natural to provide a self-contained introduction to these results, specifically
aimed at people who approach the field for the first time.
With these two main ideas as a sort of roadmap, we set out to work, and
revised the 2010 second edition. Some changes are perhaps limited in terms of
number of pages, but they are not less important than other topics, to which
full new chapters are devoted.
In the Complements of Chapter 0, we offered a detailed description of Einstein’s approach to the Brownian motion; in Chapter 1 we added Holmgren’s
uniqueness theorem, which in our opinion represents a natural complement
to the Cauchy–Kowalewski theorem; in Chapter 5, Section 2.2c in the Complements about the porous medium equation was largely expanded, following
suggestions by students who used the monograph as textbook in their PDE
class in the past; as for Chapter 6 we briefly revised the sections devoted
XXIII
XXIV Preface
to Problems and Complements, adding further exercises, examples, and even
some explicit hints of solutions.
The first substantial novelty is represented by Chapters 11 and 12, where
linear parabolic equations in divergence form with bounded and measurable
coefficients, and parabolic DeGiorgi classes are respectively discussed. These
two chapters represent a sort of parabolic analogue of Chapters 9 and 10, and
indeed a certain degree of similarity in their structures is apparent and explicitly sought for. Even though the notion of parabolic DeGiorgi class might not
be as widely accepted as the corresponding elliptic one, nevertheless they are
being used more and more, and they are of considerable help in highlighting
the deep structural nature of many results. Chapter 11 ends with the Gaussian
bounds on the fundamental solution, which lie at the heart of the celebrated
Nash’s result, but are not so easy to find in the available literature.
Chapter 13 is devoted to parabolic equations in nondivergence form, modelled on the prototype class of linear parabolic equations with bounded and
measurable coefficients. Here the focus is on solutions in Sobolev spaces, and
not on solutions in the viscosity sense. The strict analogy with the corresponding results of Chapter 12 is clear, as is the deep difference in terms of
the analytical tools needed to achieve them. We discussed at length whether
it was the case to add also an analogous chapter about elliptic equations in
nondivergence form, but we decided in the negative, due to the enormous
amount of literature already available on this topic.
The Navier–Stokes equations are discussed in Chapter 14. We tried and
concentrated on the main results, at the same time giving a brief introduction
to the most important and interesting open problems.
Chapter 15 is the last one and deals with quasi-linear first order systems.
We first consider smooth solutions, and then treat the main classical results
due to the pioneering work of Lax, Glimm, and collaborators. We obviously
do not dwell on the recent accomplishments in this field, where new and
important results are still regularly achieved.
Vanderbilt University, Emmanuele DiBenedetto,
University of Pavia, Ugo Gianazza,
May 2021
Besides the sheer scientific aspect described above, there is another difference
with respect to the previous edition. When Emmanuele DiBenedetto decided
to prepare a new edition of his 2010 manuscript, he asked me to help him; I
was very surprised by this request, but at the same time very glad to accept
it. The plan was discussed together, as well as the distribution of who should
take care of what. The pandemic made things a bit more difficult than what
we had originally expected, because a lot of work had to be done at distance,
but eventually it turned out to be not so invasive. What really impacted the
Preface
XXV
Emmanuele DiBenedetto, 1947–2021
work was Emmanuele’s illness, which cut too short an extraordinary life and a
great friendship. Up to his last days, Emmanuele kept on suggesting, advising,
correcting his part, and mine as well.
When Emmanuele passed away, Chris Eder and Springer were very supportive and confident that I could finish the work by myself, simply following
the original plan. I am very grateful for that. The work you now have in your
hands is precisely the result of such a roadmap. Emmanuele’s understanding
and vision of PDEs theory in general, and of its classical aspects in particular
were impressive, and always freely shared, without pretending to keep things
for himself in any way: although this monograph is mostly a fruit of his work,
at the same time it is dedicated to him.
When Emmanuele completed a monograph, he used to send a copy to
people active in the field, in order to collect opinions, suggestions, criticisms,
etc. It seemed to me that it is a very good way to proceed and that it was worth
doing the same in this case as well. Therefore, I am indebted to many friends,
XXVI Preface
collaborators, colleagues and students, who read large parts of this revised
version, with particular care for the new chapters, and suggested changes,
improvements, corrections, or simply used the previous editions in their classes
and reported about the comments by students.
In particular, I am grateful with Naian Liao, who read the parabolic chapters and suggested various improvements, Luc Tartar, who provided very useful and interesting literature about quasi-linear first-order systems, Giuseppe
Savaré and Sandro Salsa for their kind remarks, Enrico Vitali for a lot of
invaluable bibliographic references, Giorgio Metafune, who provided the example of Section 10c of the Complements of Chapter 11. As for students,
special thanks go to G. Cavalleri, M. Ferrari, B. Minniti, and E. Tolotti.
University of Pavia, Ugo Gianazza,
July 2023
Preface XXVII
Preface to the Second Edition
This is a revised and extended version of my 1995 elementary introduction
to partial differential equations. The material is essentially the same except
for three new chapters. The first (Chapter 8) is about nonlinear equations
of first order and in particular Hamilton–Jacobi equations. It builds on the
continuing idea that PDEs, although a branch of mathematical analysis, are
closely related to models of physical phenomena. Such underlying physics
in turn provides ideas of solvability. The Hopf variational approach to the
Cauchy problem for Hamilton–Jacobi equations is one of the clearest and
most incisive examples of such an interplay. The method is a perfect blend
of classical mechanics, through the role and properties of the Lagrangian and
Hamiltonian, and calculus of variations. A delicate issue is that of identifying
“uniqueness classes.” An effort has been made to extract the geometrical
conditions on the graph of solutions, such as quasi-concavity, for uniqueness
to hold.
Chapter 9 is an introduction to weak formulations, Sobolev spaces, and
direct variational methods for linear and quasi-linear elliptic equations. While
terse, the material on Sobolev spaces is reasonably complete, at least for a
PDE user. It includes all the basic embedding theorems, including their proofs,
and the theory of traces. Weak formulations of the Dirichlet and Neumann
problems build on this material. Related variational and Galerkin methods,
as well as eigenvalue problems, are presented within their weak framework.
The Neumann problem is not as frequently treated in the literature as the
Dirichlet problem; an effort has been made to present the underlying theory
as completely as possible. Some attention has been paid to the local behavior of these weak solutions, both for the Dirichlet and Neumann problems.
While efficient in terms of existence theory, weak solutions provide limited
information on their local behavior. The starting point is a sup bound for the
solutions and weak forms of the maximum principle. A further step is their
local Hölder continuity.
An introduction to these local methods is in Chapter 10 in the framework of DeGiorgi classes. While originating from quasi-linear elliptic equations, these classes have a life of their own. The investigation of the local
and boundary behavior of functions in these classes, involves a combination
of methods from PDEs, measure theory, and harmonic analysis. We start by
tracing them back to quasi-linear elliptic equations, and then present in detail some of these methods. In particular, we establish that functions in these
classes are locally bounded and locally Hölder continuous, and we give conditions for the regularity to extend up to the boundary. Finally, we prove that
non-negative functions on the DeGiorgi classes satisfy the Harnack inequality.
This, on the one hand, is a surprising fact, since these classes require only
some sort of Caccioppoli-type energy bounds. On the other hand, this raises
the question of understanding their structure, which to date is still not fully
understood. While some facts about these classes are scattered in the litera-
XXVIIIPreface
ture, this is perhaps the first systematic presentation of DeGiorgi classes in
their own right. Some of the material is as recent as last year. In this respect,
these last two chapters provide a background on a spectrum of techniques in
local behavior of solutions of elliptic PDEs, and build toward research topics
of current active investigation.
The presentation is more terse and streamlined, than in the first edition.
Some elementary background material (Weierstrass Theorem, mollifiers, Theorem of Ascoli–Arzelá, Jensen’s inequality, etc..) has been removed.
I am indebted to many colleagues and students who, over the past twelve
years, have offered critical suggestions and pointed out misprints, imprecise
statements, and points that were not clear on a first reading. Among these
Giovanni Caruso, Xu Guoyi, Hanna Callender, David Petersen, Mike O’Leary,
Changyong Zhong, Justin Fitzpatrick, Abey Lopez, Haichao Wang. Special
thanks go to Matt Calef for reading carefully a large portion of the manuscript
and providing suggestions and some simplifying arguments. The help of U.
Gianazza, has been greatly appreciated. He has read the entire manuscript
with extreme care and dedication, picking up points that needed to be clarified. I am very much indebted to Ugo.
I would like to thank Avner Friedman, James Serrin, Constantine Dafermos, Bob Glassey, Giorgio Talenti, Luigi Ambrosio, Juan Manfredi, John
Lewis, Vincenzo Vespri, and Gui Qiang Chen for examining the manuscript in
detail and for providing valuable comments. Special thanks to David Kinderlehrer for his suggestion to include material on weak formulations and direct
methods. Without his input and critical reading, the last two chapters probably would not have been written. Finally, I would like to thank Ann Kostant
and the entire team at Birkhäuser for their patience in coping with my delays.
Vanderbilt University, Emmanuele DiBenedetto,
June 2009
Preface XXIX
Preface to the First Edition
These notes are meant to be a self contained, elementary introduction to
partial differential equations (PDEs). They assume only advanced differential
calculus and some basic Lp theory. Although the basic equations treated in
this book, given its scope, are linear, I have made an attempt to approach
then from a nonlinear perspective.
Chapter 1 is focused on the Cauchy-Kowalewski theorem. We discuss the
notion of characteristic surfaces and use it to classify partial differential equations. The discussion grows from equations of second order in two variables to
equations of second order in N variables to PDEs of any order in N variables.
In Chapters 2 and 3 we study the Laplace equation and connected elliptic theory. The existence of solutions for the Dirichlet problem is proven by
the Perron method. This method clarifies the structure of the sub(super)harmonic functions, and it is closely related to the modern notion of viscosity
solution. The elliptic theory is complemented by the Harnack and Liouville
theorems, the simplest version of Schauder’s estimates, and basic Lp -potential
estimates. Then, in Chapter 3 the Dirichlet and Neumann problems, as well
as eigenvalue problems for the Laplacean, are cast in terms of integral equations. This requires some basic facts concerning double layer potentials and
the notion of compact subsets of Lp , which we present.
In Chapter 4 we present the Fredholm theory of integral equations and
derive necessary and sufficient conditions for solving the Neumann problem.
We solve eigenvalue problems for the Laplacean, generate orthonormal systems in L2 , and discuss questions of completeness of such systems in L2 . This
provides a theoretical basis for the method of separation of variables.
Chapter 5 treats the heat equation and related parabolic theory. We introduce the representation formulas, and discuss various comparison principles.
Some focus has been placed on the uniqueness of solutions to the Cauchy
problem and their behavior as |x| → ∞. We discuss Widder’s theorem and
the structure of the non-negative solutions. To prove the parabolic Harnack estimate we have used an idea introduced by Krylov and Safonov in the context
of fully nonlinear equations.
The wave equation is treated in Chapter 6 in its basic aspects. We derive
representation formulas and discuss the role of the characteristics, propagation of signals, and questions of regularity. For general linear second-order
hyperbolic equations in two variables, we introduce the Riemann function and
prove its symmetry properties. The sections on Goursat problems represent a
concrete application of integral equations of Volterra type.
Chapter 7 is an introduction to conservation laws. The main points of the
theory are taken from the original papers of Hopf and Lax from the 1950s.
Space is given to the minimization process and the meaning of taking the
initial data in the sense of L1 . The uniqueness theorem we present is due
to Kruzhkov (1970). We discuss the meaning of viscosity solution vis-à-vis
the notion of sub-solutions and maximum principle for parabolic equations.
XXX
Preface
The theory is complemented by an analysis of the asymptotic behavior, again
following Hopf and Lax.
Even though the layout is theoretical, I have indicated some of the physical
origins of PDEs. Reference is made to potential theory, similarity solutions
for the porous medium equation, generalized Riemann problems, etc.
I have also attempted to convey the notion of ill posed problems, mainly
via some examples of Hadamard.
Most of the background material, arising along the presentation, has been
stated and proved in the complements. Examples include the theorem of
Ascoli–Arzelà, Jensen’s inequality, the characterization of compactness in Lp ,
mollifiers, basic facts on convex functions, and the Weierstrass theorem. A
book of this kind is bound to leave out a number of topics, and this book is
no exception. Perhaps the most noticeable omission here is some treatment of
numerical methods.
These notes have grown out of courses in PDEs I taught over the years at
Indiana University, Northwestern University and the University of Rome II,
Italy. My thanks go to the numerous students who have pointed out misprints
and imprecise statements. Of these, special thanks go to M. O’Leary, D. Diller,
R. Czech, and A. Grillo. I am indebted to A. Devinatz for reading a large
portion of the manuscript and for providing valuable critical comments. I
have also benefited from the critical input of M. Herrero, V. Vespri, and J.
Manfredi, who have examined parts of the manuscript. I am grateful to E.
Giusti for his help with some of the historical notes. The input of L. Chierchia
has been crucial. He has read a large part of the manuscript and made critical
remarks and suggestions. He has also worked out in detail a large number of
the problems and supplied some of his own. In particular, he wrote the first
draft of problems 2.7–2.13 of Chapter 5 and 6.10–6.11 of Chapter 6. Finally
I like to thank M. Cangelli and H. Howard for their help with the graphics.
0
PRELIMINARIES
1 Green’s Theorem
Let E be an open set in RN , and let k be a non-negative integer. Denote
by C k (E) the collection of all real-valued, k-times continuously differentiable
functions in E. A function f is in Cok (E) if f ∈ C k (E), and its support
is contained in E. A function f : Ē → R is in C k (Ē), if f ∈ C k (E) and
all partial derivatives ∂ ℓ f /∂xℓi for all i = 1, . . . , N and ℓ = 0, . . . , k, admit
continuous extensions up to ∂E. The boundary ∂E is of class C 1 if for all
y ∈ ∂Ω, there exists ε > 0 such that within the ball Bε (y) centered at y and
radius ε, ∂E can be implicitly represented, in a local system of coordinates,
as a level set of a function Φ ∈ C 1 (Bε (y)) such that |∇Φ| =
6 0 in Bε (y).
If ∂E is of class C 1 , let n(x) = n1 (x), . . . , nN (x) denote the unit normal
exterior to E at x ∈ ∂E. Each of the components nj (·) is well defined as a
continuous function on ∂E. A real vector-valued function
Ē ∋ x → f (x) = f1 (x), . . . , fN (x) ∈ RN
is of class C k (E), C k (Ē), or Cok (E) if all components fj belong to these classes.
Theorem 1.1. Let E be a bounded domain of RN with boundary ∂E of class
C 1 . Then for every f ∈ C 1 (Ē)
Z
Z
div f dx =
f · n dσ
E
∂E
where dx is the Lebesgue measure in E and dσ denotes the Lebesgue surface
measure on ∂E.
This is also referred to as the divergence theorem, or as the formula of integration by parts. It continues to hold if n is only dσ-a.e. defined in ∂E. For
example, ∂E could be a cube in RN . More generally, ∂E could be the finite
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_1
1
2
0 PRELIMINARIES
union of portions of surfaces of class C 1 . The domain E need not be bounded,
provided |f | and |∇f | decay sufficiently fast as |x| → ∞.1
1.1 Differential Operators and Adjoints
Given a symmetric matrix (aij ) ∈ RN × RN , a vector b ∈ RN , and c ∈ R,
consider the formal expression
L(·) = aij
∂2
∂
+ bi
+c
∂xi ∂xj
∂xi
(1.1)
where we have adopted the Einstein summation convention, i.e., repeated
indices in a monomial expression mean summation over those indices. The
formal adjoint of L(·) is
L∗ (·) = aij
∂2
∂
− bi
+ c.
∂xi ∂xj
∂xi
Thus L = L∗ if b = 0. If u, v ∈ C 2 (Ē) for a bounded open set E ⊂ RN with
boundary ∂E of class C 1 , the divergence theorem yields the Green’s formula
Z
Z
[vL(u) − uL∗ (v)]dx =
[(vaij uxj ni − uaij vxi nj ) + uvb · n]dσ. (1.2)
∂E
E
If u, v ∈ Co2 (E), then
Z
E
[vL(u) − uL∗ (v)]dx = 0.
(1.2)o
More generally, the entries of the matrix (aij ) as well as b and c might be
smooth functions of x. In such a case, for v ∈ C 2 (Ē), define
L∗ (v) =
∂ 2 (aij v) ∂ (bi v)
−
+ cv.
∂xi ∂xj
∂xi
The Green’s formula (1.2)o continues to hold for every pair of functions u, v ∈
Co2 (E). If u and v do not vanish near ∂E, a version of (1.2) continues to hold,
where the right-hand side contains the extra boundary integral
Z
∂aij
nj dσ.
uv
∂xi
∂E
1
Identifying precise conditions on ∂E and f for which one can integrate by parts
is part of geometric measure theory ([102]).
2 The Continuity Equation
3
2 The Continuity Equation
Let t → E(t) be a set-valued function that associates to each t in some open
interval I ⊂ R a bounded open set E(t) ⊂ RN , for some N ≥ 2. Assume
that the boundaries ∂E(t) are uniformly of class C 1 , and that there exists a
bounded open set E ⊂ RN such that E(t) ⊂ E, for all t ∈ I. Our aim is to
compute the derivative
Z
d
ρ(x, t)dx
for a given ρ ∈ C 1 (E × I).
dt E(t)
Regard points x ∈ E(t) as moving along the trajectories t → x(t) with velocities ẋ = v(x, t). Assume that the motion, or deformation, of E(·) is smooth
in the sense that (x, t) → v(x, t) is continuous in a neighborhood of E × I.
Forming the difference quotient gives
Z
d
ρ(x, t)dx
dt E(t)
Z
Z
1
ρ(x, t)dx
ρ(x, t + ∆t)dx −
= lim
∆t→0 ∆t
E(t+∆t)
E(t)
Z
(2.1)
ρ(x, t + ∆t) − ρ(x, t)
= lim
dx
∆t→0 E(t)
∆t
Z
Z
1
ρ(t)dx −
ρ(t)dx .
+ lim
∆t→0 ∆t
E(t+∆t)−E(t)
E(t)−E(t+∆t)
The first limit is computed by carrying the limit under the integral, yielding
Z
Z
ρ(x, t + ∆t) − ρ(x, t)
lim
dx =
ρt dx.
∆t→0 E(t)
∆t
E(t)
As for the second, first compute the difference of the last two volume integrals
by means of Riemann sums as follows. Fix a number 0 < ∆σ ≪ 1, and
approximate ∂E(t) by means of a polyhedron with faces of area not exceeding
∆σ and tangent to ∂E(t) at some of their interior points. Let {F1 , . . . , Fn }
for some n ∈ N be a finite collection of faces making up the approximating
polyhedron, and let xi for i = 1, . . . , n, be a selection of their tangency points
with ∂E(t). Then approximate the set
S
E(t + ∆t) − E(t)
E(t) − E(t + ∆t)
by the union of the cylinders of basis Fi and height v(xi , t) · n∆t, built with
their axes parallel to the outward normal to ∂E(t) at xi . Therefore, for ∆t
fixed
4
0 PRELIMINARIES
Fig. 2.1
1
∆t
Z
ρ(t)dx
E(t+∆t)−E(t)
E(t)−E(t+∆t)
Z
n
P
ρ(xi , t)v(xi , t) · n∆σ + O(∆t) =
= lim
ρ(t)dx −
Z
∆σ→0 i=1
∂E(t)
ρv · ndσ + O(∆t).
Letting now ∆t → 0 in (2.1) yields
Z
Z
Z
d
ρ dx =
ρt dx +
ρv · n dσ.
dt E(t)
E(t)
∂E(t)
By the Green’s theorem
Z
∂E(t)
ρv · n dσ =
Z
(2.2)
div(ρv) dx.
E(t)
Therefore (2.2) can be equivalently written as
Z
Z
d
ρ dx =
[ρt + div(ρv)] dx.
dt E(t)
E(t)
(2.3)
Consider now an ideal fluid filling a region E ⊂ R3 . Assume that the fluid is
compressible (say a gas) and let (x, t) → ρ(x, t) denote its density. At some
instant t, cut a region E(t) out of E and follow the motion of E(t) as if each of
its points were identified with the moving particles. Whatever the subregion
E(t), during the motion the mass is conserved. Therefore
Z
d
ρ dx = 0.
dt E(t)
By the previous calculations and the arbitrariness of E(t) ⊂ E
ρt + div(ρv) = 0
in
E × R.
(2.4)
This is referred to as the equation of continuity or conservation of mass.
3 The Heat Equation and the Laplace Equation
5
3 The Heat Equation and the Laplace Equation
Any quantity that is conserved as it moves within an open set E with velocity
v satisfies the conservation law (2.4). Let u be the temperature of a material
homogeneous body occupying the region E. If c is the heat capacity, the
thermal energy stored at x ∈ E at time t is cu(x, t). By Fourier’s law the
energy “moves” following gradients of temperature, i.e.,
cuv = −k∇u
(3.1)
where k is the conductivity ([78, 30]). Putting this in (2.4) yields the heat
equation
k
ut − ∆u = 0.
(3.2)
c
Now let u be the pressure of a fluid moving with velocity v through a region
E of RN occupied by a porous medium. The porosity po of the medium is the
relative infinitesimal fraction of space occupied by the pores and available to
the fluid. Let µ, k, and ρ denote respectively kinematic viscosity, permeability,
and density. By Darcy’s law ([226])
v=−
kpo
∇u.
µ
(3.3)
Assume that k and µ are constant. If the fluid is incompressible, then ρ =
const, and it follows from (2.4) that div v = 0. Therefore the pressure u
satisfies
div ∇u = ∆u = uxi xi = 0 in E.
(3.4)
The latter is the Laplace equation for the function u. A fluid whose velocity
is given as the gradient of a scalar function is a potential fluid ([268]).
3.1 Variable Coefficients
Consider now the same physical phenomena taking place in nonhomogeneous,
anisotropic media. For heat conduction in such media, temperature gradients
might generate heat propagation in preferred directions, which themselves
might depend on x ∈ E. As an example one might consider the heat diffusion
in a solid of given conductivity, in which is embedded a bundle of curvilinear
material fibers of different conductivity. Thus in general, the conductivity
of the composite medium is a tensor dependent on the location x ∈ E and
time t, represented formally by an N × N matrix k = kij (x, t) . For such
a tensor, the product on the right-hand side of (3.1) is the row-by-column
product of the matrix (kij ) and the column vector ∇u. Enforcing the same
conservation of energy (2.4) yields a nonhomogeneous, anisotropic version of
the heat equation (3.2), in the form
ut − aij (x, t)uxi
xj
=0
in E, where aij =
kij
.
c
(3.5)
6
0 PRELIMINARIES
Similarly, the permeability of a nonhomogeneous,
anisotropic porous medium
is a position-dependent tensor kij (x) . Then, analogous considerations applied to (3.3), imply that the velocity potential u of the flow of a fluid in
a heterogeneous, anisotropic porous medium satisfies the partial differential
equation
po kij
aij (x)uxi xj = 0 in E, where aij =
.
(3.6)
µ
The physical, tensorial nature of either heat conductivity or permeability of a
medium implies that (aij ) is symmetric, bounded, and positive definite in E.
However, no further information is available on these coefficients, since they
reflect interior properties of physical domains, not accessible without altering
the physical phenomenon we are modeling. This raises the question of the
meaning of (3.5)–(3.6). Indeed, even if u ∈ C 2 (E), the indicated operations
are not well defined for aij ∈ L∞ (E). A notion of solution will be given in
Chapter 10, along with solvability methods.
Equations (3.5)–(3.6) are said to be in divergence form. Equations in nondivergence form are of the type
aij (x)uxi xj = 0
in E
(3.7)
and arise in the theory of stochastic control ([146]).
4 A Model for the Vibrating String
Consider a material string of constant linear density ρ whose end points are
fixed, say at 0 and 1. Assume that the string is vibrating in the plane (x, y),
set the interval (0, 1) on the x-axis, and let (x, t) → u(x, t) be the y-coordinate
of the string at the point x ∈ (0, 1) at the instant t ∈ R. The basic physical
assumptions are:
(i) The dimensions of the cross sections are negligible with respect to the
length, so that the string can be identified, for all t, with the graph of
x → u(x, t).
(ii) Let (x, t) → T(x, t) denote the tension, i.e., the sum of the internal forces
per unit length, generated by the displacement of the string. Assume that
T at each point (x, u(x, t)) is tangent to the string. Letting T = |T|,
assume that (x, t) → T (x, t) is t-independent.
(iii) Resistance of the material to flexure is negligible with respect to the
tension.
(iv) Vibrations are small in the sense that |u|θ and |ux |θ for θ > 1 are
negligible when compared with |u| and |ux |.
y ..............
4 A Model for the Vibrating String
7
.
..
...
..
...
.....................................
...
..
....................................
..............................................
...
.......
.....................................
.
.
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
.
.
.
.
.
......
...
.
....................
.....
...
.....
.........................
..
.....
.
........................ ................................
...........................................................α
................................. ......................................................................................................................................................
...................
u(x, t)
T
x
L
Fig. 4.1
The tangent line to the graph of u(·, t) at (x, u(x, t)) forms with the x-axis an
angle α ∈ (0, π/2) given by
ux
sin α = p
.
1 + u2x
Therefore the vertical component of the tension T at (x, u(x, t)) is
ux
.
T sin α = T p
1 + u2x
Consider next, for t fixed, a small interval (x1 , x2 ) ⊂ (0, 1) and the correspond
ing portion of the string of extremities x1 , u(x1 , t) and x2 , u(x2 , t) . Such
a portion is in instantaneous equilibrium if both the x and y components of
the sum of all forces acting on it are zero. The components in the y-direction
are:
1. The difference of the y-components of T at the two extremities, i.e.,
Z x2
∂
ux
ux
ux
(x2 , t) − T p
(x1 , t) =
dx.
Tp
Tp
1 + u2x
1 + u2x
1 + u2x
x1 ∂x
2. The total load acting on the portion, i.e.,
Z x2
p(x, t)dx,
where p(·, t) = {load per unit length}.
−
x1
3. The inertial forces due to the vertical acceleration utt (x, t), i.e.,
Z x2
∂2
ρ 2 u(x, t)dx.
∂t
x1
Therefore the portion of the string is instantaneously in equilibrium if
Z x2
Z x2 ∂2
ux
∂
ρ 2 u(x, t)dx =
Tp
(x, t) + p(x, t) dx.
∂t
∂x
1 + u2x
x1
x1
Dividing by ∆x = x2 − x1 and letting ∆x → 0 gives
∂2
ux
∂
ρ 2u −
Tp
= p in (0, 1) × R.
∂t
∂x
1 + u2x
(4.1)
8
0 PRELIMINARIES
The balance of forces along the x-direction involves only the tension and gives
(T cos α)(x1 , t) = (T cos α)(x2 , t)
or equivalently
From this
Z
x2
x1
Therefore
T
T
p
(x1 , t) = p
(x2 , t).
1 + u2x
1 + u2x
∂
T
p
dx = 0
∂x
1 + u2x
T
∂
p
=0
∂x
1 + u2x
and
for all (x1 , x2 ) ⊂ (0, 1).
x→
T
p
(x, t) = To
1 + u2x
for some To > 0 independent of x. In view of the physical assumptions (ii)
and (iv), may take To also independent of t. These remarks in (4.1) yield the
partial differential equation
2
∂2
2∂ u
u
−
c
=f
∂t2
∂x2
where
c2 =
To
ρ
and
in (0, 1) × R
f (x, t) =
(4.2)
p(x, t)
.
ρ
This is the wave equation in one space variable.
Remark 4.1 The assumption that ρ is constant is a “linear” assumption in
the sense that leads to the linear wave equation (4.2). Nonlinear effects due
to variable density were already observed by D. Bernoulli ([17]), and by S.D.
Poisson ([206]).
5 Small Vibrations of a Membrane
A membrane is a rigid thin body of constant density ρ, whose thickness is
negligible with respect to its extension. Assume that, at rest, the membrane
occupies a bounded open set E ⊂ R2 , and that it begins to vibrate under the
action of a vertical load, say (x, t) → p(x, t). Identify the membrane with the
graph of a smooth function (x, t) → u(x, t) defined in E × R and denote by
∇u = (ux1 , ux2 ) the spatial gradient of u. The relevant physical assumptions
are:
(i) Forces due to flexure are negligible.
5 Small Vibrations of a Membrane
9
(ii) Vibrations occur only in the direction u normal to the position of rest of
the membrane. Moreover , vibrations are small, in the sense that uxi uxj
and uuxi for i, j = 1, 2 are negligible when compared to u and |∇u|.
(iii) The tension T has constant modulus, say |T| = To > 0.
Cut a small ideal region Go ⊂ E with boundary ∂Go of class C 1 , and let G
be the corresponding portion of the membrane. Thus G is the graph of u(·, t)
restricted to Go , or equivalently, Go is the projection on the plane u = 0 of
the portion G of the membrane. Analogously, introduce the curve Γ limiting
G and its projection Γo = ∂Go . The tension T acts at points P ∈ Γ and is
tangent to G at P and normal to Γ . If τ is the unit vector of T and n is the
exterior unit normal to G at P , let e be the unit tangent to Γ at P oriented
so that the triple {τ , e, n} is positive and τ = e ∧ n. Our aim is to compute
the vertical component of T at P ∈ Γ . If {i, j, k} is the positive unit triple
along the coordinate axes, we will compute the quantity T · k = To τ · k.
Consider a parametrization of Γo , say
for s ∈ {some interval of R}.
s → Po (s) = x1 (s), x2 (s)
The unit exterior normal to ∂Go is given by
(x′ , −x′ )
ν = p 2′2 1 ′2 .
x1 + x2
x3
..
........
..
...
..
...
...
...
...
...
...
.........................
..................................... .... .........
...
..... ... .
.
....
.... .. ..
..
...
......... ....... ...... ..
.. ..... ..... .........
.... ............. .
...
.
........
... ...
.
.
.
.
.
.. ..
...
..
...
..
.
...........
.
.
.
..
..
...
.....
..... ..............
..
..
.....
...........
...
.
.
.
.
.
.
.
.
.
..
.........
...
. ................................................. .
......................
.
.............
..........................................
...
.
...
..
.
......
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...............
.
.
......
....
....
....
....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......
.........
........
...
.............................
...
.....
....
...
....
....
....
.
...
..
....
....
...
.....
.
....
.
.
.
.
........
.............
.
.
.
.
.
....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
................................................................
....
...
....
....
.
.
....
....
....
...
.
.
.
..
.......
.......
G
P
x1
o
Go
e
n
T
x2
Fig. 5.1
Consider also the corresponding parametrization of Γ
s → P (s) = Po (s), u(x1 (s), x2 (s), t) .
The unit tangent e to Γ is
10
0 PRELIMINARIES
(x′ , x′ , x′ )
(Ṗo , Ṗo · ∇u)
e = p ′21 2′2 3 ′2 = q
x1 + x2 + x3
|Ṗo |2 + |Ṗo · ∇u|2
and the exterior unit normal to G at Γ is
(−∇u, 1)
n= p
.
1 + |∇u|2
Therefore


i
j k
1
x′2 x′3 
τ = e ∧ n = det  x′1
J
−ux1 −ux2 1
where
q
p
2
J = 1 + |∇u| |Ṗo |2 + |Ṗo · ∇u|2 .
From this
Jτ · k = (x′2 , −x′1 ) · ∇u = |Ṗo | ∇u · ν.
If β is the cosine of the angle between the vectors ∇u and Ṗo
τ ·k =
∇u · ν
,
Jβ
Jβ =
Since
p
p
1 + |∇u|2 1 + β 2 |∇u|2 .
(1 + β 2 |∇u|2 ) ≤ Jβ ≤ (1 + |∇u|2 )
by virtue of the physical assumption (ii)
T · k ≈ To ∇u · ν.
Next, write down the equation of instantaneous equilibrium of the portion G
of the membrane. The vertical loads on G, the vertical contribution of the
tension T, and the inertial force due to acceleration utt are respectively
Z
Z
Z
To ∇u · ν dσ,
ρutt dx
p(x, t) dx,
Go
Go
∂Go
where dσ is the measure along the curve ∂Go . Instantaneous equilibrium of
every portion of the membrane implies that
Z
Z
Z
p(x, t) dx.
To ∇u · ν dσ +
ρutt dx =
By Green’s theorem
Z
∂Go
Therefore
Go
∂Go
Go
To ∇u · ν dσ =
Z
Go
To div(∇u) dx.
6 Transmission of Sound Waves
Z
Go
11
[ρutt − To div ∇u − p] dx = 0
for all t ∈ R and all Go ⊂ E. Thus
utt − c2 ∆u = f
where
c2 =
To
,
ρ
f=
p
,
ρ
in E × R
(5.1)
∆u = div(∇u).
Equation (5.1), modeling small vibrations of a stretched membrane, is the
two-dimensional wave equation.
6 Transmission of Sound Waves
An ideal compressible fluid is moving within a region E ⊂ R3 . Let ρ(x, t) and
v(x, t) denote its density and velocity at x ∈ E at the instant t. Each x can
be regarded as being in motion along the trajectory t → x(t) with velocity
x′ (t). Therefore, denoting by vi (x, t) the components of v along the xi -axes,
then
ẋi (t) = vi (x(t), t),
i = 1, 2, 3.
The acceleration has components
ẍi =
∂vi
∂vi
∂vi
+
ẋj =
+ (v · ∇)vi
∂t
∂xj
∂t
where ∇ denotes the gradient with respect to the space variables only. Cut any
region Go ⊂ E with boundary ∂Go of class C 1 . Since Go is instantaneously
in equilibrium, the balance of forces acting on Go must be zero. These are:
(i) The inertial forces due to acceleration
Z
ρ vt + (v · ∇)v dx.
Go
(ii) The Kelvin forces due to pressure. Let p(x, t) be the pressure at x ∈ E
at time t. The forces due to pressure on G are
Z
pν dσ,
ν = {outward unit normal to ∂Go }.
∂G0
(iii) The sum of the external forces, and the internal forces due to friction
Z
f dx.
−
Go
12
0 PRELIMINARIES
Therefore
Z
Go
ρ[vt + (v · ∇)v] dx = −
By Green’s theorem
Z
pν dσ =
Z
pν dσ +
Go
∂Go
f dx.
Go
∂Go
Z
Z
∇p dx.
Therefore by the arbitrariness of Go ⊂ E
in E × R.
ρ [vt + (v · ∇)v] = −∇p + f
(6.1)
Assume the following physical, modeling assumptions:
(a) The fluid moves with small relative velocity and small time variations
of density. Therefore second-order terms of the type vi vj,xh and ρt vi are
negligible with respect to first order terms.
(b) Heat transfer is slower than pressure drops, i.e., the process is adiabatic
and ρ = h(p) for some h ∈ C 2 (R).
Expanding h(·) about the equilibrium pressure po , renormalized to be zero,
gives
ρ = ao p + a1 p 2 + · · · .
Assume further that the pressure is close to the equilibrium pressure, so that
all terms of order higher than one are negligible when compared to ao p. These
assumptions in (6.1) yield
∂
(ρv) = −∇p + f
∂t
in E × R.
Now take the divergence of both sides to obtain
∂
div(ρv) = −∆p + div f
∂t
in E × R.
From the continuity equation
div(ρv) = −ρt = −ao pt .
Combining these remarks gives the equation of the pressure in the propagation
of sound waves in a fluid, in the form
∂2p
− c2 ∆p = f
∂t2
in E × R
1
,
ao
div f
.
ao
where
c2 =
f =−
Equation (6.2) is the wave equation in three space dimensions ([205]).
(6.2)
8 The Euler Equations
13
7 The Navier–Stokes System
The system (6.1) is rather general and holds for any ideal fluid. If the fluid is
incompressible, then ρ = const, and the continuity equation (2.4) gives
div v = 0.
(7.1)
If in addition the fluid is viscous, the internal forces due to friction can be
represented by µρ∆v, where µ > 0 is the kinematic viscosity ([268]). Therefore
(6.1) yields the Navier–Stokes system
∂
1
v − µ∆v + (v · ∇)v + ∇p = fe
∂t
ρ
(7.2)
where fe = f /ρ are the external forces acting on the system. The unknowns
are the three components of the velocity and the pressure p, to be determined
from the system of four equations (7.1) and (7.2).
8 The Euler Equations
Let S denote the entropy function of a gas undergoing an adiabatic process.
The pressure p and the density ρ are linked by the equation of state
p = f (S)ρ1+α ,
α>0
(8.1)
for some smooth function f (·). The entropy S x(t), t of an infinitesimal portion of the gas moving along the Lagrangian path t → x(t) is conserved.
Therefore ([268])
d
S=0
dt
where formally
d
∂
=
+v·∇
dt
∂t
is the total derivative. The system of the Euler equations of the process is
ρ vt + (v · ∇)v = −∇p + f
(8.2)
ρt + div(vρ) = 0
(8.3)
p
∂ p
+ v · ∇ 1+α = 0.
(8.4)
∂t ρ1+α
ρ
The first is the pointwise balance of forces following Newton’s law along the
Lagrangian paths of the motion. The second is the conservation of mass, and
the last is the conservation of entropy.
14
0 PRELIMINARIES
9 Isentropic Potential Flows
A flow is isentropic if S = const. In this case, the equation of state (8.1)
permits one to define the pressure as a function of the density alone. Let
u ∈ C 2 (R3 × R) be the velocity potential, so that v = ∇u. Assume that f = 0,
and rewrite (8.2) as
∂
px
ux + uxj uxi xj = − i ,
∂t i
ρ
i = 1, 2, 3.
(9.1)
From this
∂
∂xi
Z p
1
ds
= 0,
ut + |∇u|2 +
2
0 ρ(s)
i = 1, 2, 3.
From the equation of state
Z
0
p
ds
1+αp
=
.
ρ(s)
α ρ
Combining these calculations, gives the Bernoulli law for an isentropic potential flow2
1+αp
1
=g
(9.2)
ut + |∇u|2 +
2
α ρ
where g(·) is a function of t only. The positive quantity
c2 =
dp
p
= (1 + α)
dρ
ρ
has the dimension of the square of a velocity, and c represents the local speed
of sound. Notice that c need not be constant. Next multiply the ith equation
in (9.1) by uxi and add for i = 1, 2, 3 to obtain
1
1
1 ∂
|∇u|2 + ∇u · ∇|∇u|2 = − ∇p · ∇u.
2 ∂t
2
ρ
(9.3)
Using the continuity equation
1
p
p1
− ∇p · ∇u = −∇ ∇u −
∇ρ · ∇u
ρ
ρ
ρρ
p
p1
p
= −∇ · ∇u +
ρt + ∆u.
ρ
ρρ
ρ
From the equation of state
2
Daniel Bernoulli, 1700–1782, botanist and physiologist, made relevant discoveries in hydrodynamics. His father, Johann B. 1667–1748, and his uncle Jakob 1654–
1705, brother of Johann, were both mathematicians. Jakob and Johann are known
for their contributions to the calculus of variations.
10 Partial Differential Equations
15
d
d p
1 d p p d 1
= α
= 0.
f (S) =
+
dt
dt ρ1+α
ρ dt ρ ρ dt ρα
From this, expanding the total derivative
∂ p
p
p
+ ∇u · ∇ − α 2 [ρt + ∇u · ∇ρ] = 0.
∂t ρ
ρ
ρ
Again, by the equation of continuity
ρt + ∇u · ∇ρ = ρ∆u.
Therefore
∂ p
p
p
+ α ∆u.
−∇ · ∇u =
ρ
∂t ρ
ρ
Combining these calculations in (9.3) gives
1
1 ∂
1 ∂
c2 ∆u − ∇u · ∇|∇u|2 =
|∇u|2 −
p.
2
2 ∂t
ρ ∂t
(9.4)
9.1 Steady Potential Isentropic Flows
For steady flows, rewrite (9.4) in the form
(c2 δij − uxi uxj )uxi xj = 0.
(9.5)
The matrix of the coefficients of the second derivatives uxi xj is
ux ux δij − i 2 j
c
and its eigenvalues are
λ1 = 1 −
|∇u|2
c2
and
λ2 = 1.
Using the steady-state version of the Bernoulli law (9.2) gives the first eigenvalue in terms only of the pressure p and the density ρ. The ratio M = |∇u|/c
of the speed of a body to the speed of sound in the surrounding medium is
called the Mach number.3
10 Partial Differential Equations
The equations and systems of the previous sections are examples of PDEs.
Let u ∈ C m (E) for some m ∈ N, and for j = 1, 2, . . . , m, let Dj u denote
3
Ernst Mach, 1838–1916. Mach one is the speed of sound; Mach two is twice the
speed of sound; ...
16
0 PRELIMINARIES
the vector of all the derivatives of u of order j. For example, if N = m = 2,
denoting (x, y) the coordinates in R2
D1 u = (ux , uy )
and
D2 u = (uxx , uxy , uyy ).
A partial differential equation is a functional link among the variables
x, u, D1 u, D2 u, . . . , Dm u
that is
F (x, u, D1 u, D2 u, . . . , Dm u) = 0.
The PDE is of order m if the gradient of F with respect to Dm u is not
identically zero. It is linear if for all u, v ∈ C m (E) and all α, β ∈ R
F x, (αu + βv), D1 (αu + βv), D2 (αu + βv), . . . , Dm (αu + βv)
= αF x, u, D1 u, D2 u, . . . , Dm u + βF x, v, D1 v, D2 v, . . . , Dm v .
It is quasi-linear if it is linear with respect to the highest order derivatives.
Typically a quasi-linear PDE takes the form
P
m1 +···+mN =m
am1 ,...,mN
∂ m1 x1
∂m u
+ Fo = 0
· · · ∂ mN xN
where mj are non-negative integers and the coefficients am1 ,...,mN , and the
forcing term Fo , are given smooth functions of (x, u, D1 u, D2 u, . . . , Dm−1 u).
If the PDE is quasi-linear, the sum of the terms involving the derivatives of
highest order, is the principal part of the PDE.
Problems and Complements
3c The Heat Equation and the Laplace Equation
It is worth devoting some space to Einstein’s description of the Brownian
Motion [64, 63, 240] and to its surprising connection with the Heat Equation.
3.1c Basic Physical Assumptions
Particles suspended in a fluid undergo random motions owing to disordered
collisions generated by local thermal gradients. The trajectories are irregular
and intertwined, and cannot be efficiently representable by the classical concepts of piecewise differentiable curves. Instead, the motion will be described
by the density or distribution of particles, u(x, t) providing the number of
particles, per unit volume, present at x at time t. The basic assumptions are:
3c The Heat Equation and the Laplace Equation
17
H1 The motion of each particle is independent of the motion of the others.
This is in general false if the motion is caused by mutual collisions, which
by their own nature mutually affect the motion of each particle. It can
only be justified as an “average” property of the particles, in the sense that
the motion, computed and averaged, in some sense, over a large number
of particles, is not affected by the addition of a further particle, however
it is positioned within the system.
H2 For each time t, the motion of a particle at times later than t depends
only on its position at time t and not on its kinematic at times preceding
and up to t. This is also false in general, in the classical formulation of
Mechanics, where the motion after a time t depends on the kinematic at
time t. Einstein justifies this assumption provided we look at times t + τ
where 0 < τ ≪ 1 is so small that the only mechanics affecting the particle
is a possible collision at time t and not its velocity at t. In the words of
Einstein ... τ is negligible with respect to observable times...
The multitude of particles suggests looking at the particles only as a statistical
average. Moreover, the numerous collisions, and consequent lack of differentiability of the intertwined paths, suggest renouncing to the classical notions
of velocity and acceleration.
For y ∈ RN let p(y, ∆t)dy be the probability of each particle undergoing
a displacement y in the time interval (t, t + ∆t). In view of H1–H2, such a
probability is the same for each particle. Moreover,
Z
p(y, ∆t)dy = 1 for all ∆t > 0.
(3.1c)
RN
It is also assumed that the medium is isotropic, that is, the probability density
is independent of the direction of y so that p(y, ∆t) = p(|y|, ∆t).
3.2c The Diffusion Equation
The particles that are in x at time t + ∆t are those that were in x − y at time
t and that have been displaced by y in the time interval (t, t + ∆t). Thus,
Z
u(x − y, t)p(y, ∆t)dy.
u(x, t + ∆t) =
RN
By Taylor’s expansion (formal at this stage)
u(x − y, t) = u(x, t) −
Therefore,
N
P
j=1
uxj (x, t)yj +
N
1 P
ux x (x, t)yi yj + R(x, y, t).
2 i,j=1 i j
18
0 PRELIMINARIES
u(x, t + ∆t) =u(x, t)
Z
p(y, ∆t)dy
Z
N
P
−
uxj (x, t)
yj p(y, ∆t)dy
RN
j=1
RN
Z
N
1 P
+
uxi xj
yi yj p(y, ∆t)dy
2 i,j=1
RN
Z
+
R(x, y, t)p(y, ∆t)dy.
RN
Using that p(y, ∆t) = p(|y|, ∆t) one verifies that
R
y p(y, ∆t)dy = 0
for i = 1, . . . , N ;
RN j
R
R
RN
yi yj p(y, ∆t)dy = 0
for all i 6= j;
RN
yj2 p(y, ∆t)dy = 2k(∆t)
for j = 1, . . . , N
for a function k(·). Combining these calculations
Z
u(x, t + ∆t) − u(x, t) k(∆t)
1
−
∆u =
R(x, y, t)p(y, ∆t)dy.
∆t
∆t
∆t RN
(3.2c)
Einstein then postulates that
k(∆t)
= k for a given positive constant k, and
∆tZ
1
lim
R(x, y, t)p(y, ∆t)dy = 0.
∆t→0 ∆t RN
lim
∆t→0
(3.3c)
(3.4c)
Assuming these two postulates for the moment, and letting ∆t → 0 in (3.2c),
imply that the density u of particles in Brownian motion suspended in a fluid
satisfies the heat equation
ut − k∆u = 0
within the fluid being observed.
(3.5c)
3.3c Justifying the Postulates (3.3c)–(3.4c)
The first postulate is simply accepted by Einstein. The second is essentially a
consequence of the first. By the definition of the remainder R(x, y, t)
Z
Z
1
A(x, t)
lim
|R(x, y, t)|p(y, ∆t)dy ≤ lim
|y|3 |p(y, ∆t)dy
∆t→0 ∆t RN
∆t→0
∆t
N
R
for a constant A(x, t) depending on (x, t), which, however, in this process, are
fixed. Now it is reasonable to postulate that if 0 < ∆t ≪ 1, those displacements y that have an appreciable probability of actually occurring are small
in length, proportionally to ∆t. Thus, it is postulated that
3c The Heat Equation and the Laplace Equation
Z
RN
|y|3 |p(y, ∆t)dy ≤ B
Z
|y|≤C∆t
19
|y|3 p(y, ∆t)dy
for given constants B and C. This implies (3.4c).
3.4c More on the Postulates (3.3c)–(3.4c)
These postulates are verified for the probability density p(·, ·) given by
Γ (y, t) =
|y|2
1
e− 4kt .
N/2
(4πkt)
(3.6c)
Verify that such a p(·, ·) satisfies (3.1c) for all t > 0, and (3.3c)–(3.4c) for
all ∆t > 0. The probability that a particle in Brownian motion, is at y at
time t, starting from y = 0 at time t = 0, could be statistically computed by
effecting a large number of experiments, and then by taking their arithmetic
average. It can be shown that as the number of experiments tends to infinity
the corresponding probability tends to the function Γ in (3.6c), for a physical
constant k that depends on the medium. Such a constant can be given a
precise physical meaning by the following considerations. The mean distance
ℓ(t) traveled by the particle in time t, originating at the origin is the squareroot of the expected value of |y|2 , that is
Z
Z
2
4kt
|y|2 Γ (y, t)dy = N/2
ℓ2 (t) =
|y|2 e−|y| dy = const(N )kt.
(3.7c)
π
N
N
R
R
3.5 Verify that RN × R+ ∋ (y, t) → Γ (y, t) satisfies the heat equation (3.5c).
3.6 Justify the calculations in (3.7c) and compute explicitly the constant
const(N ).
3.7 Verify that the postulates (3.4c)–(3.4c) are verified for such a Γ .
3.8 Let uo (·) be a bounded and continuous function in RN . Verify that the
function u = Γ ∗ uo satisfies the heat equation (3.5c) and that
over compact subsets of RN .
lim u(·, t) = uo
t→0
3.9 Let
KN (x, y) =

1
 (N −2)ωN

1
2π
=
for N = 2.
2
1
.
ωN +1 [|x − y|2 + t2 ] N2+1
∆x KN = ∆y Kn = 0
′
∆x;t KN
+1
for N > 2;
ln |x − y|
′
KN
+1 (x, y; t) =
Verify that
1
|x−y|N −2
in x 6= y
′
∆y;t KN
+1
=0
for t > 0.
20
0 PRELIMINARIES
Verify that
′
KN
+1 =
∂
KN +1 (x, y; t, τ )
∂τ
.
τ =0
1
QUASI-LINEAR EQUATIONS AND
ANALYTIC DATA
1 Quasi-Linear Second-Order Equations in Two
Variables
Let (x, y) denote the variables in R2 , and consider the quasi-linear equation
Auxx + 2Buxy + Cuyy = D
(1.1)
where (x, y, ux , uy ) → A, B, C, D(x, y, ux , uy ) are given smooth functions of
their arguments. The equation is of order two if at least one of the coefficients A, B, C is not identically zero. Let Γ be a curve in R2 of parametric
representation
x = ξ(t)
Γ =
∈ C 2 (−δ, δ) for some δ > 0.
y = η(t)
On Γ , prescribe the Cauchy data
u
Γ
= v,
ux
Γ
= ϕ,
uy
Γ
=ψ
(1.2)
where t → v(t), ϕ(t), ψ(t) are given functions in C 2 (−δ, δ). Since
d
u ξ(t), η(t) = ux ξ ′ + uy η ′ = ϕξ ′ + ψη ′ = v ′
dt
of the three functions v, ϕ, ψ, only two can be assigned independently. The
Cauchy problem for (1.1) and Γ , consists in finding u ∈ C 2 (R2 ) satisfying the
PDE and the Cauchy data (1.2). Let u be a solution of the Cauchy problem
(1.1)–(1.2), and compute its second derivatives on Γ . By (1.1) and the Cauchy
data
Auxx + 2Buxy + Cuyy = D
ξ ′ uxx + η ′ uxy
= ϕ′
′
′
ξ uxy + η uyy = ψ ′ .
Here A, B, C are known, since they are computed on Γ . Precisely
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_2
21
22
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
A, B, C
Γ
= A, B, C(ξ, η, v, ϕ, ψ).
Therefore, uxx , uxy , and uyy can be computed on Γ , provided


A 2B C
det  ξ ′ η ′ 0  6= 0.
0 ξ ′ η′
(1.3)
We say that Γ is a characteristic curve if (1.3) does not hold, i.e., if
Aη ′2 − 2Bξ ′ η ′ + Cξ ′2 = 0.
(1.4)
In general, the property of Γ being a characteristic depends on the Cauchy
data assigned on it. If Γ admits a local representation of the type
y = y(x)
in a neighborhood of some xo ∈ R
(1.5)
the characteristics are the graphs of the possible solutions of the differential
equation
√
B ± B 2 − AC
y′ =
.
A
Associate with (1.1) the matrix of the coefficients
AB
M=
.
BC
Using (1.4) and the matrix M , we classify, locally, the family of quasi-linear
equations (1.1) as elliptic if det M > 0, i.e., if there exists no real characteristic; parabolic if det M = 0, i.e., if there exists one family of real characteristics; hyperbolic if det M < 0, i.e., if there exist two families of real
characteristics. The elliptic, parabolic, or hyperbolic nature of (1.1) may be
different in different regions of R2 . For example, the Tricomi equation ([259])
yuxx − uyy = 0
is elliptic in the region [y < 0], parabolic on the x-axis and hyperbolic in the
√
upper half-plane [y > 0]. The characteristics are solutions of yy ′ = ±1 in
the upper half-plane [y > 0].
The elliptic, parabolic, or hyperbolic nature of the PDE may also depend
upon the solution itself. As an example, consider the equation of steady compressible fluid flow of a gas of density u and velocity ∇u = (ux , uy ) in R2 ,
introduced in (9.5) of the Preliminaries
(c2 − u2x )uxx − 2ux uy uxy + (c2 − u2y )uyy = 0
where c > 0 is the speed of sound. Compute
2
c − u2x −ux uy
= c2 (c2 − |∇u|2 ).
det M = det
−ux uy c2 − u2y
2 Characteristics and Singularities
23
Therefore the equation is elliptic for sub-sonic flow (|∇u| < c), parabolic for
sonic flow (|∇u| = c), and hyperbolic for super-sonic flow (|∇u| > c). The
Laplace equation
∆u = uxx + uyy = 0
is elliptic. The heat equation
H(u) = uy − uxx = 0
is parabolic with characteristic lines y = const. The wave equation
u = uyy − c2 uxx = 0 c ∈ R
is hyperbolic with characteristic lines x ± cy = const.
2 Characteristics and Singularities
If Γ is a characteristic, the Cauchy problem (1.1)–(1.2) is in general not solvable, since the second derivatives of u cannot be computed on Γ . We may
attempt to solve the PDE (1.1) on each side of Γ and then piece together the
functions so obtained. Assume that Γ divides R2 into two regions E1 and E2
and let ui ∈ C 2 (Ēi ), for i = 1, 2, be possible solutions of (1.1) in Ei satisfying
the Cauchy data (1.2). These are taken in the sense of
lim
(x,y)→(ξ(t),η(t))
(x,y)∈Ei
ui (x, y), ui,x (x, y), ui,y (x, y) = v(t), ϕ(t), ψ(t).
Setting
u=
u1
u2
in E1
in E2
the function u is of class C 1 across Γ . If fi ∈ C(Ēi ), for i = 1, 2, and
f1
in E1
f=
f2
in E2
let [f ] denote the jump of f across Γ , i.e.,
[f ](t) =
lim
(x,y)→(ξ(t),η(t))
(x,y)∈E1
f1 (x, y) −
lim
(x,y)→(ξ(t),η(t))
(x,y)∈E2
f2 (x, y).
From the assumptions on u,
[u] = [ux ] = [uy ] = 0.
(2.1)
A[uxx ] + 2B[uxy ] + C[uyy ] = 0.
(2.2)
From (1.1),
24
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
Assume that Γ has the local representation (1.5). Then using (2.1), compute
[uxx ] + [uxy ]y ′ = 0,
[uxy ] + [uyy ]y ′ = 0.
Therefore
[uxx ] = [uyy ]y ′2 ,
[uxy ] = −[uyy ]y ′ .
(2.3)
Let J = [uyy ] denote the jump across Γ of the second y-derivative of u. From
(2.2) and (2.3)
J(Ay ′2 − 2By ′ + C) = 0.
If Γ is not a characteristic, then (Ay ′2 − 2By ′ + C) 6= 0. Therefore J = 0,
and u is of class C 2 across Γ . If J 6= 0, then Γ must be a characteristic. Thus
if a solution of (1.1) in a region E ⊂ R2 suffers discontinuities in the second
derivatives across a smooth curve, these must occur across a characteristic.
2.1 Coefficients Independent of ux and uy
Assume that the coefficients A, B, C and the term D are independent of ux
and uy , and that u ∈ C 3 (Ēi ), i = 1, 2. Differentiate (1.1) with respect to y in
Ei , form differences, and take the limit as (x, y) → Γ to obtain
A[uxxy ] + 2B[uxyy ] + C[uyyy ] = 0.
(2.4)
Differentiating the jump J of uyy across Γ gives
J ′ = [uxyy ] + [uyyy ]y ′ .
(2.5)
From the second jump condition in (2.3), by differentiation
−y ′ J ′ − y ′′ J = [uxxy ] + [uxyy ]y ′ .
(2.6)
We eliminate [uxxy ] from (2.4) and (2.6) and use (2.5) to obtain
A(y ′ J ′ + y ′′ J) = (2B − Ay ′ )[uxyy ] + C[uyyy ]
= (2B − Ay ′ )J ′ + (Ay ′2 − 2By ′ + C)[uyyy ].
Therefore, if Γ is a characteristic
2(B − Ay ′ )J ′ = Ay ′′ J.
This equation describes how the jump J of uyy at some point P ∈ Γ propagates along Γ . In particular, either J vanishes identically, or it is never zero
on Γ .
3 Quasi-Linear Second-Order Equations
25
3 Quasi-Linear Second-Order Equations
Let E be a region in RN , and let u ∈ C 2 (E). A quasi-linear equation in E
takes the form
Aij uxi xj = F
(3.1)
where we have adopted the summation notation and
(x, u, ∇u) → Aij , F (x, u, ∇u)
for
i, j = 1, 2, . . . , N
are given smooth functions of their arguments. The equation is of order two if
not all the coefficients Aij are identically zero. By possibly replacing Aij with
Aij + Aji
2
we may assume that the matrix (Aij ) of the coefficients is symmetric. Let Γ
be a hypersurface of class C 2 in RN , given as a level set of Φ ∈ C 2 (E); say for
example, Γ = [Φ = 0]. Assume that ∇Φ 6= 0 and let ν = ∇Φ/|∇Φ| be the unit
normal to Γ oriented in the direction of increasing Φ. For x ∈ Γ , introduce
a local system of N − 1 mutually orthogonal unit vectors {τ 1 , . . . , τ N −1 }
chosen so that the n-tuple {τ 1 , . . . , τ N −1 , ν} is congruent to the orthonormal
Cartesian system {e1 , . . . , eN }. Given f ∈ C 1 (E), compute the derivatives of
f , normal and tangential to Γ from
Dν f = ∇f · ν,
Dτ j f = ∇f · τ j ,
j = 1, . . . , N − 1.
If τi,j = τ i · ej , and νj = ν · ej , are the projections of τ i and ν on the
coordinate axes
Dτ j f = (τj,1 , . . . , τj,N ) · ∇f
Introduce the unitary matrix

τ1,1
τ2,1
..
.
Dν f = (ν1 , . . . , νN ) · ∇f.
and
τ1,2
τ2,2
..
.
...
...
..
.
τ1,N
τ2,N
..
.







T =



τN −1,1 τN −1,2 . . . τN −1,N 
ν1
ν2 . . . νN
and write
∇f = T
−1
Dτ f
Dν f
(3.2)
where T −1 = T t .
(3.3)
The Cauchy data of u on Γ are
u
Γ
= v,
Dτ j u
Γ
= ϕj , j = 1, . . . , N − 1,
Dν u
Γ
=ψ
(3.4)
regarded as restrictions to Γ of smooth functions defined on the whole of E.
These must satisfy the compatibility conditions
26
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
j = 1, . . . , N − 1.
Dτ j v = ϕj ,
(3.5)
Therefore, only v and ψ can be given independently. The Cauchy problem for
(3.1) and Γ consists in finding a function u ∈ C 2 (E) satisfying the PDE and
the Cauchy data (3.4). If u is a solution to the Cauchy problem (3.1)–(3.4),
compute the second derivatives of u on Γ
uxi = τk,i uτk + νi uν
∂
∂
uxi xj = τk,i
u τ k + νi
uν
∂xj
∂xj
(3.6)
= τk,i τl,j uτk τl + τk,i νj uτk ν + τk,j νi uτk ν + νi νj uνν .
From the compatibility conditions (3.4) and (3.5)
Dτ i (Dτ j u) = Dτ i ϕj ,
Dτ i (Dν u) = Dτ i ψ.
Therefore, of the terms on the right-hand side of (3.6), all but the last are
known on Γ . Using the PDE, one computes
Aij νi νj uνν = F̃
on Γ
(3.7)
where F̃ is a known function of Γ and the Cauchy data on it. We conclude
that uνν , and hence all the derivatives uxi xj , can be computed on Γ provided
Aij Φxi Φxj 6= 0.
(3.8)
Both (3.7) and (3.8) are computed at fixed points P ∈ Γ . We say that Γ is a
characteristic at P if (3.8) is violated, i.e., if
(∇Φ)t (Aij )(∇Φ) = 0
at P.
Since (Aij ) is symmetric, its eigenvalues are real and there is a unitary matrix
U such that


λ1 0 . . . 0
 0 λ2 . . . 0 


U −1 (Aij )U =  . . .
. .
 .. .. . . .. 
0 0 . . . λN
Let ξ = U xt denote the coordinates obtained from x by the rotation induced
by U . Then
(∇Φ)t (Aij )(∇Φ) = [U −1 (∇Φ)]t U −1 (Aij )U [U −1 (∇Φ)] = λi Φ2ξi .
Therefore Γ is a characteristic at P if
λi Φ2ξi = 0.
(3.9)
Writing this for all P ∈ E gives a first-order PDE in Φ. Its solutions permit
us to find the characteristic surfaces as the level sets [Φ = const].
3 Quasi-Linear Second-Order Equations
27
3.1 Constant Coefficients
If the coefficients Aij are constant, (3.9) is a first-order PDE with constant
coefficients. The PDE in (3.1) is classified according to the number of positive
and negative eigenvalues of (Aij ). Let p and n denote the number of positive
and negative eigenvalues of (Aij ), and consider the pair (p, n). The equation
(3.1) is classified as elliptic if either (p, n) = (N, 0) or (p, n) = (0, N ). In
either of these cases, it follows from (3.9) that
0 = |λi Φξ2i | ≥ min |λi ||∇Φ|2 .
1≤i≤N
Therefore, there exist no characteristic surfaces.
Equation (3.1) is classified as hyperbolic if p + n = N and p, n ≥ 1.
Without loss of generality, we may assume that eigenvalues are ordered so
that λ1 , . . . , λp are positive and that λp+1 , . . . , λN are negative. In such a
case, (3.9) takes the form
p
P
i=1
This is solved by
Φ± (ξ) =
λi Φξ2i =
N
P
j=p+1
q
p p
N
P
P
λi ξi ± k
|λj |ξj
i=1
where
k2 =
|λj |Φ2ξj .
p
P
i=1
j=p+1
λ2i
.
N
P
j=p+1
λ2j .
The hyperplanes [Φ± = const] are two families of characteristic surfaces for
(3.1). In the literature these PDE are further classified according to the values
of p and n. Namely they are called hyperbolic if either p = 1 or n = 1.
Otherwise they are called ultra-hyperbolic.
Equation (3.1) is classified as parabolic if p + n < N . Then at least one
of the eigenvalues is zero. If, say, λ1 = 0, then (3.9) is solved by any function
of ξ1 only, and the hyperplane ξ1 = const is a characteristic surface.
3.2 Variable Coefficients
In analogy with the case of constant coefficients we classify the PDE in (3.1)
at each point P ∈ E as elliptic, hyperbolic, or parabolic according to the
number of positive and negative eigenvalues of (Aij ) at P . The classification
is local, and for coefficients depending on the solution and its gradient, the
nature of the equation may depend on its own solutions.
28
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
4 Quasi-Linear Equations of Order m ≥ 1
An N -dimensional multi-index α, of size |α|, is an N -tuple of non-negative
integers whose sum is |α|, i.e.,
α = (α1 , . . . , αN ),
αi ∈ N ∪ {0}, i = 1, . . . , N,
|α| =
N
P
αi .
i=1
If f ∈ C m (E) for some m ∈ N, and α is a multi-index of size m, let
Dα f =
∂ α1 ∂ α2
∂ αN
f.
α1
α2 · · ·
N
∂x1 ∂x2
∂xα
N
If |α| = 0 let Dα f = f . By Dm−1 f denote the vector of all the derivatives
Dα f for 0 ≤ |α| ≤ m − 1. Consider the quasi-linear equation
P
Aα Dα u = F
(4.1)
|α|=m
where (x, Dm−1 u) → Aα , F (x, Dm−1 u) are given smooth functions of their
arguments. The equation is of order m if not all the coefficients Aα are identically zero. If v = (v1 , . . . , vN ) is a vector in RN and α is an N -dimensional
multi-index, let
αN
vα = v1α1 v2α2 · · · vN
.
Prescribe a surface Γ as in the previous section and introduce the matrix
T as in (3.2), so that the differentiation formula (3.3) holds. Denoting by
β = (β1 , . . . , βN −1 ) an (N − 1)-dimensional multi-index of size |β| ≤ m, set
β
Dτβ f = Dτβ11 Dτβ22 · · · DτNN−1
f.
−1
Write N -dimensional multi-indices as α = (β, s), where s is a non-negative
integer, and for |α| ≤ m, set
s
Dτα ,ν f = Dτβ Dν
f.
The Cauchy data of u on Γ are
Dτα ,ν u
Γ
= fα ∈ C m (E)
for all |α| < m.
(4.2)
Among these we single out the Dirichlet data
u
Γ
= fo ,
(4.2)D
the normal derivatives
s
Dν
u
Γ
= fs ,
and the tangential derivatives
|α| = s ≤ m − 1,
(4.2)ν
4 Quasi-Linear Equations of Order m ≥ 1
Dτβ u
Γ
= fβ ,
|β| < m.
29
(4.2)τ
Of these, only (4.2)D and (4.2)ν can be given independently. The remaining
ones must be assigned to satisfy the compatibility conditions
s
Dν
fβ = Dτβ fs
for all |β| ≥ 0,
|β| + s ≤ m − 1.
(4.3)
The Cauchy problem for (4.1) consists in finding a function u ∈ C m (E) satisfying (4.1) in E and the Cauchy data (4.2) on Γ .
4.1 Characteristic Surfaces
If u is a solution of the Cauchy problem, compute its derivatives of order m
on Γ , by using (4.1), the Cauchy data (4.2) and the compatibility conditions
(4.3). Proceeding as in formula (3.6), for a multi-index α of size |α| = m
αN m
Dα u = ν1α1 · · · νN
Dν u + g
on Γ
where g is a known function that can be computed a priori in terms of Γ ,
the Cauchy data (4.2), and the compatibility conditions (4.3). Putting this in
(4.1) gives
P
αN
m
ν α = ν1α1 · · · νN
u = F̃ ,
Aα ν α Dν
|α|=m
where F̃ is known in terms of Γ and the data. Therefore all the derivatives,
normal and tangential, up to order m can be computed on Γ , provided
P
Aα ν α 6= 0.
(4.4)
|α|=m
We say that Γ is a characteristic surface if (4.4) is violated, i.e.,
P
Aα (DΦ)α = 0.
(4.5)
|α|=m
In general, the property of Γ being a characteristic depends on the Cauchy
data assigned on it, unless the coefficients Aα are independent of Dm−1 u.
Condition (4.5) was derived at a fixed point of Γ . Writing it at all points
of E gives a first-order nonlinear PDE in Φ whose solutions permit one to
find the characteristics associated with (4.1) as the level sets of Φ. To (4.1)
associate the characteristic form
L(ξ) = Aα ξ α .
If L(ξ) 6= 0 for all ξ ∈ RN −{0}, then there are no characteristic hypersurfaces,
and (4.1) is said to be elliptic.
30
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
5 Analytic Data and the Cauchy–Kowalewski Theorem
A real-valued function f defined in an open set G ⊂ Rk , for some k ∈ N,
is analytic at η ∈ G, if in a neighborhood of η, f (y) can be represented as
a convergent power series of y − η. The function f is analytic in G if it is
analytic at every η ∈ G.
Consider the Cauchy problem for (4.1) with analytic data. Precisely, assume that Γ is noncharacteristic and analytic about one of its points xo ; the
Cauchy data (4.2) satisfy the compatibility conditions (4.3) and are analytic
at xo . Finally, the coefficients Aα and the free term F are analytic about the
point (xo , u(xo ), Dm−1 u(xo )).
The Cauchy–Kowalewski Theorem asserts that under these circumstances,
the Cauchy problem (4.1)–(4.2) has a solution u, analytic at xo . Moreover,
the solution is unique within the class of analytic solutions at xo .
5.1 Reduction to Normal Form ([32])
Up to an affine transformation of the coordinates, we may assume that xo
coincides with the origin and that Γ is represented by the graph of xN =
Φ(x̄), with x̄ = (x1 , . . . , xN −1 ), where x̄ → Φ(x̄) is analytic at the origin
of RN −1 . Flatten Γ about the origin by introducing new coordinates (x̄, t)
where t = xN − Φ(x̄). In this way Γ becomes a (N − 1)-dimensional open
neighborhood of the origin lying on the hyperplane t = 0. Continue to denote
by u, Aα , and F the transformed functions and rewrite (4.1) as
A(0,...,m)
∂m
u=
∂tm
P
|β|+s=m
0<s<m
A(β,s)
P
∂s β
A(β,0) Dβ u + F.
D u+
s
∂t
|β|=m
(5.1)
Here Dβ operates only on the variables x̄. Since [t = 0] is not a characteristic
surface, (4.4) implies
A(0,...,m) x̄, 0, Dm−1 u(x̄, 0) 6= 0
and this continues to hold in a neighborhood of 0, 0, Dm−1 u(0, 0) , since the
functions Aα are analytic near such a point. Divide (5.1) by the coefficient of
Dtm u and continue to denote by the same letters the transformed terms on the
right-hand side. Next, introduce the vector u = Dm−1 u, and let uα = Dα u
for |α| ≤ m − 1, be one component of this vector. If α = (0, . . . , m − 1), then
the derivative
∂
∂m
uα = m u
∂t
∂t
satisfies (5.1). If α = (β, s) and s ≤ (m − 2), then
∂
∂
ũα
uα =
∂t
∂xi
5 Analytic Data and the Cauchy–Kowalewski Theorem
31
for some i = 1, . . . , N − 1 and some component ũα of the vector u. Therefore,
the PDE of order m in (5.1) can be rewritten as a first-order system in the
normal form
∂
∂
ui = Aijk
uk + Fi (x̄, t, u).
∂t
∂xj
The Cauchy data on Γ reduce to u(x̄, 0) = f (x̄). Therefore setting
v(x̄, t) = u(x̄, t) − f (0)
and transforming the coefficients Aijk and the function F accordingly, reduces
the problem to one with Cauchy data vanishing at the origin.
The coefficients Aijk as well as the free term F , can be considered independent of the variables (x̄, t). Indeed these can be introduced as auxiliary
dependent variables by setting
uj = xj ,
satisfying
∂
uj = 0 and uj (x̄, 0) = xj
∂t
for j = 1, . . . , N − 1, and
uN = t,
satisfying
∂
uN = 1 and uN (x̄, 0) = 0.
∂t
These remarks permit one to recast the Cauchy problem (4.1)–(4.2), in the
normal form
u = (u1 , u2 , . . . , uℓ ), ℓ ∈ N
∂
∂
u = Aj (u)
u + F(u)
(5.2)
∂t
∂xj
u(x, 0) = uo (x),
uo (0) = 0
where Aj = (Aik )j are ℓ×ℓ matrices and F = (F1 , . . . , Fℓ ) are known functions
of their arguments. We have also renamed and indexed the space variables,
on the hyperplane t = 0, as x = (x1 , . . . , xN ).
Theorem 5.1 (Cauchy–Kowalewski). Assume that u → Aikj (u), Fi (u)
and x → uo,i (x) for i, j, k = 1, . . . , ℓ are analytic in a neighborhood of the
origin. Then there exists a unique analytic solution of the Cauchy problem
(5.2) in a neighborhood of the origin.
For linear systems, the theorem was first proved by Cauchy ([33]). It was
generalized to nonlinear systems by Sonja Kowalewskaja1 ([140]). A generalization is also due to G. Darboux ([41]).
1
Sonja Kowalewskaja, 1850–1891, was born in Russia into an upper middle class
family. Karl Weierstrass was the advisor of her PhD thesis, which was devoted to the
analyticity of solutions of the Cauchy Problem. She obtained a chair of mathematics
in Stockholm in 1884, where she died at 41 of pneumonia.
32
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
Theorem 5.1 is a fundamental achievement, which lies at the heart of a lot
of different results. However, as Lars Gårding wrote in [95] about a general
linear partial differential equation of the kind
P (x, Dx u) = f,
(5.3)
it was pointed out very emphatically by Hadamard that it is not natural to consider only analytic solutions and source functions f even P has analytic coefficients. This reduces the interest of the Cauchy-Kowalewski theorem which
says that (5.3) has locally analytic solutions if P and f are analytic. The
Cauchy-Kowalewski theorem does not distinguish between classes of differential operators which have, in fact, very different properties such as the Laplace
operator and the wave operator.
6 Proof of the Cauchy–Kowalewski Theorem
First use the system in (5.2) to compute all the derivatives, at the origin, of
a possible solution. Then using these numbers, construct the formal Taylor’s
series of an anticipated solution, say
u(x, t) =
Dβ Dts u(0, 0) β s
x t
β!s!
|β|+s≥0
P
(6.1)
where xβ = xβ1 1 · · · xβNN , and β! = β1 ! · · · βN !. If this series converges in a neighborhood of the origin, then (6.1) defines a function u, analytic near (0, 0). Such
a u is a solution to (5.2). Indeed, substituting the power series (6.1) on the
left- and right-hand sides of the system in (5.2), gives two analytic functions
whose derivatives of any order coincide at (0, 0). Thus they must coincide in a
neighborhood of the origin. Uniqueness within the class of analytic solutions
follows by the same unique continuation principle. Therefore the proof of the
theorem reduces to showing that the series in (6.1), with all the coefficients
Dβ Ds u(0, 0) computed from (5.2), converges about the origin.
The convergence of the series could be established, indirectly, by the
method of the majorant ([131], 73–78). This was the original approach of
A. Cauchy, followed also by S. Kowalewskaja and G. Darboux. The convergence of the series, can also be established by a direct estimation of all the
derivatives of u. This is the method we present here. This approach, originally due to Lax ([157]), has been further elaborated and extended by A.
Friedman [82]. It has also been generalized to an infinite dimensional setting
by M. Shimbrot and R.E. Welland ([238]).
Let α and β be N -dimensional multi-indices, denote by m a non-negative
m
integer and set αm = αm
1 · · · αN , and
(α + β)!
α+β
α + β = (α1 + β1 , . . . , αN + βN ),
.
=
α
α!β!
7 Auxiliary Inequalities
33
Denote by ι the multi-index ι = (1, . . . , 1), so that
β + ι = (β1 + 1, β2 + 1, . . . , βN + 1).
The convergence of the series in (6.1) is a consequence of the following
Lemma 6.1 There exist constants Co and C such that,
|Dβ Dts u(0, 0)| ≤ Co C |β|+s−1
β!s!
(β + ι)2 (s + 1)2
(6.2)
for all N -dimensional multi-indices β, and all non-negative integers s.
6.1 Estimating the Derivatives of u at the Origin
We will establish first the weaker inequality
Lemma 6.2 There exist constants Co and C such that
|Dβ Dts u(0, 0)| ≤ Co C |β|+s−1
(|β| + s)!
(β + ι)2 (s + 1)2
(6.3)
for all N -dimensional multi-indices β and all non-negative integers s.
This inequality holds for s = 0, since Dβ u(0, 0) = Dβ uo (0), and uo is analytic.
We will show that if it does hold for s it continues to hold for s + 1.
Let γ = (γ1 , . . . , γℓ ) be an ℓ-dimensional multi-index. Since Aj (·) and F(·)
are analytic at the origin
P i γ
P ikj γ
Aikj (u) =
Aγ u ,
Fi (u) =
Fγ u
(6.4)
|γ|≥0
|γ|≥0
where uγ = uγ11 · · · uγℓ ℓ , and Aikj
and Fγi are constants satisfying
γ
|γ|
|Fγi | + |Aikj
γ | ≤ Mo M
(6.5)
for all i, k = 1, . . . , ℓ and all j = 1, . . . , N , for two given positive constants Mo
and M . From (5.2) and (6.4), compute
P ikj β s
P i β s γ
∂uk
Aγ D Dt uγ
Dβ Dts+1 ui =
Fγ D Dt u .
(6.6)
+
∂x
j
|γ|≥1
|γ|≥1
The induction argument requires some preliminary estimates.
7 Auxiliary Inequalities
Lemma 7.1 Let m, n, i, j be non-negative integers such that n ≥ i and m ≥ j.
Then
n
P
16
1
≤
(7.1)
2 (i + 1)2
(n
−
i
+
1)
(n
+
1)2
i=0
and
n
m
n+m
and
≤
.
(7.2)
i
j
i+j
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
34
Proof. If 1 ≤ i ≤ 21 n, the argument of the sum is majorized by
4
1
(n + 1)2 (i + 1)2
for all integers
1 ≤ i ≤ 12 n.
If 12 n < i ≤ n, it is majorized by
4
1
(n + 1)2 (n − i + 1)2
for all integers
1
2n
< i ≤ n.
Thus in either case
n
P
∞
P
1
8
1
≤
.
2
2
2
(n + 1) i=0 (i + 1)2
i=0 (n − i + 1) (i + 1)
Inequality (7.2) is proved by induction on m, by making use of the identity
m+1
m
m
=
+
.
j
j
j−1
Next, we establish a multi-index version of Lemma 7.1. If β and σ are N dimensional multi-indices, we say that σ ≤ β, if and only if σi ≤ βi for all
i = 1, . . . , N .
Lemma 7.2 Let α and β be N -dimensional multi-indices. Then
P
σ≤β
and
16N
1
≤
.
(β − σ + ι)2 (σ + ι)2
(β + ι)2
α+β
α
≤
|α + β|
.
|α|
(7.3)
(7.4)
Proof (of (7.3)). For 1-dimensional multi-indices, (7.3) is precisely (7.1). Assuming that (7.3) holds true for multi-indices of dimension k ∈ N, will show
that it continues to hold for multi-indices of dimension k + 1. Let β and σ be
k-dimensional multi-indices and let
β̃ = (β1 , . . . , βk , βk+1 ),
σ̃ = (σ1 , . . . , σk , σk+1 )
denote multi-indices of dimension k + 1. Then by the induction hypothesis
and (7.1)
P
σ̃≤β̃
1
(β̃ − σ̃ + ι)2 (σ̃ + ι)2
=
P
σ≤β
×
≤
1
(β − σ + ι)2 (σ + ι)2
βP
k+1
1
(β
−
σ
+
1)2 (σk+1 + 1)2
k+1
k+1
σk+1 =0
16k
16
16k+1
=
.
2
2
(β + ι) (βk+1 + 1)
(β̃ + ι)2
8 Auxiliary Estimations at the Origin
35
Proof (of (7.4)). By (7.2), the inequality holds for 2-dimensional multi-indices.
Assuming that it holds for k-dimensional multi-indices, we show that it continues to hold for multi-indices of dimension k + 1. Let α and β be (k + 1)dimensional multi-indices. Then by the induction hypothesis
α+β
α
Qk
(α + β)!
j=1 (αj + βj ) (αk+1 + βk+1 )!
=
= Qk
α!β!
αk+1 !βk+1 !
j=1 αj !βj !
#
"
k
P
(αj + βj ) !
j=1
(αk+1 + βk+1 )!
# "
#
≤ "
αk+1 !βk+1 !
k
k
P
P
αj !
βj !
j=1
j=0

k+1

P
(α
(α
j + βj )
j + βj )
 αk+1 + βk+1
 j=1

j=1
 = |α + β| .


≤
≤
k



 k+1
P
P
|α|
αk+1
αj
αj

k
P
j=1
j=1
8 Auxiliary Estimations at the Origin
Lemma 8.1 Let u = (u1 , . . . , bℓ ) satisfy (6.3). Then for every 1 ≤ p, q ≤ ℓ
and every N -dimensional multi-index β
2
cCo
(|β| + s)!
|Dβ Dts (up uq )(0, 0)| ≤
C |β|+s
(8.1)
C
(β + ι)2 (s + 1)2
where c = 16N +1 . For every ℓ-dimensional multi-index γ
|Dβ Dts uγ (0, 0)| ≤
cCo
C
|γ|
C |β|+s
(|β| + s)!
.
(β + ι)2 (s + 1)2
(8.2)
Moreover, for all indices h = 1, . . . , N and k = 1, . . . , ℓ
|Dβ Dts (uγ Dxh uk )(0, 0)| ≤ Co
cCo
C
|γ|
C |β|+s
(|β| + s + 1)!
.
(β + ι)2 (s + 1)2
Proof (of (8.1)). By the generalized Leibniz rule
s
P
s P β
β s
D Dt u p u q =
(Dβ−σ Dts−j up )(Dσ Dtj uq )
j=0 j σ≤β σ
(8.3)
where σ is an N -dimensional multi-index of size |σ| ≤ |β|. Compute this at
the origin and estimate the right-hand side by using (6.3), which is assumed
to hold for all the t-derivatives of order up to s. By (7.4)
β
s
|β|
s
|β| + s
≤
≤
.
σ
j
|σ|
j
|σ| + j
36
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
Therefore
|Dβ Dts (up uq )(0, 0)| ≤Co2 C |β|+s−2 (|β| + s)!
s
P
P
1
1
×
.
2 (j + 1)2
2 (σ + ι)2
(s
−
j
+
1)
(β
−
σ
+
ι)
j=0
σ≤β
To prove (8.1) estimate the two sums on the right-hand side with the aid of
(7.1) of Lemma 7.1 and (7.3) of Lemma 7.2.
Proof (of (8.2)). The proof is by induction. If γ is a ℓ-dimensional multi-index
of either form
(. . . , 1, . . . , 1, . . . ),
(. . . , 2, . . . )
then (8.2) is precisely (8.1) for such a multi-index. Therefore (8.2) holds for
multi-indices of size |γ| = 2. Assuming it does hold for multi-indices of size
|γ|, we will show that it continues to hold for all ℓ-dimensional multi-indices
1≤p≤ℓ
γ̃ = (γ1 , . . . , γp + 1, . . . , γℓ ),
of size |γ| + 1. By the Leibniz rule
s
P
s P β
β s γ̃
D Dt u =
(Dβ−σ Dts−j uγ )(Dσ Dtj up ).
j=0 j σ≤β σ
First compute this at the origin. Then estimate Dβ−σ Dts−j uγ (0, 0) by the
induction hypothesis, and the terms Dσ Dtj up (0, 0) by (6.3).
The estimation is concluded by proceeding as in the proof of (8.1).
Proof (of (8.3)). By the generalized Leibniz rule
s
P
s P β
β s
γ
D Dt (u Dxh uk ) =
(Dβ−σ Dts−j uγ )(Dσ Dtj Dxh uk ).
j=0 j σ≤β σ
First compute this at the origin. Then majorize the terms involving uγ , by
means of (8.2), and the terms Dσ Dxh Dtj uk (0, 0) by (6.3). This gives
|γ|
cCo
C |β|+s
|D
≤ Co
C
s P
P
(|β − σ| + s − j)!(|σ| + j + 1)!
|β| + s
×
|σ|
+
j
(β
−
σ + ι)2 (σ̃ + ι)2 (s − j + 1)2 (j + 1)2
j=0 σ≤β
β
Dts (uγ Dxh uk )(0, 0)|
(8.4)
where σ̃ is the N -dimensional multi-index σ̃ = (σ1 , . . . , σh + 1, . . . , σN ). Estimate
(|σ| + j + 1)! ≤ (|σ| + j)!(|β| + s + 1)
These estimates in (8.4) yield
and
(σ̃ + ι)2 ≥ (σ + ι)2 .
9 Proof of the Cauchy–Kowalewski Theorem (Concluded)
|D
β
Dts (uγ Dxh uk )(0, 0)|
≤ Co
cCo
C
|γ|
37
C |β|+s (|β| + s)!(|β| + s + 1)
P
1
1
2
2
2
2
j=0 (s − j + 1) (j + 1) σ≤β (β − σ + ι) (σ + ι)
|γ|
cCo
≤ cCo
C |β|+s (|β| + s + 1)!.
C
×
s
P
9 Proof of the Cauchy–Kowalewski Theorem
(Concluded)
To prove (6.3) return to (6.6) and estimate
|Dβ Dts+1 u(0, 0)| ≤2(cCo + 1)Mo N ℓ2 C |β|+s
×
(|β| + s + 1)! P
(β + ι)2 (s + 2)2 |γ|≥1
cCo M
C
|γ|
where M and Mo are the constants appearing in (6.5). It remains to choose
C so large that
2(Co + 1)Mo N ℓ2
P
|γ|≥1
cCo M
C
|γ|
≤ Co .
(9.1)
Remark 9.1 The choice of C gives a lower estimate of the radius of convergence of the series (6.1) and (6.4).
9.1 Proof of Lemma 6.1
Given the inequality (6.3), the proof of the Cauchy–Kowalewski theorem is a
consequence of the following algebraic lemma
Lemma 9.1 Let α be a N -dimensional multi-index. Then |α|! ≤ N |α| α!.
Proof. Let xi for i = 1, . . . , N be given real numbers, and let k be a positive
integer. If α denotes an N -dimensional multi-index of size k, by the Leibniz
version of Newton’s formula (9.2 of the Complements)
N
P
i=1
xi
k
=
N
P k! Y
i
xα
i .
|α|=k α! i=1
From this, taking xi = 1 for all i = 1, . . . , N
N |α| =
P |α|!
.
|α|=k α!
(9.2)
38
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
10 Holmgren’s Uniqueness Theorem
As we have seen in Section 5, if Γ is noncharacteristic and analytic about one of
its points xo , the Cauchy data (4.2) satisfy the compatibility conditions (4.3)
and are analytic at xo , the coefficients Aα and the free term F are analytic
about the point (xo , u(xo ), Dm−1 u(xo )), then the Cauchy–Kowalewski theorem ensures that the Cauchy problem (4.1)–(4.2) has a solution u, analytic at
xo . Moreover, the solution is unique within the class of analytic solutions at
xo . However, the possibility that a non-analytic solution to (4.1)–(4.2) exists
is left open.
For linear equations, Holmgren’s theorem denies this possibility. We shall
use this result in Chapter 15, and therefore, we are going to present a fairly
general version of it.
Before stating the theorem, and giving its full proof, let us briefly discuss
the heuristic idea behind it. Let
X
x ∈ RN ,
Aα (x)Dα
(10.1)
L=
|α|≤m
where the coefficients Aα are N ×N matrices, having C m+2 entries (we will be
more precise about the regularity of these elements later on). Extending what
we have already seen in Section 4, we associate with (10.1) the characteristic
matrix
X
Aα (x)ξ α ,
L (x, ξ) =
|α|=m
αN
where ξ = (ξ1 , . . . , ξN ) and ξ α = ξ1α1 · · · · · ξN
, and the characteristic form
L(x, ξ) = det L (x, ξ),
which is a form of degree N m.
The extension of the notion of characteristic surface to the vector-valued
framework is quite straightforward. Indeed, a surface Γ , which is the zerolevel set of a function Φ = Φ(x), is called characteristic with respect to L at
xo ∈ Γ , if
L(xo , DΦ(xo )) = 0.
Moreover, we say that Γ is characteristic with respect to L, if it is characteristic at each single point. For a scalar operator, it is apparent that the previous
condition reduces to (4.5).
Now consider G ⊂ RN with boundary ∂G = Γ1 ∪ Γ2 , and let u, v be two
vectors such that
Dβ u = 0
β
D v=0
on Γ1 , |β| ≤ m − 1,
on Γ2 , |β| ≤ m − 1.
We are interested in the uniqueness of the solutions to the problem
(10.2)
(10.3)
10 Holmgren’s Uniqueness Theorem
(
in G,
on Γ1 , |β| ≤ m − 1.
Lu = 0
Dβ u = 0
39
(10.4)
In particular, we would like to conclude that u ≡ 0 in G. If we integrate the
first one of (10.4) by parts, we have
Z
Z
X
X
(10.5)
(−1)|α| Dα (vAα ) dx.
u
Aα Dα u dx =
v
G
G
|α|≤m
We let M v be given by
any w the problem
(
X
|α|≤m
(−1)|α| Dα (vAα ). Let us now assume that for
|α|≤m
Mv = w
Dβ v = 0
in G,
on Γ2 , |β| ≤ m − 1,
(10.6)
has a solution. In such a case, relying on (10.5) yields
Z
Z
Z
w u dx = (M v)u dx =
v(Lu) dx = 0,
G
G
G
whence we conclude that u ≡ 0 in G. Therefore, if (10.6) has a solution for
any w, we conclude that the solution of (10.4) is uniquely determined and
identically vanishes.
As a matter of fact, we do not need to have existence for any function
w; assuming that w ∈ C o (RN ), it suffices to show that we have existence for
each element wk of a sequence {wk }, which converges uniformly to w.
Indeed, if the problem
(
in G,
M vk = wk
(10.7)
Dβ vk = 0
on Γ2 , |β| ≤ m − 1,
has a solution vk , then
Z
u w dx = lim
G
k→∞
Z
u wk dx = 0,
G
and we conclude as we want.
The set of function {w}k dense in the set of continuous functions has
been left unspecified here. In the proof, the approximating functions will be
assumed to be analytic, so that it will be possible to apply Theorem 5.1.
Theorem 10.1 (Holmgren). The equation
X
Aα (x)Dα u = f (x),
Lu =
|α|≤m
x ∈ RN ,
40
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
with analytic coefficients Aα and data
D β u = gβ
on
Γ, |β| ≤ m − 1,
on an analytic, noncharacteristic surface Γ has at most one solution in a
neighborhood of Γ .
Remark 10.1 As we have already discussed above, owing to the linearity of
the equation, it suffices to show that for f = g = 0, the only solution is u = 0.
Remark 10.2 The theorem was originally proved by Holmgren (see [121]) in
1901; here, we follow a modern presentation of the original argument as given,
for example, in Smoller [239].
Remark 10.3 Notice that we are not assuming f or g to be analytic.
Remark 10.4 It is not difficult to see that Γ is characteristic for the operator
L if and only if Γ is characteristic for the operator M .
Remark 10.5 As already hinted at before, the analyticity of the coefficients
Aα suggest taking w analytic as well, so that we can apply Theorem 5.1.
However, this raises a problem: the Cauchy–Kowalewski theorem ensures that
for w analytic, the problem
(
in G,
Mv = w
β
D v =0
on Γ, |β| ≤ m − 1,
has a local solution on Γ if such a surface is analytic, but the region where v is
defined, depends on w. Therefore, we need to determine a region of existence,
which is independent of w.
11 Proof of the Holmgren Uniqueness Theorem
Let
Γ = {x ∈ RN : Φ(x) = 0}
be a noncharacteristic surface with respect to L, and denote by Γo a compact
subset. We now deform Γo by an analytic homotopy, keeping its boundary
∂Γo fixed.
In order to do that, we consider a family of surfaces
Γλ = {x ∈ Rn : Φ(x, λ) = 0}
with λ ∈ [0, λ̄], all having the same boundary and such that all the functions
Φ are analytic with respect to both x and λ.
We choose λ1 , λ2 such that 0 ≤ λ1 < λ2 ≤ λ̄ and we let Eλ1 λ2 be the
region determined by Γλ1 and Γλ2 as in Figure 11.1.
11 Proof of the Holmgren Uniqueness Theorem
41
Γλ 2
Γλ 1
E λ1 λ2
Γ
Γo
Fig. 11.1
If we let
∂Γo = {x ∈ RN : Φ(x) = 0} ∩ {x ∈ RN : α(x) = 0},
where α is a proper analytic function, then the family of surfaces Γλ can be
built by
def
Φ(x, λ) = (1 − λ)Φ(x) + λα(x).
It is apparent that for λ = 0, we have precisely Γo ; the issue at hand here is
the choice of λ̄. Provided λ is taken sufficiently small, we have that
L (x, DΦ(x, λ)) 6= 0
for x ∈ Γλ ,
L (x, DΦ(x, λ)) 6= 0
for x ∈ Γo .
if
Hence, λ̄ is chosen in such a way that for any λ ∈ [0, λ̄] all the surfaces Γλ are
def
noncharacteristic. We also let E = E0λ̄ . What we want to show is that the
problem
Lu = f
in E
is uniquely determined by its data on Γo .
Let us first state the following auxiliary result.
Lemma 11.1 There exists ǫ > 0 such that for λ, µ satisfying
0 ≤ λ < µ < λ̄,
the problem
(
Mv
Dβ v
=w
=0
µ < λ + ǫ,
in Eλµ ,
on Γλ , |β| ≤ m − 1,
has solutions v for w in a dense set of C o (Eλµ ).
(11.1)
42
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
We postpone the proof of the lemma, and we first show how it allows us to
conclude.
Suppose that u is a solution of the problem
(
=0
in E,
Lu
β
D u =0
on Γo , |β| ≤ m − 1.
Since λ̄ > 0 is given, once we fix ǫ > 0, we can find a finite set
S = {0 < λ1 < λ2 < · · · < λj = λ̄}
such that λi+1 − λi < ǫ for i = 1, . . . , j − 1. If ǫ is the value given by
Lemma 11.1, we know that for w in a dense set of C o (E0λ1 ) the problem
(
in E0λ1 ,
Mv = w
Dβ v = 0
on Γλ1 , |β| ≤ m − 1
has solutions. Since ∂E0λ1 = Γo ∪ Γλ1 , u vanishes with its derivatives on
Γo , and v vanishes with its derivatives on Γλ1 , working as in the heuristic
argument, we conclude that u = 0 in E0λ1 . We can now repeat the previous
argument finitely many times, to conclude that u ≡ 0 in E.
.
The proof has an intrinsic local feature, since u ≡ 0 in a small region
around Γo . It might be interesting to study conditions, that ensure that u
vanishes in a prescribed region, but this goes beyond the scope of these notes.
On the other hand, we have shown that the solution of the initial value
problem vanishes in a region, which is obtained by holomorphic deformation of
Γo ; therefore, the region of uniqueness is determined by the noncharacteristic
surface.
11.1 Proof of Lemma 11.1
We consider an entire function w = w(x, ξ), where ξ = (ξ1 , . . . , ξN ) is a
parameter.
As above, Γλ is a family of noncharacteristic surfaces, with λ ∈ [0, λ̄]. For
any parameter ξ with |ξ| ≤ 1 and any µ ∈ [0, λ̄] we want to solve the problem
(
M v = w(x, λ)
in R2N +1 ,
(11.2)
Dβ v = 0
on Γµ , |β| ≤ m − 1.
We can consider M v = w as a system for the unknown v, where v is a
function of 2N + 1 variables, namely x1 , . . . , xN , ξ1 , . . . , ξN , µ and there are
no derivatives with respect to the N + 1 variables ξ1 , . . . , ξN , µ.
Since
S = {(x, ξ, µ) ∈ R2N +1 : x ∈ Γµ , 0 ≤ µ ≤ λ̄, |ξ| ≤ 1}
1c Quasi-Linear Second-Order Equations in Two Variables
43
is a compact, noncharacteristic, analytic surface in the variables (x, ξ, µ), by
the Cauchy–Kowalewski theorem there exists an analytic solution to (11.2) in
a neighborhood of S. In other words, there exists δ > 0 such that a solution
v = v(x, ξ, µ) with |ξ| ≤ 1 and µ ∈ [0, λ̄] is defined with x in a neighborhood
of width δ of the surface Γµ .
Provided we choose µ sufficiently close to λ such that the neighborhoods
of Γλ and Γµ intersect each other, we conclude that v(x, ξ, µ) is defined in
Eλµ for all |ξ| ≤ 1. The closeness of µ and λ is precisely quantified by ǫ, that
is, 0 < µ − λ < ǫ.
Altogether, we have found a solution v of
(
Mv = w
in Eλµ ,
(11.3)
Dβ v = 0
on Γµ , |β| ≤ m − 1,
with µ − λ < ǫ and |λ| ≤ 1. In particular, ξ is still completely arbitrary. If
we can choose |ξ| ≤ 1 in such a way that the corresponding data w = w(x, ξ)
give linear combinations that are dense in C o , then the problem (11.3) has
a solution for a set of w that is dense in C o (Eλµ ), and this would finish the
proof.
In order to achieve this, we pick
w(x, ξ) = ex·ξ .
N
This allows us to uniformly approximate every monomial xα = x1α1 · · · · · xα
N
in every bounded set by linear combinations of such functions w(x, ξ) with
|ξ| ≤ 1. Indeed,
α
N xk ξk
Y
−1 k
e
xα = lim
.
ξ→0
ξk
k=1
Moreover, since the linear combinations are dense in the space of continuous
functions, the same occurs for linear combinations of our w(x, ξ) with |ξ| ≤ 1.
Problems and Complements
1c Quasi-Linear Second-Order Equations in Two
Variables
1.1. Assume that the functions A, B, C, D in (1.1) are of class C ∞ . Assume
also that Γ is of class C ∞ . Prove that all the derivatives
44
1 QUASI-LINEAR EQUATIONS AND ANALYTIC DATA
∂k u
,
∂xh ∂y ℓ
k, h, ℓ ∈ N,
h+ℓ=k
can be computed on Γ provided (1.4) holds.
1.2. Assume that in (1.1), A, B, C are constants and D = 0. Introduce an
affine transformation of the coordinate variables that transforms (1.1) into
either the Laplace equation, the heat equation, or the wave equation.
1.3. Prove the last statement of Section 2.1. Discuss the case of constant
coefficients.
1.4 Consider the equation
(y − 1)2 uxx − y 2 uyy = 0.
a) Classify the equation and determine its characteristics.
b) Show that if data are prescribed on the line y = 2, the Cauchy problem
(1.2) can be solved.
1.5 Consider the equation
p
x uxx + 2 x(1 + y) uxy + y uyy = 0
in the open set E = {(x, y) ∈ R2 : x(1 + y) > 0}. Classify the equation
and determine its characteristics in the hyperbolic region.
1.6 Consider the equation
x3 y uxx − 4xy 3 uyy = 0.
a) Classify the equation and determine its characteristics.
b) Show that if data are prescribed on the half-line y = 41 , x > 0, the
Cauchy problem (1.2) can be solved.
1.7 Consider the equation
x2 uxx + 4xy uxy + (4y 2 − x2 + 2x − 1) uyy = 0.
a) Classify the equation and determine its characteristics.
b) Show that if data are prescribed on the line x = 14 , the Cauchy problem
(1.2) can be solved.
5c Analytic Data and the Cauchy–Kowalewski Theorem
5.1. Denote points in RN +1 by (x, t) where x ∈ RN and t ∈ R. Let ϕ and ψ
be analytic in RN . Find an analytic solution, about t = 0 of
∆u = 0,
u(x, 0) = ϕ(x),
ut (x, 0) = ψ(x).
5.2. Let f1 and f2 be analytic and periodic of period 2π in R. Solve the
problem
1 − ε < |x| < 1 + ε
∂u
u |x|=1 = f1 (θ),
= f2 (θ)
∂|x| |x|=1
∆u = 0
in
for some ε ∈ (0, 1). Compare with the Poisson integral (3.11) of Chapter 2.
9c Proof of the Cauchy–Kowalewski Theorem Concluded
45
6c Proof of the Cauchy–Kowalewski Theorem
6.1. Prove that (6.2) ensures the convergence of the series (6.1) and give an
estimate of the radius of convergence.
6.2. Let β be a N −dimensional multi-index of size |β|. Prove that the number
of derivatives Dβ of order |β| does not exceed |β|N .
6.3. Let u : RN → Rℓ be analytic at some point xo ∈ RN . Prove that there
exist constants Co and C such that for all N −dimensional multi-indices
β
|β|!
|Dβ u(xo )| ≤ Co C |β|−1
.
(β + ι)2
6.4. Prove (6.4)–(6.4).
8c The Generalized Leibniz Rule
8.1. Let u, v ∈ C ∞ (R) be real-valued. The Leibniz rule states that for every
n∈N
n
P
n
Dn (uv) =
Dn−i uDi v.
i=0 i
In particular if u, v ∈ C ∞ (RN ), then
n
P
n
n
uDxi r v.
Dxr (uv) =
Dxn−i
r
i=0 i
Prove, by induction, the generalized Leibniz rule
P β
Dβ (uv) =
Dβ−σ uDσ v.
σ
σ≤β
9c Proof of the Cauchy–Kowalewski Theorem Concluded
9.1. Prove that C can be chosen such that (9.1) holds.
9.2. Prove (9.2) by induction, starting from the binomial formula
k
P
k
k
xj1 x2k−j .
(x1 + x2 ) =
k
−
j
j=0
2
THE LAPLACE EQUATION
1 Preliminaries
Let E be a domain in RN for some N ≥ 2, with boundary ∂E of class C 1 .
Points in E are denoted by x = (x1 , . . . , xN ). A function u ∈ C 2 (E) is harmonic in E if
N ∂2
P
∆u = div ∇u =
in E.
(1.1)
2u = 0
i=1 ∂xi
The formal operator ∆ is called the Laplacean.1 The interest in this equation
stems from its connection to physical phenomena such as
1. Steady-state heat conduction in a homogeneous body with constant heat
capacity and constant conductivity.
2. Steady-state potential flow of an incompressible fluid in a porous medium
with constant permeability.
3. Gravitational potential in RN generated by a uniform distribution of
masses.
The interest is also of pure mathematical nature in view of the rich structure
exhibited by (1.1). The formal operator in (1.1) is invariant under rotations or
translations of the coordinate axes. Precisely, if A is a (unitary, orthonormal)
rotation matrix and y = A(x − ξ) for some fixed ξ ∈ RN , then formally
∆x =
N ∂2
N ∂2
P
P
=
2
2 = ∆y .
i=1 ∂xi
i=1 ∂yi
This property is also called spherical symmetry of the Laplacean in RN .
1
Pierre Simon, Marquis de Laplace, 1749–1827. Author of Traité de Mécanique
Céleste (1799–1825). Also known for the frequent use of the phrase il est aisé de voir
which has unfortunately become all too popular in modern mathematical writings.
The same equation had been introduced, in the context of potential fluids, by Joseph
Louis, Compte de Lagrange, 1736–1813, author of Traité de Mécanique Analytique
(1788).
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_3
47
48
2 THE LAPLACE EQUATION
1.1 The Dirichlet and Neumann Problems
Given ϕ ∈ C(∂E), the Dirichlet problem for the operator ∆ in E consists in
finding a function u ∈ C 2 (E) ∩ C(Ē) satisfying
∆u = 0 in E,
and
u
∂E
= ϕ.
(1.2)
Given ψ ∈ C(∂E), the Neumann problem consists in finding a function u ∈
C 2 (E) ∩ C 1 (Ē) satisfying
∂
u = ∇u · n = ψ on ∂E
(1.3)
∂n
where n denotes the outward unit normal to ∂E. The Neumann datum ψ is
also called variational.
We will prove that if E is bounded, the Dirichlet problem is always
uniquely solvable. The Neumann problem, on the other hand, is not always
solvable. Indeed, integrating the first of (1.3) in E, we arrive at the necessary
condition
Z
ψ dσ = 0
(1.4)
∆u = 0 in E,
and
∂E
where dσ denotes the surface measure on ∂E. Thus ψ cannot be assigned
arbitrarily.
Lemma 1.1 Let E be a bounded open set with boundary ∂E of class C 1 and
assume that (1.2) and (1.3) can both be solved within the class C 2 (Ē). Then
the solution of (1.2) is uniquely determined by ϕ, and the solution of (1.3) is
uniquely determined by ψ up to a constant.
Proof. We prove only the statement regarding the Dirichlet problem. If ui for
i = 1, 2, are two solutions of (1.2), the difference w = u1 − u2 is a solution of
the Dirichlet problem with homogeneous data
∆w = 0
in E,
w
∂E
= 0.
Multiplying the first of these by w and integrating over E gives
Z
|∇w|2 dx = 0.
E
Remark 1.1 Arguments of this kind are referred to as energy methods. The
assumption w ∈ C 2 (Ē) is used to justify the various calculations in the integration by parts. The lemma continues to hold for solutions in the class
C 2 (E) ∩ C 1 (Ē). Indeed, one might first carry the integration over an open,
proper subset E ′ ⊂ E, with boundary ∂E ′ of class C 1 , and then let E ′ expand to E. We will show later that uniqueness for the Dirichlet problem holds
within the class C 2 (E) ∩ C(Ē), required by the formulation (1.2).
Remark 1.2 A consequence of the lemma is that the problem
∆u = 0 in E
in general is not solvable.
and u
∂E
= ϕ, ∇ · n = ψ
1 Preliminaries
49
1.2 The Cauchy Problem
Let Γ be an (N − 1)-dimensional surface of class C 1 contained in E and
prescribe N +1 functions ψi ∈ C 2 (Γ ), for i = 0, 1, . . . , N . The Cauchy problem
consists in finding u ∈ C 2 (E) satisfying
∆u = 0 in E
and
u = ψo , uxi = ψi , i = 1, . . . , N on Γ.
(1.5)
The Cauchy problem is not always solvable. First the data ψi , must be compatible, i.e., derivatives of u along Γ computed using ψo and computed using
ψi must coincide. Even so, in general, the solution, if any, can only be found
near Γ . The Cauchy–Kowalewski theorem gives some sufficient conditions to
ensure local solvabilty of (1.5).
1.3 Well-Posedness and a Counterexample of Hadamard
A boundary value problem for the Laplacean, say the Dirichlet, Neumann or
Cauchy problem, is well posed in the sense of Hadamard if one can identify
a class of boundary data, say C, such that each datum in C yields a unique
solution, and small variations of the data within C yield small variations on
the corresponding solutions. The meaning of small variation is made precise
in terms of the topology suggested by the problem. This is referred to as the
problem of stability. A problem that does not meet any one of these criteria
is called ill posed.
Consider the problem of finding a harmonic function in E taking either
Dirichlet data or Neumann data on a portion Σ1 of ∂E and both Dirichlet
and variational data on the remaining part Σ2 = ∂E − Σ1 . Such a problem is
ill posed. Even if a solution exists, in general it is not stable in any reasonable
topology, as shown by the following example due to Hadamard ([111]).
The boundary value problem
uxx + uyy = 0
u(± π2 , y) = 0
u(x, 0) = 0 √
uy (x, 0) = e− n cos nx
in (− π2 < x < π2 ) × (y > 0)
for y > 0
for − π2 < x < π2
for − π2 < x < π2
admits the family of solutions
un (x, y) =
1 −√n
cos nx sinh ny,
e
n
where n is an odd integer.
One verifies that
kun,y (·, 0)k∞,(− π2 , π2 ) → 0 as n → ∞
and that for all y > 0
kun (·, y)k∞,(− π2 , π2 ) ,
kun (·, y)k2,(− π2 , π2 ) → ∞ as n → ∞.
50
2 THE LAPLACE EQUATION
1.4 Radial Solutions
The invariance of ∆ under orthonormal linear transformations suggests that
we look for solutions of (1.1) in RN depending only on ρ = |x − y|, for any
fixed y ∈ RN . Any such solution ρ → V (ρ; y) must satisfy
V ′′ +
N −1 ′
V = 0,
ρ
y ∈ RN fixed
where the derivatives are meant with respect to ρ. By integration this gives,
up to additive and multiplicative constants

1

if N ≥ 3

|x − y|N −2
N
(1.6)
R − {y} ∋ x →


ln |x − y|
if N = 2.
These are the potentials of the Laplacean in RN with a pole at y. Consider a
finite distribution {(ei , yi )} for i = 1, . . . , n, of electrical charges ei , concentrated at the points yi . The function
n
P
ei
RN − {y1 , . . . , yn } ∋ x →
,
x 6= yi , i = 1, 2, . . . , n
N −2
i=1 |x − yi |
is harmonic, and it represents the potential generated by the charges (ei , yi )
outside them.
Let E be a bounded, Lebesgue measurable set in RN , and let µ ∈ C(Ē).
The function
Z
µ(y)
dy,
N ≥3
RN − Ē ∋ x →
|x
−
y|N −2
E
is harmonic in RN − Ē, and it represents the Newtonian potential generated
outside Ē, by the distribution of masses (or charges) µ(y)dy in E. Let Σ be an
(N − 1)-dimensional bounded surface of class C 1 in RN , for some N ≥ 3, and
let n(y) denote the unit normal at y ∈ Σ. The orientation of n(y) is arbitrary
but fixed, so that y → n(y) is continuous on Σ. Given ϕ, ψ ∈ C(Σ) the two
functions
Z
ϕ(y)


dσ


 Σ |x − y|N −2
RN − Σ̄ ∋ x → Z


ψ(y)


(x − y) · n(y)dσ

N
Σ |x − y|
are harmonic in RN − Σ̄. The first is called single-layer potential, and it gives
the potential generated, outside Σ̄, by a distribution of charges (or masses) on
Σ, of density ϕ(·). The second is called double-layer potential and it represents
the electrical potential generated, outside Σ, by a distribution of dipoles on
Σ, with density ψ(·).
Analogous harmonic functions can be constructed for N = 2, by using the
second of (1.6). These would be called logarithmic potentials.
2 The Green and Stokes Identities
51
2 The Green and Stokes Identities
Let E be a bounded open set in RN with boundary ∂E of class C 1 , and let
u, v ∈ C 2 (Ē). By the divergence theorem we obtain the Green’s identities
Z
Z
Z
∂u
v
v∆u dx = −
∇v · ∇u dx +
dσ
(2.1)
∂n
E
∂E
ZE
Z ∂u
∂v
dσ.
(2.2)
v
−u
(v∆u − u∆v) dx =
∂n
∂n
E
∂E
Remark 2.1 By approximation, (2.1)–(2.2) continue to hold for functions
u, v ∈ C 2 (E) ∩ C 1 (Ē) such that ∆u and ∆v are essentially bounded in E.
Remark 2.2 If u is harmonic in E, then
Z
Z
Z
∂u
∂u
dσ = 0 and
|∇u|2 dx =
u dσ.
∂E ∂n
E
∂E ∂n
2.1 The Stokes Identities
Let u ∈ C 2 (Ē) and let ωN denote the area of the unit sphere in RN for N ≥ 3.
Then for all x ∈ E
Z ∂|x − y|2−N
1
2−N ∂u
|x − y|
dσ
− u(y)
u(x) =
ωN (N − 2) ∂E
∂n
∂n
(2.3)
Z
1
−
|x − y|2−N ∆u dy.
ωN (N − 2) E
If N = 2
1
u(x) =
2π
Z
∂E
1
+
2π
∂u
∂ ln |x − y|
dσ
− ln |x − y|
u
∂n
∂n
E
ln |x − y|∆u dy.
Z
(2.4)
Remark 2.3 These are implicit representation formulas of smooth functions
in Ē.
Proof. We prove only (2.3). Fix x ∈ E and let Bε (x) be the ball of radius ε
centered at x. Assume that ε is so small that Bε (x) ⊂ E, and apply (2.2) in
E − Bε (x) for y → v(y) equal to the potential with pole at x introduced in
(1.6). Since V (·; x) is harmonic in E − Bε (x), (2.2) yields
Z Z
N −2
∂|x − y|2−N
2−N ∂u
|x − y|
dσ
−u
u(y) dσ =
εN −1 |x−y|=ε
∂n
∂n
∂E
(2.5)
Z
Z
x−y
2−n
2−N
+ε
∇u ·
dσ −
|x − y|
∆u dy.
|x − y|
|x−y|=ε
E−Bε (x)
52
2 THE LAPLACE EQUATION
As ε → 0
Z
E−Bε (x)
|x − y|
and
ε2−N
Z
2−N
∆u dy −→
|x−y|=ε
∇u ·
Z
E
|x − y|2−N ∆u dx
x−y
dσ −→ 0.
|x − y|
As for the left-hand side of (2.5)
Z
Z
1
1
u(y)dσ = ωN u(x) + N −1
[u(y) − u(x)]dσ.
εN −1 |x−y|=ε
ε
|x−y|=ε
The last integral tends to zero as ε → 0, since
Z
Z
1
k∇uk∞,E
|u(y)
−
u(x)|dσ
≤
|x − y|dσ
εN −1 |x−y|=ε
εN −1
|x−y|=ε
≤ εωN k∇uk∞,E .
These remarks in (2.5) prove (2.3) after we let ε → 0.
Motivated by the Stokes identities, set

1
1



 ωN (N − 2) |x − y|N −2
F (x; y) =



 −1 ln |x − y|
2π
if N ≥ 3
(2.6)
if N = 2.
The function F (·; y) is called the fundamental solution of the Laplacean with
pole at y.
Corollary 2.1 Let E be a bounded open set in RN with boundary ∂E of class
C 1 and let u ∈ C 2 (Ē) be harmonic in E. Then for all x ∈ E
Z ∂u
∂F (x; ·)
F (x; ·)
u(x) =
dσ.
(2.7)
−u
∂n
∂n
∂E
A consequence of this corollary, and the structure of the fundamental solution
F (·, y) is the following
Proposition 2.1 Let E be an open set in RN and let u ∈ C 2 (E) be harmonic
in E. Then u ∈ C ∞ (E), and for every multi-index α, the function Dα u is
harmonic in E.
Proof. If E is bounded, ∂E is of class C 1 , and u ∈ C 2 (Ē), the statement
follows from the representation (2.7). Otherwise, apply (2.7) to any bounded
open subset E ′ ⊂ E with boundary of class C 1 .
R
Corollary 2.2 u ∈ Co2 (E) =⇒ u(x) = − E F (x; y)∆u dy for all x ∈ E.
3 Green’s Function and the Dirichlet Problem for a Ball
53
3 Green’s Function and the Dirichlet Problem for a Ball
Given a bounded open set E ⊂ RN with boundary ∂E of class C 1 , consider
the problem of finding, for each fixed x ∈ E, a function y → Φ(x; y) ∈ C 2 (Ē)
satisfying
∆y Φ(x; ·) = 0 in E,
and
Φ(x; ·)
∂E
= F (x; ·)
(3.1)
where F (x; y) is the fundamental solution of the Laplacean introduced in
(2.6). Assume for the moment that (3.1) has a solution. Assume also that the
Dirichlet problem (1.2) has a solution u ∈ C 2 (Ē). Then the second Green’s
identity (2.2) written for y → u(y) and y → Φ(x; y) gives
Z ∂u
∂Φ(x; ·)
Φ(x; ·)
0=
dσ for all fixed x ∈ E.
−u
∂n
∂n
∂E
Subtract this from the implicit representation (2.7), to obtain
Z
∂G(x; ·)
dσ
u(x) = −
ϕ
∂n
∂E
(3.2)
where
(x, y) → G(x; y) = F (x; y) − Φ(x; y).
(3.3)
The function G(·; ·) is the Green’s function for the Laplacean in E. Its relevance is in that every solution u ∈ C 2 (Ē) of the Dirichlet problem (1.2) admits
the explicit representation (3.2), through the Dirichlet data ϕ and G(·; ·). Its
relevance is also in that it permits a pointwise representation of a smooth
function u defined in E and vanishing near ∂E.
R
Corollary 3.1 u ∈ Co2 (E) =⇒ u(x) = − E G(x; y)∆u dy for all x ∈ E.
Lemma 3.1 The Green’s function is symmetric, i.e., G(x; y) = G(y; x).
Proof. Fix x1 , x2 ∈ E and let ε > 0 be small enough that
Bε (xi ) ⊂ E for i = 1, 2, and Bε (x1 ) ∩ Bε (x2 ) = ∅.
Apply Green’s identity (2.2) to the pair of functions G(xi ; ·) for i = 1, 2 in the
domain E − [Bε (x1 ) ∪ Bε (x2 )]. Since G(xi ; ·) are harmonic in such a domain,
and vanish on ∂E
Z
∂
∂
G(x2 ; y) − G(x2 ; y)
G(x1 ; y) dσ
−
G(x1 ; y)
∂n(y)
∂n(y)
∂Bε (x1 )
Z
∂
∂
G(x1 ; y)
=
G(x2 ; y) − G(x2 ; y)
G(x1 ; y) dσ.
∂n(y)
∂n(y)
∂Bε (x2 )
Let ε → 0 and observe that
54
2 THE LAPLACE EQUATION
Z
∂
G(x2 ; y)dσ = 0
∂n(y)
∂B (x )
Z ε 1
∂
lim
G(x2 ; y)
G(x1 ; y)dσ = 0.
ε→0 ∂B (x )
∂n(y)
ε
2
lim
ε→0
G(x1 ; y)
Therefore
Z
lim
ε→0
∂G(x1 ; y)
G(x2 ; y)
dσ = lim
ε→0
∂n(y)
∂Bε (x1 )
Z
G(x1 ; y)
∂Bε (x2 )
∂G(x2 ; y)
dσ.
∂n(y)
From the definition of G(·; ·)
∂G(xi ; y)
∂F (xi ; y) ∂Φ(xi ; y)
=
−
,
∂n(y)
∂n(y)
∂n(y)
and observe that
Z
∂
Φ(x2 ; y)dσ = 0
ε→0 ∂B (x )
∂n(y)
ε
1
Z
∂
lim
G(x2 ; y)
Φ(x1 ; y)dσ = 0
ε→0 ∂B (x )
∂n(y)
ε
2
lim
G(x1 ; y)
since y → Φ(xi ; y) are regular. This implies that
Z
Z
∂F (x1 ; y)
∂F (x2 ; y)
lim
G(x2 ; y)
dσ = lim
G(x1 ; y)
dσ.
ε→0 ∂B (x )
ε→0 ∂B (x )
∂n(y)
∂n(y)
ε
1
ε
2
Computing the limits as in the the proof of the Stokes identity gives
Z
∂F (xj ; y)
lim
G(xi ; y)
dσ = G(xi ; xj ) for xi =
6 xj .
ε→0 ∂B (x )
∂n(y)
ε
j
Thus G(x1 ; x2 ) = G(x2 ; x1 ).
Corollary 3.2 The functions G(·; y) and ∂G(·; y)/∂n(y), for fixed y ∈ ∂E,
are harmonic in E.
To solve the Dirichlet problem (1.2) we may find G(·; ·), and write down (3.2).
This would be a candidate for a solution. By Corollary 3.2 it is harmonic. It
would remain to show that
lim u(x) = ϕ(x∗ )
x→x∗
for all x∗ ∈ ∂E.
(3.4)
We will show that this is indeed the case if (3.1) has a solution, i.e., if the
Green’s function for ∆ in E can be determined. Thus solving the Dirichlet
problem (1.2) reduces to solving the family of Dirichlet problems (3.1). The
advantage in dealing with the latter is that the boundary datum F (x; ·) is
specific and given by (2.6). Nevertheless, (3.1) can be solved explicitly only
for domains E exhibiting a simple geometry.
3 Green’s Function and the Dirichlet Problem for a Ball
55
3.1 Green’s Function for a Ball
Let BR be the ball of radius R centered at the origin of RN . The map2
ξ=
R2
x,
|x|2
x 6= 0
(3.5)
transforms BR −{0} into RN −BR , and ∂BR into itself. Referring to Figure 3.1,
the two triangles ∆(y, 0, ξ) and ∆(x, 0, y), are similar whenever y ∈ ∂BR .
Indeed, they have in common the angle θ, and in view of (3.5), the ratios
|x|/|y| and |y|/|ξ| are equal, provided |y| = R. Therefore
Fig. 3.1
R
|x|
= |ξ − y| ,
R
|ξ|
x ∈ BR and y ∈ ∂BR .
(3.6)
Then for each fixed x ∈ BR − {0}, the solution of (3.1) is given by

N −2

1
R
1


if N ≥ 3

 ωN (N − 2) |x|
|ξ − y|N −2
Φ(x; y) =




 −1 ln |ξ − y| |x|
if N = 2.
2π
R
(3.7)
|x − y| = |ξ − y|
While constructed for x 6= 0, the function Φ(x; ·) is well defined also for x = 0,
modulo taking the limit as |x| → 0. For all x ∈ BR , the function Φ(x; ·) is
harmonic in BR since its pole ξ lies outside BR . Moreover, by virtue of (3.6),
the boundary conditions in (3.1) are satisfied. Thus the Green’s function for
the ball BR is
N −2
1
R
1
1
−
(3.8)N ≥3
G(x; y) =
ωN (N − 2) |x − y|N −2
|x|
|ξ − y|N −2
2
called also the Kelvin transform ([257]).
56
2 THE LAPLACE EQUATION
for N ≥ 3, and
1
|x|
G(x; y) =
ln |ξ − y|
− ln |x − y|
2π
R
for N = 2.
(3.8)N =2
The derivative of G(x; ·), normal to the sphere |y| = R, is computed from
∂
G(x; y)
∂|y|
y∈∂BR
.
From Figure 3.1, by elementary trigonometry
|x − y|2 = |y|2 + |x|2 − 2|y||x| cos θ
|ξ − y|2 = |y|2 +
R4
R2
− 2|y|
cos θ.
2
|x|
|x|
Therefore for θ fixed and y ∈ ∂BR
∂|x − y|
|y| − |x| cos θ
=
,
∂|y|
|x − y|
∂|ξ − y|
=
∂|y|
R2
cos θ
|x|
.
|ξ − y|
|y| −
First let N ≥ 3. Computing from (3.8)N =2 and (3.8)N ≥3 , and using (3.6),
gives
∂
1 R2 − |x|2
−
G(x; y) y∈∂BR =
.
∂|y|
RωN |x − y|N
Such a formula also holds for N = 2 with ω2 = 2π. Put this in (3.3) to derive
the following Poisson representation.
Lemma 3.2 Let u ∈ C 2 (Ē) be a solution of the Dirichlet problem (1.2) in
the ball BR . Then
Z
1
R2 − |x|2
u(x) =
ϕ(y)
dσ,
N ≥ 2.
(3.9)
ωN R ∂BR
|x − y|N
Setting u ≡ 1 in (3.9) gives
Z
1
R2 − |x|2
dσ = 1
ωN R ∂BR |x − y|N
for all x ∈ BR .
(3.10)
Even though the representation (3.9) has been derived for solutions of (1.2)
of class C 2 (B̄R ), it actually gives the unique solution of the Dirichlet problem
for the sphere, as shown by the following existence theorem.
Theorem 3.1. The Dirichlet problem (1.2) for E = BR has a unique solution
given by (3.9).
4 Sub-Harmonic Functions and the Mean Value Property
57
Proof (existence). By Corollary 3.2, the function u given by (3.9) is harmonic
in BR . To prove (3.4), fix x∗ ∈ ∂BR , choose an arbitrarily small positive
number ε, and let δ ∈ (0, 1) be so small that
|ϕ(y) − ϕ(x∗ )| < ε for all y ∈ Σδ = y ∈ ∂BR |y − x∗ | < δ . (3.11)
By (3.10)
1
ϕ(x∗ ) =
RωN
Z
∂BR
ϕ(x∗ )
Therefore
u(x) − ϕ(x∗ ) =
1
RωN
1
=
RωN
+
=
Z
∂BR
Z
1
RωN
Σδ
Z
R2 − |x|2
dσ.
|x − y|N
[ϕ(y) − ϕ(x∗ )]
[ϕ(y) − ϕ(x∗ )]
R2 − |x|2
dσ
|x − y|N
R2 − |x|2
dσ
|x − y|N
[ϕ(y) − ϕ(x∗ )]
∂BR −Σδ
(1)
(2)
Iδ (x, x∗ ) + Iδ (x, x∗ ).
R2 − |x|2
dσ
|x − y|N
(2)
For δ fixed, Iδ (x, x∗ ) → 0 as x → x∗ . Moreover, in view of (3.10) and (3.11),
(1)
|Iδ (x, x∗ )| < ε. Therefore limx→x∗ |u(x) − ϕ(x∗ )| ≤ ε for all ε > 0.
Remark 3.1 By Lemma 1.1 and the Remark 1.1, such a solution is unique
in the class C 2 (BR ) ∩ C 1 (B̄R ). It will be shown in the next section that
uniqueness holds for solutions u ∈ C 2 (BR ) ∩ C(B̄R ).
4 Sub-Harmonic Functions and the Mean Value
Property
Let E be a bounded open set in RN with boundary ∂E of class C 1 . Let
u ∈ C 2 (Ē), and assume that the solution Φ(x; ·) of (3.1) exists for all x ∈ E.
Subtracting the second Green’s identity (2.2) for u and Φ(x; ·) from the Stokes
identity (2.3) gives
Z
Z
∂G(x; y)
dσ −
G(x; y)∆u(y)dy for all x ∈ E.
u(x) = −
u
∂n
E
∂E
In particular, if E is a ball BR (xo ) of radius R centered at xo , by setting
x = xo , we obtain
Z
Z
udσ −
G(0; y − xo )∆u(y)dy
(4.1)
u(xo ) = −
∂BR (xo )
BR (xo )
where for a measurable set D ⊂ RN of finite measure and f ∈ L1 (D)
58
2 THE LAPLACE EQUATION
Z
Z
1
− f dy =
f dy,
|D| D
D
|D| = meas (D).
Let E ⊂ RN be open. A function u ∈ C(E) is sub-harmonic in E if
Z
u(xo ) ≤ −
u dσ
for all BR (xo ) ⊂ E.
(4.2)
∂BR (xo )
This implies that if u ∈ C(E) is sub-harmonic in E, then (4.1 of the Complements)
Z
u(xo ) ≤ −
for all BR (xo ) ⊂ E.
u dy
BR (xo )
(4.3)
A function u ∈ C(E) is super-harmonic if −u is sub-harmonic in E.
The Green’s function G(·; ·) for a ball, as defined in (3.8)N =2 and (3.8)N ≥3 ,
is non-negative. A consequence is that if u ∈ C 2 (E) is such that ∆u ≥ (≤)0
in E, then, by (4.1), u is sub(super)-harmonic in E. Conversely if u ∈ C 2 (E)
is sub(super)-harmonic, then ∆u ≥ (≤)0 in E. Indeed, (4.1) implies
Z
G(0; y − xo )∆u(y)dy ≥ 0
for all BR (xo ) ⊂ E.
BR (xo )
From this
Z
∆u(xo )
BR (xo )
G(0; y − xo )dy ≥
Z
BR (xo )
G(0; y − xo )[∆u(xo ) − ∆u(y)]dy
and the assertion follows upon dividing by the coefficient of ∆u(xo ), and
letting R → 0.
Lemma 4.1 Let E be a bounded, connected, open set in RN . If u ∈ C(Ē) is
sub-harmonic in E, then either u is constant, or
u(x) < sup u
∂E
for all x ∈ E.
Proof. Let xo ∈ Ē be a point where u(xo ) = supĒ u, and assume that u is not
identically equal to u(xo ). If xo ∈ E, for every ball BR (xo ) ⊂ E
u(y) ≤ u(xo )
which implies
Z
−
∂BR (xo )
for all y ∈ BR (xo )
[u(y) − u(xo )]dσ ≤ 0.
On the other hand, since u is sub-harmonic, this same integral must be nonnegative. Therefore
Z
−
[u(y) − u(xo )]dσ = 0 and u(y) ≤ u(xo ).
∂BR (xo )
4 Sub-Harmonic Functions and the Mean Value Property
59
Thus there exists a ball Br (xo ) for some r > 0 such that u(y) = u(xo ) for
all y ∈ Br (xo ). Consider the set Eo = {y ∈ E|u(y) = u(xo )}. The previous
remarks prove that Eo is open. By the continuity of u, it is closed in the relative
topology of E. Therefore, since E is connected, Eo = E and u ≡ u(xo ). The
contradiction implies that xo ∈ ∂E.
A function u ∈ C(E) satisfies the mean value property in E if
Z
u(xo ) = −
u dσ
for all BR (xo ) ⊂ E,
(4.4)
∂BR (xo )
equivalently if (4.2 of the Complements)
Z
u(xo ) = −
u dy
for all BR (xo ) ⊂ E.
(4.5)
BR (xo )
Functions satisfying such a property are both sub- and super-harmonic. By
(4.1), harmonic functions in E satisfy the mean value property.
Lemma 4.2 Let E be a bounded, connected, open set in RN . If u ∈ C(Ē)
satisfies the mean value property in E, then either it is constant or
sup |u| = sup |u|.
E
∂E
Proof (Theorem 3.1, uniqueness). If u, v are two solutions of the Dirichlet
problem (1.2) for E = BR , the difference w = u − v is harmonic in BR and
vanishes on ∂BR . Thus w ≡ 0 by Lemma 4.2.
Lemma 4.3 The following are equivalent:
u ∈ C(E) satisfies the mean value property
2
u ∈ C (E) and
∆u = 0.
(i)
(ii)
Proof. We have only to prove (i)=⇒(ii). Having fixed BR (xo ) ⊂ E, let v ∈
C 2 (BR (xo )) ∩ C(B̄R (xo ) be the unique solution of the Dirichlet problem
∆v = 0 in BR (xo )
and
v
∂BR (xo )
= u.
Such a solution is given by the Poisson formula (3.9) up to a change of variables
that maps xo into the origin. The difference w = u − v satisfies the mean value
property in BR (xo ), and by Lemma 4.2, w ≡ 0.
4.1 The Maximum Principle
We restate some of these properties in a commonly used form. Let E be a
bounded, connected, open set in RN with boundary ∂E of class C 1 , and let
u ∈ C 2 (E) ∩ C(Ē) be nonconstant in E. Then
∆u ≥ 0 in E =⇒ u(x) < sup∂E u
∆u ≤ 0 in E =⇒ u(x) > inf ∂E u
∆u = 0 in E =⇒ |u(x)| < sup∂E |u|
∀x ∈ E
∀x ∈ E
∀x ∈ E.
60
2 THE LAPLACE EQUATION
Remark 4.1 The assumption that ∂E is of class C 1 can be removed by
applying the maximum principle to a family of expanding connected open
sets with smooth boundary, exhausting E.
Remark 4.2 The assumption of E being bounded cannot be removed, as
shown by the following counterexample. Let E be the sector x2 > |x1 | in R2 .
The function u(x) = x22 − x21 is harmonic in E, vanishes on ∂E, and takes
arbitrarily large values in E.
4.2 Structure of Sub-Harmonic Functions
Set
σ(E) = {v ∈ C(E) v is sub-harmonic in E}
Σ(E) = {v ∈ C(E) v is super-harmonic in E}.
(4.6)
Proposition 4.1 Let v, vi ∈ σ(E) and ci ∈ R+ for i = 1, . . . , n. Then
v ∈ σ(E ′ ) for every open subset E ′ ⊂ E
n
P
ci vi ∈ σ(E)
(i)
(ii)
i=1
max{v1 , v2 , . . . , vn } ∈ σ(E)
For every nondecreasing convex function f (·) in R
(iii)
(iv)
v ∈ σ(E) =⇒ f (v) ∈ σ(E).
Proof. The statements (i)–(ii) are obvious. To prove (iii), observe that having
fixed BR (xo ) ⊂ E, for some 1 ≤ i ≤ n
Z
max{v1 (xo ), . . . , vn (xo )} = vi (xo ) ≤ −
vi dσ
∂BR (xo )
Z
≤−
max{v1 , . . . , vn }dσ.
∂BR (xo )
To prove (iv), write (4.2) for v and apply f (·) to both sides. By Jensen’s
inequality
Z
Z
vdσ ≤ −
f (v)dσ.
f v(xo ) ≤ f −
∂BR (xo )
∂BR (xo )
Remark 4.3 For simplicity, (iii) and (iv) have been stated separately. In fact
(iv) implies (iii).
An important subclass of σ(E) is that of the sub-harmonic functions in E
that actually are harmonic in some sphere contained in E. Given v ∈ σ(E),
fix Bρ (ξ) ⊂ E and solve the Dirichlet problem
∆Hv = 0 in Bρ (ξ)
and
Hv
∂Bρ (ξ)
= v.
5 Estimating Harmonic Functions and Their Derivatives
61
The unique solution Hv is the harmonic extension of v ∂Bρ (ξ) into Bρ (ξ). The
function that coincides with v in E − Bρ (ξ) and that equals Hv in Bρ (ξ) is
denoted by vξ,ρ , i.e.,
v(x) if x ∈ E − Bρ (ξ)
vξ,ρ (x) =
(4.7)
Hv (x) if x ∈ Bρ (ξ).
Since v ∈ σ(Bρ (ξ)) and Hv is harmonic in Bρ (ξ), we have v − Hv ∈ σ(Bρ (ξ)).
Therefore v ≤ Hv in Bρ (ξ). The definition of vξ,ρ then implies
v ≤ vξ,ρ
in E.
(4.8)
Proposition 4.2 Let v ∈ σ(E). Then vξ,ρ ∈ σ(E).
Proof. One needs to verify that vξ,ρ satisfies (4.2) for all BR (xo ) ⊂ E. This is
obvious for xo ∈ E − Bρ (ξ) in view of (4.8). Fix xo ∈ Bρ (ξ) and assume, by
contradiction, that there is a ball BR (xo ) ⊂ E such that
Z
vξ,ρ (xo ) > −
vξ,ρ dσ.
∂BR (xo )
Construct the function
w = (vξ,ρ )xo ,R =
vξ,ρ in E − BR (xo )
Hvξ,ρ in BR (xo ).
Since vξ,ρ ≥ v, by the maximum principle w ≥ vxo ,R . Since w satisfies the
mean value property in BR (xo ), the contradiction assumption implies that
vξ,ρ (xo ) − w(xo ) > 0.
(4.9)
The difference vξ,ρ − w is harmonic in BR (xo ) ∩ Bρ (ξ). The boundary of such
a set is the union of ∂1 and ∂2 , where
∂1 = ∂BR (xo ) ∩ B̄ρ (ξ)
and
∂2 = ∂Bρ (ξ) ∩ B̄R (xo ).
Because of (4.9), the function x → (vξ,ρ − w)(x), restricted to BR (xo ) ∩ Bρ (ξ),
must take its positive maximum at some point x∗ ∈ ∂1 ∪ ∂2 . Since it vanishes
on ∂1 , there exists some x∗ ∈ ∂2 such that vξ,ρ (x∗ ) > w(x∗ ). By construction,
vξ,ρ = v on ∂2 . Therefore v(x∗ ) > w(x∗ ). Since w ≥ vxo ,R , we conclude that
v(x∗ ) > vxo ,R (x∗ ). This contradicts (4.8) and proves the proposition.
Remark 4.4 Analogous facts hold for super-harmonic functions.
5 Estimating Harmonic Functions and Their Derivatives
We will prove that if u is harmonic and is non-negative in E, then in any
compact subset K ⊂ E, its maximum and minimum value are comparable.
We also establish sharp estimates for the derivatives of u in the interior of E.
62
2 THE LAPLACE EQUATION
5.1 The Harnack Inequality and the Liouville Theorem
Theorem 5.1 (Harnack ([118])). Let u be a non-negative harmonic function in E. Then for all x ∈ Bρ (xo ) ⊂ BR (xo ) ⊂ E
R
R+ρ
N −2
R−ρ
u(xo ) ≤ u(x) ≤
R+ρ
R
R−ρ
N −2
R+ρ
u(xo ).
R−ρ
(5.1)
Proof. Modulo a translation, we may assume that xo = 0. By the Poisson
formula (3.9) and the mean value property (4.7), for all x ∈ BR
Z
Z
R2 − |x|2
u(y)
u(y)
R2 − |x|2
dσ ≤
dσ
u(x) =
N
RωN
|x
−
y|
Rω
(|y|
− |x|)N
N
∂BR
∂BR
N −2
Z
R2 − |x|2 N −2
R
R + |x|
=
u
dσ
=
R
−
u(0).
(R − |x|)N
R
−
|x|
R
− |x|
∂BR
This proves the estimate above in (5.1). For the estimate below, observe that
Z
Z
R2 − |x|2
u(y)
u(y)
R2 − |x|2
dσ ≥
dσ
u(x) =
N
RωN
|x
−
y|
Rω
(|y|
+ |x|)N
N
∂BR
∂BR
and conclude as above.
Corollary 5.1 (Harnack Inequality ([118])) For every compact and connected subset K ⊂ E, there exists a constant C depending only on N and
dist(K; ∂E), such that
C min u ≥ max u.
(5.2)
K
K
Proof. Let x1 , x2 ∈ K be such that minK u = u(x1 ) and maxK u = u(x2 ). Fix
a path Γ in K connecting x1 and x2 , and cover Γ with finitely many spheres
for each of which (5.1) holds.
Corollary 5.2 (Liouville Theorem) A non-negative harmonic function in
RN is constant.
Proof. In (5.1) fix xo ∈ RN and ρ > 0. Letting R → ∞ gives u(x) = u(xo ) for
all x ∈ Bρ (xo ). Since xo and ρ > 0 are arbitrary, u = const in RN .
Corollary 5.3 Let u be harmonic in RN and such that u ≥ k for some
constant k. Then u is constant.
Proof. The function u − k is harmonic and non-negative in RN .
5 Estimating Harmonic Functions and Their Derivatives
63
Remark 5.1 The proof of Theorem 5.1 shows that in (5.1), for x ∈ BR (xo )
fixed, the number ρ can be taken to be |x|. This permits us to estimate from
below the normal derivative of any harmonic function u in BR (xo ) at points
x∗ ∈ ∂BR (xo ) where u attains its minimum.
Proposition 5.1 Let u ∈ C 2 (BR (xo )) ∩ C(B̄R (xo )) be harmonic in BR (xo ),
let x∗ ∈ ∂BR (xo ) be a minimum point of u in B̄R (xo ), and set
u(x∗ ) = min u
B̄R (xo )
Then
−
and
n=
x∗ − xo
.
|x∗ − xo |
∂u
u(xo ) − u(x∗ )
(x∗ ) ≥ 21−N
.
∂n
R
(5.3)
Proof. The function u−u(x∗ ) is harmonic and non-negative in BR (xo ). Apply
(5.1) to such a function, with ρ = |x|, to get
u(xo ) − u(x∗ )
u(x) − u(x∗ )
≥ 21−N
.
R − |x|
R
Letting now x → x∗ along n proves (5.3).
5.2 Analyticity of Harmonic Functions
If u is harmonic in E, by Proposition 2.1, Dα u is also harmonic in E, for every
multi-index α. Therefore Dα u satisfies the mean value property (4.5) for all
multi-indices α. In particular, for all i = 1, . . . , N and all BR (xo ) ⊂ E
Z
Z
∂u
N
(y − xo )i
uxi (y)dy =
(xo ) = −
dσ.
u
∂xi
ωN RN ∂BR (xo ) |y − xo |
BR (xo )
From this
N
∂u
(xo ) ≤
sup |u|.
∂xi
R BR (xo )
(5.4)
This estimate is a particular case of the following
Theorem 5.2. Let u be harmonic in E. Then for all BR (xo ) ⊂ E, and for
all multi-indices α
|α|
Ne
|α|!
|Dα u(xo )| ≤
sup |u|.
(5.5)
R
e BR (xo )
Proof. By (5.4) the estimate holds for multi-indices of size 1. It will be shown
by induction that if (5.5) holds for multi-indices of size |α|, it continues to
hold for multi-indices β of size |β| = |α| + 1. For any such β
Dβ u =
∂ α
D u
∂xi
for some 1 ≤ i ≤ N.
64
2 THE LAPLACE EQUATION
Fix τ ∈ (0, 1) and apply (4.5) to Dβ u in the ball Bτ R (xo ). This gives
Z
∂ α
β
D u(xo ) = −
D u dy
∂x
i
Bτ R (xo )
Z
N
(y − xo )i
=
Dα u(y)
dσ.
ωN τ N RN ∂Bτ R (xo )
|y − xo |
By (5.5) applied over balls centered at y ∈ ∂Bτ R (xo ) and radius (1 − τ )R
|α|
Ne
|α|!
α
|D u(y)| ≤
sup |u| for all y ∈ ∂Bτ R (xo ).
(1 − τ )R
e BR (xo )
Therefore
β
|D u(xo )| ≤
Ne
R
|α|+1
1
|α|!
sup |u|.
|α|
(1 − τ ) τ e2 BR (xo )
To prove the theorem, choose
τ=
1
1
=
|α| + 1
|β|
so that
1 |α|
≤ e.
(1 − τ )−|α| ≤ 1 +
|α|
Corollary 5.4 Let u be harmonic in E. Then u is locally analytic in E.
Proof. Let k be a positive number to be chosen, and having fixed xo ∈ E, let
R be so small that B(k+1)R (xo ) ⊂ E. The Taylor expansion of u in BR (xo )
about xo is
u(x) =
P Dβ u(ξ)
P Dα u(xo )
(x − xo )α +
(x − xo )β
α!
β!
|β|=n+1
|α|≤n
for some ξ ∈ BR (xo ). Estimate the terms of the remainder by applying (5.5)
to the ball centered at ξ and radius kR. This gives
|β|
|Dβ u(ξ)|
|β|! R|β|
Ne
β
|(x − xo ) | ≤
sup
|u|
β!
kR
β! e B(k+1)R (xo )
|β|
N eN +1
≤
sup
|u|
k
B(k+1)R (xo )
where we have also used the inequality |β|! ≤ eN |β| β!. Set
N eN +1
=θ
k
and
sup
B(k+1)R (xo )
|u| = M.
Choose k so that θ < 1, and majorize the remainder of the Taylor series by
P
Dβ u(ξ)
θ|β| ≤ M |β|N θ|β| .
(x − xo )β ≤ M
β!
|β|=n+1
|β|=n+1
P
Since this tends to zero as |β| → ∞, the Taylor series of u about xo converges
to u uniformly in BR (xo ).
6 The Dirichlet Problem
65
6 The Dirichlet Problem
We will establish that the boundary value problem (1.2) has a unique solution
for any given ϕ ∈ C(∂E). In the statement of the Dirichlet problem (1.2), the
boundary ∂E was assumed to be of class C 1 . In particular ∂E satisfies the
exterior sphere condition, i.e.,
for all x∗ ∈ ∂E there exists an exterior ball
BR (xo ) ⊂ RN − Ē such that ∂BR (xo ) ∩ ∂E = x∗ .
(6.1)
The ball BR (xo ) is exterior to E, and its boundary ∂BR (xo ) touches ∂E
only at x∗ . Such a property is shared by domains whose boundary could be
irregular. For example, it is satisfied if ∂E exhibits corners or even spikes
pointing outside E.
Theorem 6.1. Let E be a bounded domain in RN whose boundary ∂E satisfies the exterior sphere condition (6.1). Then for every ϕ ∈ C(∂E) there
exists a unique solution to the Dirichlet problem
u ∈ C 2 (E) ∩ C(Ē),
∆u = 0 in E,
and
u
∂E
= ϕ.
(6.2)
Proof (Perron ([201])). Recall the definition (4.6) of the classes σ(E) and
Σ(E) and for a fixed ϕ ∈ C(∂E), consider the two classes
σ(ϕ; E) = v ∈ σ(E) ∩ C(Ē) and v ∂E ≤ ϕ
Σ(ϕ; E) = v ∈ Σ(E) ∩ C(Ē) and v ∂E ≥ ϕ .
Any constant k ≤ min∂E ϕ is in σ(ϕ; E), and any constant h ≥ max∂E ϕ is
in Σ(ϕ; E). Therefore σ(ϕ; E) and Σ(ϕ; E) are not empty. If a solution u to
(6.2) exists, it must satisfy
v≤u≤w
for all v ∈ σ(ϕ; E) and for all w ∈ Σ(ϕ; E).
This suggests to look for u as the unique element of separation of the two
classes σ(ϕ; E) and Σ(ϕ; E), i.e.,
sup
v∈σ(ϕ;E)
def
def
v(x) = u(x) =
inf
w∈Σ(ϕ;E)
w(x),
∀x ∈ Ē.
To prove the theorem we have to prove the following two facts.
Lemma 6.1 The function u defined by (6.3) is harmonic in E.
Lemma 6.2 u ∈ C(Ē) and u|∂E = ϕ.
(6.3)
66
2 THE LAPLACE EQUATION
Proof (of Lemma 6.1). Fix xo ∈ E and select a sequence {vn } ⊂ σ(ϕ; E) such
that vn (xo ) → u(xo ). The functions
Vn = max{v1 , v2 , . . . , vn }
(6.4)
belong to σ(ϕ; E), and the sequence {Vn } satisfies
Vn ≤ Vn+1
and
lim Vn (xo ) = u(xo ).
n→∞
Let Bρ (ξ) ⊂ E be a ball containing xo , and construct the functions Vn;ξ,ρ as
described in (4.7). By Proposition 4.2, Vn;ξ,ρ ∈ σ(ϕ; E), and by the previous
remarks
Vn;ξ,ρ ≤ Vn+1;ξ,ρ and Vn;ξ,ρ (xo ) → u(xo ).
Thus {Vn;ξ,ρ } converges monotonically to some function z(·), which we claim
is harmonic in Bρ (ξ). Indeed, Vn;ξ,ρ − V1;ξ,ρ are all harmonic and non-negative
in Bρ (ξ), and the sequence {Vn;ξ,ρ (xo )−V1;ξ,ρ (xo )} is equi-bounded. Therefore
by the Harnack inequality (5.1), {Vn;ξ,ρ − V1;ξ,ρ } is equi-bounded on compact
subsets of Bρ (ξ). By Theorem 5.2, also all the derivatives Dα (Vn;ξ,ρ − V1;ξ,ρ )
are equi-bounded on compact subsets of Bρ (ξ). Therefore, by possibly passing
to a subsequence, {Dα (Vn;ξ,ρ −V1;ξ,ρ )} converge uniformly on compact subsets
of Bρ (ξ), for all multi-indices α. Thus z(·) is infinitely differentiable in Bρ (ξ)
and {Dα Vn;ξ,ρ } → Dα z uniformly on compact subsets of Bρ (ξ), for all multiindices α. Since all Vn;ξ,ρ are harmonic in Bρ (ξ), also z is harmonic in Bρ (ξ).
By construction, z(xo ) = u(xo ). To prove that z(x) = u(x) for all x ∈
Bρ (ξ), fix x̃ ∈ Bρ (ξ) and construct sequences {ṽn } and {Ṽn } as follows:
ṽn ∈ σ(ϕ; E)
and ṽn (x̃) → u(x̃)
Ṽn (x) = max{Vn (x); ṽ1 (x), ṽ2 (x), . . . , ṽn (x)}
∀x ∈ E
where Vn are defined in (6.4). Starting from Ṽn , construct the corresponding
functions Ṽn;ξ,ρ as indicated in (4.7). Arguing as before, these satisfy
Ṽn;ξ,ρ ≤ Ṽn+1;ξ,ρ ,
Ṽn ≤ Ṽn;ξ,ρ ,
Ṽn;ξ,ρ (x̃) → u(x̃).
Moreover, {Ṽn;ξ,ρ } converges monotonically in Bρ (ξ) to a harmonic function
z̃(·) satisfying
z̃(x) ≥ z(x)
for all x ∈ Bρ (ξ)
and
z̃(x̃) = u(x̃).
By the construction (6.3) of u
u(xo ) = z(xo ) ≤ z̃(xo ) = u(xo ).
Thus the function z̃ −z is non-negative and harmonic in Bρ (ξ), and it vanishes
in an interior point xo of Bρ (ξ). This is impossible unless z̃(x) = z(x) for all
x ∈ Bρ (ξ). In particular
z̃(x̃) = z(x̃) = u(x̃).
Since x̃ ∈ Bρ (ξ) is arbitrary, we conclude that u is harmonic in a neighborhood
of xo and hence in the whole of E, since xo is an arbitrary point of E.
6 The Dirichlet Problem
67
Proof (of Lemma 6.2). Fix x∗ ∈ ∂E and let BR (xo ) be the ball exterior to E
and touching ∂E only at x∗ claimed by (6.1). The function

1
1


−
if N ≥ 3

 RN −2
|x − xo |N −2
H(x) =
(6.5)


|x
−
x
|

o
 ln
if N = 2
R
is harmonic in a neighborhood of E and positive on ∂E except at x∗ , where it
vanishes. Fix an arbitrarily small positive number ε and determine δ = δ(ε) ∈
(0, 1) so that
|ϕ(x) − ϕ(x∗ )| ≤ ε
∀[|x − x∗ | ≤ δ] ∩ ∂E.
We claim that for all ε > 0 there exists a constant Cε , depending only on
kϕk∞,∂E , R, N , and δ(ε), such that
|ϕ(x) − ϕ(x∗ )| < ε + Cε H(x)
∀x ∈ ∂E.
(6.6)
This is obvious if |x − x∗ | ≤ δ. If x ∈ ∂E and |x − x∗ | > δ
|ϕ(x) − ϕ(x∗ )| ≤ 2kϕk∞,∂E
H(x)
,
Hδ
where
Hδ =
min
[|x−x∗ |≥δ]∩∂E
H(x).
To prove (6.6), we have only to observe that Hδ > 0. It follows from (6.6)
that for all x ∈ ∂E
ϕ(x∗ ) − ε − Cε H(x) ≤ ϕ(x) ≤ ϕ(x∗ ) + ε + Cε H(x).
This implies that
ϕ(x∗ ) − ε − Cε H ∈ σ(ϕ; E)
and
ϕ(x∗ ) + ε + Cε H ∈ Σ(ϕ; E).
Therefore for all x ∈ Ē
ϕ(x∗ ) − ε − Cε H(x) ≤ u(x) ≤ ϕ(x∗ ) + ε + Cε H(x).
This in turn implies
|u(x) − ϕ(x∗ )| ≤ ε + Cε H(x)
∀x ∈ Ē.
We now let x → x∗ for ε ∈ (0, 1) fixed. Since H ∈ C(Ē) and H(x∗ ) = 0
lim sup |u(x) − ϕ(x∗ )| ≤ ε
x→x∗
∀ε ∈ (0, 1).
68
2 THE LAPLACE EQUATION
7 About the Exterior Sphere Condition
The existence theorem is based on an interior statement (Lemma 6.1) and a
boundary statement concerning the behavior of u near ∂E (Lemma 6.2). The
first can be established regardless of the structure of ∂E. The second relies
on the construction of the function H(·) in (6.5). Such a construction is made
possible by the exterior sphere condition (6.1). Indeed, this is the only role
played by (6.1). Keeping this in mind, we might impose on ∂E the
Barrier Postulate: ∀x∗ ∈ ∂E, ∃H(x∗ ; ·) ∈ C(Ē) satisfying
H(x∗ ; ·) is super-harmonic in a neighborhood of E
H(x∗ ; x) > 0 ∀x ∈ Ē − {x∗ }, and H(x∗ ; x∗ ) = 0.
(7.1)
Any such function H(x∗ ; ·) is a barrier for the Dirichlet problem (6.2) at x∗ .
Assume that ∂E satisfies the barrier postulate. Arguing as in the proof of
Lemma 6.2, having fixed x∗ ∈ ∂E, for all ε > 0 there exists a constant
Cε = Cε (kϕk∞,∂E , N, H(x∗ ; ·), ε)
such that
|ϕ(x) − ϕ(x∗ )| ≤ ε + Cε H(x∗ ; x)
∀x ∈ ∂E.
Therefore, for all x ∈ ∂E
ϕ(x∗ ) − ε − Cε H(x∗ ; x) ≤ ϕ(x) ≤ ϕ(x∗ ) + ε + Cε H(x∗ ; x).
Since H(x∗ ; ·) is super-harmonic
ϕ(x∗ ) − ε − Cε H(x∗ ; x) ∈ σ(ϕ; E)
and
ϕ(x∗ ) + ε + Cε H(x∗ ; x) ∈ Σ(ϕ; E).
Therefore for all x ∈ Ē
ϕ(x∗ ) − ε − Cε H(x∗ ; x) ≤ u(x) ≤ ϕ(x∗ ) + ε + Cε H(x∗ ; x)
and
|u(x) − ϕ(x∗ )| < ε + Cε H(x∗ ; x)
∀x ∈ Ē.
This proves Lemma 6.2 if the exterior sphere condition (6.1) is replaced by
the barrier postulate (7.1). We conclude that the Dirichlet problem (6.2) is
uniquely solvable for every domain E satisfying the barrier postulate.
7 About the Exterior Sphere Condition
69
7.1 The Case N = 2 and ∂E Piecewise Smooth
Let E be a bounded domain in R2 whose boundary ∂E is the finite union
of portions of curves of class C 1 . Domains of this kind permit corners and
even spikes pointing outside or inside E. Fix x∗ ∈ ∂E and assume, modulo
a translation, that x∗ coincides with the origin. We may also assume, up to
a homothetic transformation, that E is contained in the unit disc about the
origin. Identifying R2 with the complex plane C, points z = ρeiθ of E, are
determined by a unique value of the argument θ ∈ (−π, π). Therefore ln z is
uniquely defined in E. A barrier at the origin is
1
ln ρ
H(x) = −Re
=− 2
.
ln z
ln ρ + θ2
7.2 A Counterexample of Lebesgue for N = 3 ([163])
If N ≥ 3, spikes pointing outside E are permitted, since any such point would
satisfy the exterior sphere condition (6.1). Spikes pointing inside E are, in
general, not permitted as shown by the following example of Lebesgue.
Denote points in R3 by (x, z), where x = (x1 , x2 ) and z ∈ R. The function
v(x, z) =
Z
1
0
s ds
p
2
|x| + (s − z)2
(7.2)
is harmonic outside [|x| = 0] ∩ [0 ≤ z ≤ 1]. By integration by parts one
computes
p
p
v(x, z) = |x|2 + (1 − z)2 − |x|2 + z 2
p
p
+ z ln (1 − z) + |x|2 + (1 − z)2 z + |x|2 + z 2
− 2z ln |x|.
As (x, z) → 0, the sum of the first three terms on the right-hand tends to 1,
whereas the last term is discontinuous at zero. It tends to zero if (x, z) → 0
along the curve |z|β = |x| for all β > 0. However, if (x, z) → 0 along |x| =
e−γ/2z , for z > 0 and γ > 0, it converges to γ. We conclude that
lim
(x,z)→0
along |x|=e−γ/2z
v(x, z) = 1 + γ.
Therefore, all the level surfaces [v = 1 + γ] for all γ > 0 go through the origin,
and as a consequence, v is not continuous at the origin.
Fix c > 0 and consider the domain
E = [v < 1 + c] ∩ [|x, z| < 1].
70
2 THE LAPLACE EQUATION
R2 ....
........
... ... ..... .... ..............
.
.
...
.....
..............................u.....=..... 1 +
z
... O ........
....
...
.
.....
.............................
Fig. 7.1
There exists no solution to the Dirichlet problem
u ∈ C 2 (E) ∩ C(Ē),
∆u = 0 in E,
and
u
∂E
=v
∂E
.
(7.3)
Notice that even though v is not continuous in Ē, the restriction v|∂E is
continuous on ∂E. The idea of the counterexample is based on showing that
any solution of (7.3) must coincide with v, which itself is not a solution.
Fix any ε > 0, and consider the domain Eε = E ∩ [|x, z| > ε]. Assume that
u is a solution of (7.3) and let C be a constant such that |u − v| < C in E.
The functions
ε
± (u − v)
wε = C
|x, z|
are harmonic in Eε and non-negative on ∂Eε . Thus by the maximum principle
ε
|u(x, z) − v(x, z)| ≤ C
in Eε .
|x, z|
8 The Poisson Integral for the Half Space
Denote points in RN +1 by (x, t), where x ∈ RN and t ∈ R. Consider the
Dirichlet problem

 u ∈ C 2 (RN × R+ ) ∩ C(RN × R+ )
(8.1)
∆u = 0 in RN × R+

u(x, 0) = ϕ(x) ∈ C(RN ) ∩ L∞ (RN ).
A solution to (8.1) is called the harmonic extension of ϕ in the upper halfspace RN × R+ . Consider the fundamental solution (2.6) of the Laplacean in
RN +1 with pole at (y, 0)

1
1

if N ≥ 2


 (N − 1)ωN +1 [|x − y|2 + t2 ] N2−1
F (x, t; y) =



 − 1 ln |x − y|2 + t2 1/2
if N = 1.
2π
8 The Poisson Integral for the Half Space
71
The Poisson kernel for the half-space is defined for all N ≥ 1 by
K(x; y) = −2
∂F (x, t; y)
2t
=
N +1 .
∂t
ωN +1 [|x − y|2 + t2 ] 2
(8.2)
Theorem 8.1. Every ϕ ∈ C(RN ) ∩ L∞ (RN ) has a unique bounded harmonic
extension Hϕ in RN × R+ , given by
Z
2t
ϕ(y)
Hϕ (x, t) =
dy.
(8.3)
ωN +1 RN [|x − y|2 + t2 ] N2+1
Proof (Uniqueness). If u and v are both bounded solutions of (8.1), the difference w = u − v is harmonic in RN × R+ and vanishes for t = 0. By reflection
about the hyperplane t = 0, the function
w(x, t)
if t > 0
w̃(x, t) =
−w(x, −t) if t ≤ 0
is bounded and harmonic in RN +1 . Therefore, by Liouville’s theorem (Corollary 5.3), it is constant. Since w(·, 0) = 0, it vanishes identically.
Remark 8.1 The statement of uniqueness in Theorem 8.1 holds only within
the class of bounded solutions. Indeed, the two functions u = 0 and v = t are
both harmonic extensions of ϕ = 0.
Proof (Existence). The function Hϕ defined in (8.3) is harmonic in RN × R+ .
The boundedness of Hϕ follows from the boundedness of ϕ and the following
lemma.
Lemma 8.1 For all ε > 0 and all x ∈ RN
Z
2ε
dy
= 1.
ωN +1 RN [|x − y|2 + ε2 ] N2+1
(8.4)
Proof. Assume N ≥ 2. The change of variables y − x = εξ transforms the
integral in (8.4) into
2
ωN +1
Z
RN
dξ
(1 + |ξ|2 )
N +1
2
ωN
=2
ωN +1
Z
0
∞
ρN −1
(1 + ρ2 )
N +1
2
dρ = 1.
(8.5)
The case N = 1 is treated analogously (8.1 of the Complements).
To conclude the proof of Theorem 8.1, it remains to show that for all x∗ ∈ RN
lim
(x,t)→x∗
Hϕ (x, t) = ϕ(x∗ ).
This is established as in Theorem 3.1 by making use of (8.4).
72
2 THE LAPLACE EQUATION
9 Schauder Estimates of Newtonian Potentials
Let E be a bounded open set in RN and continue to denote by F (·; ·) the
fundamental solution of the Laplacean, introduced in (2.6). The Newtonian
potential generated in RN by a density distribution f ∈ Lp (E) for some p > 1
is defined by
Z
AF f =
(9.1)
F (·; y)f (y)dy
E
provided the right-hand side is finite. If f ∈ L∞ (E)
kAF f k∞,RN + k∇AF f k∞,RN ≤ γkf k∞,E
(9.2)
where γ is a constant depending only on N and diam(E). Further regularity
of AF f can be established if f is Hölder continuous and compactly supported
in E. For m ∈ N ∪ {0}, η ∈ (0, 1), and ϕ ∈ C ∞ (E), set
P
def
|||ϕ|||m,η;E =
|α|≤m
kDα ϕk∞,E +
P
sup
|α|=m x,y∈E
|Dα ϕ(x) − Dα ϕ(y)|
.
|x − y|η
(9.3)
Denote by C m,η (E) the space of functions ϕ ∈ C m (E) with finite norm
|||ϕ|||m,η;E and by Com,η (E) the space of functions ϕ ∈ C m,η (E) compactly
supported in E. If m = 0, we let C 0,η (E) = C η (E) and |||ϕ|||0,η;E = |||ϕ|||η;E .
Proposition 9.1 Let f ∈ Com,η (E). Then AF f ∈ C m+2,η (E), and there exists a constant γ depending upon N , m, η, and diam(E), such that
|||AF f |||m+2,η;E ≤ γ|||f |||m,η;E .
(9.4)
Proof. It suffices to prove (9.4) for m = 0 and for f ∈ Co∞ (E). Assume N ≥ 3,
the proof for N = 2 being analogous, and rewrite (9.1) as
Z
ωN (N − 2)AF f = v(·) =
|ξ|2−N f (· + ξ)dξ
RN
and compute
vxi xj (x) =
Z
RN
=−
=
−
Z
Z
Z
|ξ|2−N fξi ξj (x + ξ)dξ = −
Z
RN
(|ξ|2−N )ξj fξi (x + ξ)dξ
(|ξ|2−N )ξj fξi (x + ξ)dξ
|ξ|>r
|ξ|<r
(|ξ|2−N )ξj [f (x + ξ) − f (x)]ξi dξ
(|ξ|2−N )ξj ξj f (x + ξ)dξ
|ξ|>r
Z
(|ξ|2−N )ξj ξj (f (x + ξ) − f (x))dξ
Z
ξi
+ f (x)
(|ξ|2−N )ξj dσ.
|ξ|
|ξ|=r
+
|ξ|<r
(9.5)
9 Schauder Estimates of Newtonian Potentials
73
In this representation, r is any positive number, dσ is the surface measure
over the sphere ∂Br , and the integral extended over the ball |ξ| < r is meant
in the sense of the limit
Z
(|ξ|2−N )ξj ξj (f (x + ξ) − f (x))dξ
|ξ|<r
Z
def
(|ξ|2−N )ξj ξj (f (x + ξ) − f (x))dξ.
= lim
ε→0
ε<|ξ|<r
Such a limit exists, since f is Hölder continuous. From (9.5), by taking r =
diam(E), we estimate
!
Z
P
−N
α
|ξ| dξ
kD vk∞,RN ≤ γkf k∞,E 1 +
|α|=2
+γ
Z
|ξ|<r
r<|ξ|<2r
|ξ|−N +η
|f (x + ξ) − f (x)|
dξ
|ξ|η
(9.6)
≤ γ (1 + diam(E)) |||f |||η;E .
Next we fix y ∈ RN and represent vxi xj (y). By calculations analogous to those
leading to (9.5)
Z
|ξ|2−N fξi ξj (y + ξ)dξ
vxi xj (y) =
RN
Z
|(x − y) + ξ|2−N fξi ξj (x + ξ)dξ
=
RN
Z
|(x − y) + ξ|2−N ξi ξj f (x + ξ)dξ
=
(9.7)
|ξ|>r
Z
|(x − y) + ξ|2−N ξ ξ [f (x + ξ) − f (y)]dξ
+
i j
|ξ|<r
+ f (y)
Z
|ξ|=r
|(x − y) + ξ|2−N
ξj
ξi
dσ.
|ξ|
From the representations (9.5) and (9.7), we obtain by difference
74
2 THE LAPLACE EQUATION
vxi xj (x)−vxi xj (y)
Z
|ξ|2−N − |(x − y) + ξ|2−N ξ ξ [f (x + ξ) − f (y)]dξ
=
i j
|ξ|>r
Z
(|ξ|2−N )ξj ξj [f (x + ξ) − f (x)]dξ
+
|ξ|<r
Z
|(x − y) + ξ|2−N ξi ξj [f (x + ξ) − f (y)]dξ
−
|ξ|<r
Z
ξi
(|ξ|2−N )ξj dσ
+ [f (x) − f (y)]
|ξ|
|ξ|=r
Z
ξi
2−N
|ξ|
− |(x − y) + ξ|2−N ξj
+ f (y)
dσ
|ξ|
|ξ|=r
Z
|ξ|2−N − |(x − y) + ξ|2−N ξi ξj dξ.
+ f (y)
|ξ|>r
The sum of the last two integrals is zero, and the integral extended over the
shell |ξ| = r is majorized by
γ|||f |||η;E |x − y|η
where γ is a constant depending only upon the dimension N . In the estimates
below we denote by γ a constant that can be different in different contexts,
and it can be computed quantitatively a priori in terms of N alone. Estimating
the first integral extended over the ball |ξ| < r we have
Z
Z
(|ξ|2−N )ξj ξj [f (x + ξ) − f (x)]dξ ≤ γ(N )|||f |||η;E
|ξ|−N +η dξ
|ξ|<r
≤ γ|||f |||η;E rη .
Analogously
Z
|ξ|<r
|(x − y) + ξ|2−N
ξi ξj
|ξ|<r
[f (x + ξ) − f (y)]dξ
Z
≤ γ|||f |||η;E
|(x − y) + ξ|−N +η dξ
|ξ|<r
Z
≤ γ|||f |||η;E
|z|−N +η dz
|z|<r+|x−y|
≤ γ|||f |||η;E (rη + |x − y|η ).
Combining these estimates, we conclude that there is a constant γ depending
only upon N , such that for every r > 0
|vxi xj (x) − vxi xj (y)| ≤ γ|||f |||η;E (rη + |x − y|η )
Z
+γ
|ξ|2−N − |(x − y) + ξ|2−N ξi ξj [f (x + ξ) − f (y)]dξ .
|ξ|>r
(9.8)
10 Potential Estimates in Lp (E)
75
Choose r = 2|x − y| so that over the set |ξ| > r
2|ξ| ≥ |(x − y) + ξ| ≥
1
|ξ|.
2
Then by direct calculation and the mean value theorem
|ξ|2−N − |(x − y) + ξ|2−N
ξi ξj
≤
δij
δij
−
|ξ|N
|(x − y) + ξ|N
ξi ξj
[(x − y) + ξ]i [(x − y) + ξ]j
+
−
|ξ|N +2
|(x − y) + ξ|N +2
|x − y|
≤ γ N +1 .
|ξ|
Therefore the last integral in (9.8) can be majorized by
Z
|ξ|η |f (x + ξ) − f (y)|
γ|x − y|
dξ
N +1 |(x − y) + ξ|η
|ξ|>r |ξ|
Z
|ξ|−(N +1)+η dξ = γ|||f |||η;E rη .
≤ γ|||f |||η;E r
|ξ|>r
Combining this with (9.2) and (9.6) proves the proposition.
Remark 9.1 The proof shows that
sup
x,y∈RN
|vxi xj (x) − vxi xj (y)|
|f (x) − f (y)|
≤ γ sup
η
|x − y|
|x − y|η
x,y∈E
where the constant γ depends only on N and η and is independent of |E|.
Remark 9.2 The dependence on diam(E) on the right-hand side of (9.3)
enters only through (9.2) and (9.6). Therefore the constant γ in (9.4) depends
on diam(E) as
γ = γo (1 + diam(E))
for some γo = γo (N, m, η).
10 Potential Estimates in Lp (E)
The estimate (9.2) implies that A, as defined by (9.1), is a map from L∞ (E)
into L∞ (E). More precisely, it maps L∞ (E) into the subspace of the Lipschitz
continuous functions defined in E. It is natural to ask whether f ∈ Lp (E) for
some p ≥ 1 would imply that AF f ∈ Lq (E) for some q ≥ 1, and what is the
relation between p and q.
If f ∈ Coη (E), then AF f , as defined by (9.1), is differentiable and
Z
1
(x − y)
∇AF f (x) =
f (y)dy.
ωN E |x − y|N
76
2 THE LAPLACE EQUATION
If however f ∈ Lp (E) for some p ≥ 1, the symbol ∇AF f does not have
the classical meaning of derivative, and it is simply defined by its right-hand
side. If this is finite a.e. in E, we say that ∇AF f is the weak gradient of the
potential AF f . We will give sufficient integrability conditions on f to ensure
that the weak gradient ∇AF f is in Lq (E) for some q ≥ 1.
Both issues are addressed by investigating the integrability of the Riesz
potential
Z
f (y)
E ∋ x → wα (x) =
dy
for some α > 0.
(10.1)
|x
−
y|N −α
E
Proposition 10.1 Let f ∈ Lp (E) for some p ≥ 1. Then

Np


1,
if p <


N − αp





q
|wα | ∈ L (E) where q ∈ [1, ∞)
if p =








 [1, ∞]
if p >
N
α
N
α
(10.2)
N
.
α
Moreover, there exists a constant γ that can be determined a priori only in
terms of N , p, q, α, and diam(E), such that
kwα kq,E ≤ γkf kp,E .
(10.3)
The constant γ → ∞ as either p → N/α or as diam(E) → ∞.
Proof. Assume first p < N/α, and choose s > 1 from
1 1
1
=1+ − ,
s
q
p
1<s<
N
.
N −α
By Hölder’s inequality
Z q1
p
s
|x − y|(α−N )s |f |p |f |1− q |x − y|(α−N )(1− q ) dy
|wα (x)| =
E
≤
≤
Z
E
Z
E
(α−N )s
|x − y|
(α−N )s
|x − y|
p
|f | dy
p
|f | dy
q1 Z
1q
E
|x − y|
p(1− 1 )
kf kp,E s
q−s
(α−N ) q−1
Z
E
|f |
|x − y|
q−p
q−1
dy
(α−N )s
q−1
q
dy
s1 − 1q
.
The last integral involving |x − y|(α−N )s is estimated above by extending the
domain of integration to the ball of center x and radius diam(E). It gives
10 Potential Estimates in Lp (E)
Z
E
|x − y|(α−N )s dy ≤ ωN
Z
diam(E)
ρ(N −1)−(N −α)s dρ
0
N −α
ωN diam(E)N (1−s N
=
N 1 − s NN−α
provided
77
)
= γ(N, s)
1 1
α
N
i.e.,
− < .
N −α
p q
N
This determines the range of q in (10.1). In the estimates below, denote by γ
a generic positive constant that can be determined a priori only in terms of
N, p, q, and diam(E). To proceed, carry this estimate into the right-hand side
of the estimation of |wα (x)|, take the qth power of both sides and integrate
over E. Interchanging the order of integration with the aid of Fubini’s theorem
1<s<
pq(1− 1 )
|wα | dx ≤ γ
|x − y|
|f (y)| dydx kf kp,E s
E
ZE E
Z
pq(1− 1 )
p
(α−N )s
≤γ
|f (y)|
|x − y|
dx dy kf kp,E s
Z
q
Z Z
E
(α−N )s
p
E
≤ γkf kqp,E .
Let now p > N/α. Then by Hölder’s inequality
Z
p−1
p
p
(α−N ) p−1
.
|wα (x)| ≤ kf kp,E
|x − y|
dy
E
p
Corollary 10.1 Let f ∈ L (E) for some p > N . There exists a constant γ
depending only upon N, p and diam(E), such that
kvk∞,E + k∇vk∞,E ≤ γkf kp,E .
The constant γ → ∞ as p → N .
Let E a bounded open set in RN with boundary ∂E of class C 1 . For f ∈
Lp (∂E) set
Z
f (y)dσ(y)
vα (x) =
for α > 0
(10.4)
N −1−α
∂E |x − y|
where dσ is the Lebesgue surface measure on ∂E.
Corollary 10.2 Let f ∈ Lp (∂E) for some p ≥ 1. Then

(N − 1)p
N −1

1,

if p <


(N
−
1)
−
αp
α





N −1
|vα | ∈ Lq (E),
where q ∈ [1, ∞)
if p =

α







N −1
 [1, ∞]
.
if p >
α
(10.5)
78
2 THE LAPLACE EQUATION
Moreover, there exists a constant γ that can be determined a priori in terms
of N , p, q, α, and |∂E| only, such that
kvα kq,∂E ≤ γkf kp,∂E .
(10.6)
The constant γ → ∞ as either p → (N − 1)/α or as |∂E| → ∞.
11 Local Solutions
Consider formally local solutions of the Poisson equation
∆u = f,
(11.1)
in E
with no reference to possible boundary data on ∂E. Thus we assume that
m+η
(E) for some non-negative integer m and some
u ∈ C 2 (E) and f ∈ Cloc
η ∈ [0, 1). This means that |||f |||m,η;K < ∞ for every compact set K ⊂ E. Let
K ⊂ K ′ ⊂ E be such that dist(K; ∂K ′ ) > 0. We will derive estimates of the
norm C m+η (K) for u in terms of the norms kuk∞,K ′ and |||f |||m+η;K ′ .
Proposition 11.1 There exists a constant γ depending only upon N, m, and
dist(K; ∂K ′ ) such that
|||u|||m+2,η;K ≤ γ (|||f |||m,η;K ′ + kuk∞,K ′ ) .
Proof. It suffices to prove the proposition for m = 0, and for K ⊂ K ′ two
concentric balls Bσρ (xo ) ⊂ Bρ (xo ) ⊂ E, for some σ ∈ (0, 1). In such a case
the statement takes the following form.
Lemma 11.1 There exists a constant γ depending only upon N such that for
all Bσρ (xo ) ⊂ Bρ (xo ) ⊂ E
1
|||u|||2+η;Bσρ (xo ) ≤γ 1 +
|||f |||η;Bρ (xo )
(1 − σ)ρη
1
kuk∞,Bρ (xo ) .
+γ 1+
(1 − σ)N +2 ρ2+η
Proof (of Lemma 11.1). The point xo ∈ E being fixed, we may assume after a
translation that it coincides with the origin and write Bρ (0) = Bρ . Construct
a smooth non-negative cutoff function ζ ∈ Co∞ (E) such that
ζ = 1 in B (1+σ) ρ
2
and
|Dα ζ| ≤
C |α|
[(1 − σ)ρ]
|α|
(11.2)
for all multi-indices α of size |α| ≤ 2 and for some constant C. Multiplying
(11.1) by ζ and setting v = uζ, we find that v satisfies
∆v = f ζ + u∆ζ + 2∇u · ∇ζ,
v ∈ Co2 (Bρ ).
11 Local Solutions
79
Let N ≥ 3, the case N = 2 being similar. By the Stokes identity (2.3), we
may represent v as the superposition of the two Newtonian potentials
Z
1
V1 (x) = −
|x − y|2−N (f ζ)(y)dy
ωN (N − 2) RN
Z
1
V2 (x) = −
|x − y|2−N (u∆ζ + 2∇u · ∇ζ)(y)dy.
ωN (N − 2) RN
By virtue of Proposition 9.1 and Remark 9.2
|||V1 |||2,η;RN ≤ γ(1 + ρ)|||f ζ|||η;Bρ ≤ γ(1 + ρ + ρ−η )|||f |||η;Bρ .
We estimate V2 (·) within the ball Bσρ by rewriting it as
Z
1
V2 (x) =
u(y)|x − y|2−N ∆ζ dy
ωN (N − 2) RN
Z
2
u(y)∇|x − y|2−N · ∇ζ dy.
+
ωN (N − 2) RN
Since ∇ζ = 0 within the ball of radius (1+σ)
2 ρ, these integrals are not singular
for x ∈ Bσρ and we estimate
1
|||V2 |||2,η;Bσρ ≤ γ 1 +
kuk∞,Bρ .
(1 − σ)N +2 ρ2+η
11.1 Local Weak Solutions
η
The previous remarks imply that if f ∈ Cloc
(E), the classical local solution
of the Poisson equation (11.1) can be implicitly represented about any point
xo ∈ E as
Z
F (x; y)f ζdy
u(x) = −
Bρ (xo )
+
Z
Bρ (xo )
(11.3)
u(y) (F (x; y)∆ζ + 2∇F (x; y) · ∇ζ) dy
where F (·; ·) is the fundamental solution of the Laplace equation, introduced
in (2.6), and ζ satisfies (11.2). Consider now the various integrals in (11.3),
regardless of their derivation. The second is well defined for all x ∈ Bσρ (xo )
if u ∈ L1loc (E). The first defines a function
Z
F (x; y)f ζ dy ∈ L1loc (E)
x→
Bρ (xo )
Lploc (E)
provided f ∈
for some p ≥ 1. Since Bρ (xo ) ⊂ E is arbitrary, this
suggests the following
Definition: Let f ∈ Lploc (E) for some p ≥ 1. A function u ∈ L1loc (E) is a
weak solution to the Poisson equation (11.1) in E if it satisfies (11.3).
The estimates of Section 10 imply:
80
2 THE LAPLACE EQUATION
Proposition 11.2 Let f ∈ Lploc (E) for p > 1 and let u ∈ L1loc (E) be a local
weak solution of (11.1) in E. There exists a constant γ depending only on N
and p such that for all Bσρ (xo ) ⊂ Bρ (xo ) ⊂ E
N
k∇ukq,Bσρ (xo ) ≤ γkf kp,Bρ(xo ) + γ[(1 − σ)ρ]−(N +1) ρ q kuk1,Bρ (xo )
where
q∈


 1, N



N −p


[1, ∞)







[1, ∞]
if 1 ≤ p < N
if
p=N
if
p > N.
If p ∈ [1, N ], the constant γ → ∞ as q → N p/(N − p). Moreover, if p > N/2
N
kuk∞,Bσρ (xo ) ≤ γρ2− p kf kp,Bρ (xo ) + γ[(1 − σ)ρ]−N kuk1,Bρ (xo ) .
The constant γ → ∞ as p → N/2.
12 Inhomogeneous Problems
12.1 On the Notion of Green’s Function
Let E be a bounded domain in RN with boundary ∂E of class C 1 . The
construction of the Green’s function for E, introduced in (3.3), hinges on
solving the family of Dirichlet problems (3.1). These solutions y → Φ(x; y)
were required to be of class C 2 (Ē). Such a regularity has been used to justify
intermediate calculations, and it appears naturally in the explicit construction
of Green’s function for a ball.
However for each fixed x ∈ E, the Dirichlet problem (3.1) has a unique
solution if one merely requires that ∂E satisfies the barrier postulate and that
Φ(x; ·) ∈ C 2 (E) ∩ C(Ē)
for all fixed x ∈ E.
This is the content of Theorem 6.1 and the remarks of Section 7.
Proposition 12.1 Every bounded open set E ⊂ RN with boundary ∂E satisfying the barrier postulate admits a Green’s function G(·; ·). Moreover, for all
(x; y) ∈ E × E,
0 ≤ G(x; y) ≤ F (x; y)
0 ≤ G(x; y) ≤
diam(E)
1
ln
2π
|x − y|
for N ≥ 3
(12.1)
for N = 2.
12 Inhomogeneous Problems
81
Proof. For fixed x ∈ E, let ε > 0 be so small that Bε (x) ⊂ E. The function
G(x; ·) is harmonic in E − Bε (x), and it vanishes on ∂E. The number ε can be
chosen to be so small that G(x; ·) > 0 on ∂Bε (x). Therefore, by the maximum
principle, G(x; ·) ≥ 0 in E − Bε (x) and hence in E since ε is arbitrary. The
function Φ(x; ·) is harmonic in E, and by the maximum principle, it takes its
maximum and minimum values on ∂E. If N ≥ 3 it is positive on ∂E and thus
Φ(·; ·) > 0 in E. This proves the first of (12.1). If N = 2 rewrite the Green’s
function as
1
diam(E)
1
G(x; y) =
ln
− Φ(x; y) +
ln diam(E) .
2π
|x − y|
2π
For fixed x ∈ E, the function of y in [· · · ] is harmonic in E and non-negative
on ∂E.
Remark 12.1 The estimate roughly asserts that the singularity of G(·; ·) is
of the same nature as the singularity of the fundamental solution F (·; ·).
Corollary 12.1 G(x; ·) ∈ Lp (E) uniformly in x, for all p ∈ [1, NN−2 ).
12.2 Inhomogeneous Problems
Given f ∈ C η (Ē) for some η ∈ (0, 1), consider the boundary value problem
u ∈ C 2 (E) ∩ C(Ē),
∆u = f in E,
and
u
∂E
= 0.
(12.2)
Theorem 12.1. The boundary value problem (12.2) has a unique solution.
Proof (Uniqueness). If ui for i = 1, 2 solve (12.2), their difference is harmonic
in E and vanishes on ∂E. Thus it vanishes identically in E, by the maximum
principle.
Proof (Existence). Assume momentarily that (12.2) has a solution u ∈ C 2 (Ē)
and that the Green’s function G(x; ·) for E is of class C 2 (Ē). Then subtracting
Green’s identity (2.2) written for the pair of functions u and Φ(x; ·) from the
Stokes identity (2.3)–(2.3) gives
Z
G(x; y)f (y)dy
u(x) = −
Z
ZE
(12.3)
Φ(x; y)f (y)dy.
F (x; y)f (y)dy +
=−
E
E
This is a candidate for a solution of (12.2). To show that it is indeed a solution,
we have to show that it takes zero boundary data in the sense of continuous
functions in Ē, it is of class C 2 (E), and it satisfies the PDE. Fix x∗ ∈ ∂E and
write
82
2 THE LAPLACE EQUATION
lim u(x) = − lim
x→x∗
x→x∗
= − lim
x→x∗
− lim
x→x∗
Z
G(x; y)f (y)dy
ZE
G(x; y)f (y)dy
E∩Bε (x∗ )
Z
G(x; y)f (y)dy.
E−Bε (x∗ )
The second integral tends to zero by the property of the Green’s function,
since y is away from the singularity x = x∗ . The first integral is estimated by
means of (12.1), and it yields
Z
Z
G(x; y)|f (y)|dy ≤ 2 sup |f |
|F (x; y)|dy ≤ O(ε)
E
E∩Bε (x∗ )
E∩Bε (x∗ )
uniformly in x.
To establish in what sense the PDE is satisfied, we assume N ≥ 3, the arguments for N = 2 being similar.
12.3 The Case f ∈ Co∞ (E)
The set K = supp(f ) is a compact proper subset of E, and Φ(x; ·) is in C ∞ (E).
Therefore by symmetry
Z
Z
∆
Φ(x; y)f (y)dy =
∆y Φ(x; y)f (y)dy = 0.
E
E
Calculating the Laplacean of the first term on the right-hand side of (12.3)
gives
Z
Z
|x − y|2−N f (y)dy =
|ξ|2−N ∆ξ f (x − ξ)dξ
∆
N
R
E
Z
= lim
|ξ|2−N ∆ξ f (xξ)dξ.
ε→0
|ξ|>ε
Perform a double integration by parts on the last integral using that f is
compactly supported in RN . Taking into account that |ξ|2−N is harmonic in
|ξ| > ε, and proceeding as in the proof of the Stokes identity (2.3) gives
Z
1
|x − y|2−N f (y)dy = f (x).
∆
ωN (N − 2)
E
Combining these calculations shows that u defined by (12.3) satisfies the Poisson equation (12.2) in the classical sense.
1c Preliminaries
83
12.4 The Case f ∈ C η (Ē)
Let {Kj } be a family of nested compact subsets of E exhausting E. Construct
a sequence of functions {fj } ⊂ Co∞ (E) satisfying
|||fj |||η;Kj ≤ |||f |||η;E
and
lim |||fj − f |||η;K = 0
j→∞
for every compact subset K ⊂ E. Let uj be the unique classical solution of
uj ∈ C 2 (E) ∩ C(Ē),
∆uj = fj in E,
and
uj
∂E
= 0.
(12.4)
From the representation formula (12.3)
kuj k∞,E ≤ γkfj k∞;E ≤ γ|||f |||η;E ,
∀j ∈ N.
Combining this with Proposition 11.1 gives
|||uj |||2,η;K1 ≤ γ|||fj |||η;K1 ≤ γ|||f |||η;E
∀j ∈ N
where γ depends on N and dist{K1 ; ∂E} and is independent of j. By the
Ascoli–Arzelà theorem, we may select a subsequence {uj1 } out of {uj } converging in C 2,η (K1 ) to a function u1 ∈ C 2,η (K1 ). By the same process, we
may select a subsequence {uj2 } out of {uj1 } converging in C 2,η (K2 ) to a
function u2 ∈ C 2,η (K2 ), which coincides with u1 within K1 . Continuing this
2,η
(E) and a subsequence
diagonalization process, we obtain a function u ∈ Cloc
2,η
{uj ′ } out of the original sequence {uj } such that {uj ′ } → u in Cloc
(E). Letting j → ∞ in (12.4) along such a subsequence proves that u ∈ C 2,η (E) is a
classical solution of the PDE. To verify that u ∈ C(Ē) and that it vanishes
on ∂E in the sense of continuous functions, we have only to observe that u
satisfies (12.2) by the same limiting process.
Problems and Complements
1c Preliminaries
1.1c Newtonian Potentials on Ellipsoids
Compute the Newtonian potential generated by a uniform distribution of
masses, or charges, on the surface of an ellipsoid. Verify that such a potential
is constant inside the ellipsoid ([136], pages 22 and 193).
84
2 THE LAPLACE EQUATION
Theorem 1.1c. Let E be a bounded domain in RN of boundary ∂E of class
C 2 . The Newtonian potential V (·), generated by a uniform distribution of
masses on ∂E, is constant in E if and only if E is an ellipsoid.
The sufficient part of the theorem is due to Newton. The necessary part in
dimension N = 2 was established in 1931 by Dive [60]. The necessary part for
all N ≥ 2 has been recently established in [52]. The assumption of uniform
distribution cannot be removed as shown in [237].
1.2c Invariance Properties
1.4. Prove that the Laplacean is invariant under a unitary affine transformation of coordinates in RN .
1.5. Find all second-order rotation invariant operators of the type
L(u) =
N
P
aijhk uxi xj uxh xk .
i,j,h,k=1
1.6. Prove that ∆ is the only second-order, linear operator invariant under
orthogonal linear transformation of the coordinates axes.
1.7. Find all homogeneous harmonic polynomials of degree n in two and
three variables ([120]).
Hint: For N = 2 attempt z n and z̄ n , where z = x1 + ix2 and z̄ = x1 − ix2 .
For N = 3 attempt polynomials of the type z j Pn−j (|z|2 , x3 ), where Pn−j
is a polynomial of degree n − j in the variables |z|2 = x12 + x22 and x3 .
1.8. Let N = 2, and identify E with a portion of the complex plane C. Then
the real and imaginary part of a holomorphic function in E are harmonic
in E ([31], pages 124-125).
2c The Green and Stokes Identities
2.1. Prove that if u ∈ C 2 (E) ∩ C 1 (Ē) is harmonic in E, then it is locally
analytic in E. It will be a consequence of the estimates in Section 5 that
the hypothesis u ∈ C 1 (Ē), can be removed.
2.2. It follows from the Stokes identity (2.3)–(2.3) that if u ∈ C 2 (E) ∩ C 1 (Ē)
is harmonic in E, it can be represented as the sum of a single-layer, and
a double-layer potential.
2.3. Let ωN denote the surface area of the unit sphere in RN . Prove that for
N = 2, 3, . . . ,
Z π/2
ωN +1 = 2ωN
(sin t)N −1 dt.
0
3c Green’s Function and the Dirichlet Problem for the Ball
85
3c Green’s Function and the Dirichlet Problem for the
Ball
3.1. Prove that the Green’s function in (3.3) is non-negative.
3.2. Verify by direct calculation that the kernel in (3.9) is harmonic.
Hint : One needs to prove that for all y ∈ ∂BR
∆x
R2 − |x|2
= 0 in BR .
|x − y|N
From (3.8)
2|x|
2|x| ∂
|x − y|2−N =
(|x| − |y| cos θ)
2 − N ∂|x|
|x − y|N
|x|2 − |y|2 + |x − y|2
=
|x − y|N
and for y ∈ ∂BR
R2 − |x|2
2|x| ∂
= |x − y|2−N +
|x − y|2−N .
N
|x − y|
N − 2 ∂|x|
Therefore it suffices to show that the second term on the right-hand side
is harmonic.
3.3. Study the Dirichlet problem
0 if xN < 0
∆u = 0 in BR ,
and u|∂BR =
1 if xN ≥ 0.
Examine the behavior of the solution for xN = 0 near ∂BR .
3.4. Find Green’s function for the half-space xN > 0.
Hint: Set x̄ = (x1 , . . . , xN −1 ) and consider the reflection map analogous
to (3.5), i.e., ξ(x) = (x̄, −xN ).
3.5. Using the results of 3.4, discuss the solvability of the Dirichlet problem
∆u = 0 in RN −1 × [xN > 0]
u(x̄, 0) = ϕ(x̄) ∈ C(RN −1 ) ∩ L∞ (RN −1 ).
3.6. Construct Green’s function for the quadrant [x > 0] ∩ [y > 0] in R2 .
3.7. Find Green’s function for the half-ball in RN .
3.1c Separation of Variables
3.8. Solve the Dirichlet problem for the rectangle [0 < x < a]× [0 < y < b] in
R2 by looking for “separated” solutions of the form u(x, y) = X(x)Y (y).
Enforcing the PDE, derive the ODEs
X ′′ = (const)X,
Y ′′ = −(const)Y.
86
2 THE LAPLACE EQUATION
Superpose the families of solutions Xn , Yn of these ODEs, by writing
P
An ∈ R, n = 0, 1, 2, . . . ,
u = An Xn (x)Yn (y),
Finally, determine the coefficients An from the prescribed boundary data.
In the actual calculations, it is convenient to split the problem into the
sum of Dirichlet problems for each of which the data are zero on three
sides of the rectangle. For example, the problem
∆u = 0 in R = [0 < x < 1] × [0 < y < 1]
π
u(0, y) = cos y, u(x, 0) = 1 − x, u(1, y) = u(x, 1) = 0
2
can be solved by superposing the solutions of the two problems
π
∆u1 = 0 in R, and u1 (0, y) = cos y
2
u1 (x, 0) = u1 (1, y) = u1 (x, 1) = 0,
and
∆u2 = 0 in R,
and
u2 (x, 0) = 1 − x
u2 (1, y) = u2 (x, 1) = u2 (0, y) = 0.
Even though the boundary data are not continuous on ∂R, one might
solve formally for ui , i = 1, 2 and verify that u = u1 + u2 is indeed the
unique solution of the given problem.
3.9. Solve the Dirichlet problem for the disc x2 + y 2 < 1 by separation of
variables, and show that this produces the same solution as that obtained
by the Poisson formula (3.9).
Hint: Write ∆u in terms of polar coordinates (ρ, θ) to arrive at
1
1
uρρ + uρ + 2 uθθ = 0 in [0 < ρ < 1] × [0 ≤ θ < 2π].
ρ
ρ
To this equation apply the method of separation of variables.
3.10. Use a modification of this technique to solve the Dirichlet problem for
the annulus r < |x| < R in R2 .
3.11. Solve the Dirichlet problem for the rectangle with vertices A = (1, 0),
B = (2, 1), C = (1, 2), D = (0, 1).
4c Sub-Harmonic Functions and the Mean Value
Property
4.1. Prove that (4.2) implies (4.3). Hint: (4.2) implies
Z
ωN rN −1 u(xo ) ≤
u(y) dσ for all Br (xo ) ⊂ E.
∂Br (xo )
Integrate both sides in dr for r ∈ (0, R).
4c Sub-Harmonic Functions and the Mean Value Property
87
4.2. Prove that (4.4) and (4.5) are equivalent. Hint: Write (4.5) in the form
Z
ωN N
r u(xo ) =
u dy
N
Br (xo )
and take the derivative of both sides with respect to r.
4.3. Let u ∈ C 2 (E) satisfy ∆u = u in E. Prove that u has neither a positive
maximum nor a negative minimum in E.
4.4. Let u be harmonic in E. Prove that |∇u|2 is sub-harmonic in E.
4.1c Reflection and Harmonic Extension
4.5. Denote points in RN by x = (x̄, xN ) where x̄ = (x1 , . . . , xN −1 ), and let
u be the unique solution of
∆u = 0 in B1 ,
and
u
∂B1
= ϕ ∈ C(∂B1 ).
Prove that ϕ(x̄, xN ) = −ϕ(x̄, −xN ) implies u(x̄, xN ) = −u(x̄, −xN ).
4.6. Let u be harmonic in B1+ = B1 ∩ [xN > 0], and vanishing for xN = 0.
Extend it with a harmonic function in the whole of B1 .
4.7. Solve explicitly the Dirichlet problem
3
xN if xN > 0
∆u = 0 in B1+ ,
and u ∂B + =
0 if xN = 0.
1
4.2c The Weak Maximum Principle
Consider the formal differential operator
Lo = aij (x)
∂
∂2
+ bi (x)
∂xi xj
∂xi
(4.1c)
where aij , bi ∈ C(Ē) and the matrix (aij ) is symmetric and positive definite
in E, that is there exists λ > 0 such that aij (x)ξi ξj ≥ λ|ξ|2 for all ξ ∈ RN
and all x ∈ E.
Theorem 4.1c. Let u ∈ C 2 (E) ∩ C(Ē) satisfy Lo (u) ≥ 0 in E. Then
u(x) ≤ max u
∂E
for all x ∈ E.
Proof. Fix y ∈ RN − Ē, let γ be a constant to be chosen later, and consider
the function
2
v = u + εeγ|x−y|
for ε > 0.
It satisfies
88
2 THE LAPLACE EQUATION
Lo (v) ≥ 2γε [aij δij + 2γaij (x − y)i (x − y)j ] eγ|x−y|
2
2
+ 2γε [bi (x − y)i ] eγ|x−y|
2
≥ 2γε N λ + 2γλ|x − y|2 − B|x − y| eγ|x−y|
where B = max1≤i≤n maxĒ |bi |. By the Cauchy inequality
B2
N λ + 2γλ|x − y|2 − B|x − y| ≥ N λ + 2γλ −
|x − y|2 − N λ.
4N λ
Therefore γ can be chosen a priori dependent only upon B, N , and λ, such
that Lo (v) > 0 in E. If xo is an interior maximum for v, bi (xo )vxi (xo ) = 0
and
aij vxi xj x=x > 0 and vxi xj x=x ≤ 0.
(∗)
o
o
Next observe that aij vxi xj = trace (aij )(vxi xj ) . Using that (aij ) and (vxi xj )
are symmetric, and that the trace is invariant under orthogonal linear transformations, prove that
aij (xo )vxi xj (xo ) ≤ 0.
This and (∗) give a contradiction. Therefore
max v ≤ max v
∂E
Ē
and the theorem follows on letting ε → 0.
4.8. Let u ∈ C 2 (E) ∩ C(Ē) satisfy
L(u) ≡ Lo (u) + c(x)u ≥ 0
in E.
Prove that if c < 0, then u cannot have a positive maximum in the interior
of E. Give a counterexample to show that the assumption c < 0 cannot
be removed.
4.3c Sub-Harmonic Functions
4.9. Prove that v ∈ σ(E) if and only if for every open set E ′ ⊂ E and every
harmonic function u such that u ∂E ′ = v, then v ≤ u in E ′ .
4.10. Prove that x → ln |x| is sub-harmonic in RN − {0}.
4.11. Give examples of nondifferentiable sub-harmonic functions.
4.12. Let u ∈ C 2 (E) ∩ C(Ē) be a solution of
∆u = −1 in E,
and
u
∂E
Prove that for all xo ∈ E
u(xo ) ≥
1
inf |x − xo |2 .
2N x∈∂E
= 0.
5c Estimating Harmonic Functions
89
4.3.1c A More General Notion of Sub-Harmonic Functions
Certain questions in potential theory require a notion of sub-harmonic functions that does not assume continuity. First, by a real valued function in E
is meant u : E → [−∞, +∞). Thus u is defined everywhere in E and is permitted to take the “value” −∞. A real-valued function u : E → [−∞, +∞) is
upper semi-continuous if [u < s] is open for all s ∈ R.
Definition of F. Riesz ([214], see also [209]): Let E be a connected, open
subset of RN . A real valued function u : E → [−∞, +∞), is sub-harmonic in
E, if it is upper semi-continuous, and if for every compact subset K ⊂ E and
for every function H ∈ C(K) and harmonic in the interior of K
u
∂K
≤H
=⇒
∂K
u ≤ H in K.
4.13. Prove that this notion of upper semi-continuity is equivalent to
lim sup u(x) ≤ u(xo )
x→xo
for all xo ∈ E.
4.14. Prove that the function
u(x) =
ln |x|
−∞
if x 6= 0
if x = 0
is upper semi-continuous. Such a function is sub-harmonic in the sense of
F. Riesz, and it is not sub-harmonic in the sense of Section 4.
4.15. Prove that apart from the continuity requirement, Riesz notion of a
sub-harmonic function is equivalent to the notion of Section 4.
4.16. Let {un } be a decreasing sequence of sub-harmonic functions, in the
sense of F. Riesz and let
u(x) = lim un (x)
n→∞
∀ x ∈ E.
Prove that either u ≡ −∞ in E, or u is sub-harmonic in E, in the sense
of F. Riesz ([128], page 16).
5c Estimating Harmonic Functions
5.1. Let u be harmonic in E. Estimate the radius of convergence of its
Taylor’s series about xo ∈ E.
5.2. (Unique Continuation) Let u be harmonic in E and vanishing in
an open subset of E. Prove that if E is connected, u vanishes identically
in E. As a consequence, if u and v are harmonic in a connected domain
E and coincide in a open subset of E, then u ≡ v in E.
90
2 THE LAPLACE EQUATION
5.3. Find two harmonic functions in the unit ball that coincide on the set
[|x| < 12 ] ∩ [xN = 0].
N
5.4. (Phragmen–Lindelöf-Type Theorems) Let RN
+ = R ∩ [xN > 0],
N
and let u be a non-negative harmonic function in R+ . Prove that if u is
bounded and vanishes on the hyperplane xN = 0, then it is identically
zero.
Remark 5.1c The function u = xN shows that the assumption of u being
bounded cannot be removed. However, this is in some sense the only counterexample as shown by the following theorem of Serrin ([232]).
Theorem 5.1c. Let u be a non-negative harmonic function in RN
+ vanishing
for xN = 0. There exists a constant C depending only on N , such that
lim sup
|x|→∞
u(x)
≤ C.
|x|
5.1c Harnack-Type Estimates
5.5. Let u be harmonic in RN , and let G be its graph. If P ∈ G, denote by
πP be the tangent plane to G at P . Prove that [πP ∩ G] − P 6= ∅.
5.6. Prove that a non-negative harmonic function in a connected open set
E is either identically zero or strictly positive in E.
5.7. Let Q be the rectangle with vertices (0, 0), (nr, 0), (nr, 2r),
(0, 2r), for
some r > 0 and n ∈ N. Let Po = (r, r) and P∗ = (n − 1)r, r . Prove that
a non-negative harmonic function u in Q satisfies
2−2n u(Po ) ≤ u(P∗ ) ≤ 22n u(Po ).
5.8. Assume that the mixed boundary problem
u ∈ C 2 (B1 ) ∩ C 1 (B̄1 ), ∆u = −1 in B1
u = 0 in ∂B1 ∩ [xN > 0], ∇u · n = −u on ∂B1 ∩ [xN < 0]
has a unique solution. Prove that u ≥ 0 in B̄1 , and that u
∂B1 ∩[xN >0]
> 0.
5.2c Ill Posed Problems. An Example of Hadamard
The following problem is in general ill posed.
∆u = 0 in E = [0 < x < 1] × [0 < y < 1]
u(·, 0) = ϕ,
uy (·, 0) = ψ
Proposition 5.1c A solution of (∗) exists if and only if the function
Z
1 1
ψ(s) ln |x − s|ds
(0, 1) ∋ x → ϕ(x) −
π 0
is analytic.
(∗)
5c Estimating Harmonic Functions
91
Proof. If u solves (∗), write u = v + w, where
Z 1
1
v(x, y) =
ψ(s) ln[(x − s)2 + y 2 ]ds.
2π 0
Then ∆w = 0 in E and wy (·, 0) = 0. Therefore, by the reflection principle
(x, y) → w(x, |y|) is harmonic in [0 < x < 1] × [−1 < y < 1], and x → w(x, 0)
is analytic.
5.9. Prove that the following problem is ill posed.
∆u = 0 in [|x| < 1] × [0 < y < 1]
u(−1, ·) = u(·, 1) = u(1, ·) = 0
uy (·, 0) = 0, u(·, 0) = 1 − |x|.
5.3c Removable Singularities
Let xo ∈ E and let u be harmonic in E − {xo }. The function u is analytic
in E − {xo }, and it might be singular at xo . An example is the fundamental
solution F (·; xo ) of the Laplace equation with pole at xo . A point xo ∈ E is
a removable singularity for u if u can be extended continuously in xo so that
the resulting function is harmonic in the whole of E.
The pole xo is not a removable singularity for F (·; xo ). This suggests that
for a singularity at xo to be removable, the behavior of u near xo should be
better than that of F (·; xo ).
Theorem 5.2c. Assume that
lim
x→xo
u(x)
= 0.
F (x; xo )
(∗∗)
Then xo is a removable singularity.
Proof. Let v be the harmonic extension in the ball Bρ (xo ) of u ∂Bρ (xo ) . Such
an extension can be constructed by the Poisson formula (3.9) and Theorem 3.1.
The proof consists in showing that u = v in Bρ (xo ). Assume N ≥ 3, the proof
for N = 2 being similar. Consider the ball Bε (xo ) ⊂ Bρ (xo ) and set
Mε = ku − vk∞,∂Bε (xo ) .
By (∗∗), for every fixed η > 0, there exists εo ∈ (0, ρ) such that Mε ≤ ε2−N η
for all ε ≤ εo . The two functions
N −2
ε
±
w = Mε
± (u − v)
|x − xo |
are harmonic in the annulus ε < |x−xo | < ρ and non-negative for |x−xo | = ρ.
Moreover, on the sphere |x − xo | = ε
92
2 THE LAPLACE EQUATION
w±
|x−xo |=ε
= Mε ± (u − v)
|x−xo |=ε
≥ 0.
Therefore, by the maximum principle, for all ε < |x − xo | < ρ
|u − v|(x) ≤
Mε εN −2
η
≤
.
N
−2
|x − xo |
|x − xo |N −2
Theorem 5.3c. Assume that
lim |x − xo |N −1 ∇u ·
x→xo
x − xo
= 0.
|x − xo |
Then xo is a removable singularity.
Proof (Hint:). Let v be the harmonic extension of u|∂Bρ (xo ) into Bρ (xo ), and
for ε ∈ (0, ρ) set
x − xo
Dε = sup ∇(u − v) ·
.
|x
− xo |
∂Bρ (xo )
Introduce the two functions
w± =
εN −1 Dε
± (u − v)
(N − 2)|x − xo |N −2
and prove that a minimum for w± cannot occur on ∂Bε (xo ).
5.10. Prove that if
lim
x→xo
u(x)
=c
F (x; xo )
for some c ∈ R
then u = cF (·; xo ) + v, where v is harmonic in E.
5.11. The previous statements assume that u has a limit as x → xo . Prove
that if u is a non-negative harmonic function in the punctured ball B1 −
{0}, then the limit of u(x) as |x| → 0 exists, finite or infinite ([100]).
7c About the Exterior Sphere Condition
The counterexample of Lebesgue leads to a question that can be roughly
formulated as follows. How wide should the cusp in Figure 7.1 be, to ensure
the existence of solutions? The correct way of measuring “how wide” a cusp
should be is by means of the concept of capacity introduced by Wiener ([277]).
The capacity of a compact set K ⊂ RN is defined by
Z
cap(K) =
inf
|∇v|2 dx.
∞ (RN )
v∈Co
v≥1 on K
RN
8c Problems in Unbounded Domains
93
Such a definition can be extended to Borel sets. Now consider a domain E
whose boundary has a cusp pointing inside E as in Figure 7.1. Let x∗ be the
“vertex” of the cusp, and consider the compact sets
Kn = (RN − E) ∩ B̄2−n (x∗ )
n∈N
obtained by intersecting the region enclosed by the cusp, outside E, with balls
centered at x∗ and radius 2−n .
Theorem 7.1c (Wiener [278]). The following are equivalent:
(i). There existsP
a barrier H(x∗ ; ·) for the Dirichlet problem (6.2) at x∗
(ii). The series
cap(Kn ) is divergent.
For a theory of capacity and capacitable sets, see [136, 181, 154].
8c Problems in Unbounded Domains
8.1. Compute (8.5) for N = 2, 3 first. Then proceed by induction for all N .
Use 2.3 and
Z ∞
Z π/2
ρN −2
dρ =
(sin t)N −2 dt.
(1 + ρ2 )N/2
0
0
8.2. Having in mind 3.7 and the representation formula (3.2), justify the
definition (8.2) of the Poisson kernel for the half-space.
8.3. Give a solution formula for the Neumann problem in the half-space.
Discuss uniqueness.
8.4. Let E ⊂ RN be bounded, connected, and with boundary ∂E of class
C 1 . Prove that there exists at most one solution to the boundary value
problem in the exterior of E
u ∈ C 2 (RN − E) ∩ C(RN − E),
u
∂E
= ϕ ∈ C(∂E),
∆u = 0 in RN − E
lim u(x) = γ
|x|→∞
for a given constant γ.
8.1c The Dirichlet Problem Exterior to a Ball
Let N ≥ 3, set E = |x| > R for some R > 0, and consider the exterior problem
u
u ∈ C 2 (E) ∩ C(Ē),
|x|=R
for a given constant γ.
= ϕ ∈ C(∂E),
∆u = 0 in E
lim u(x) = γ
|x|→∞
(8.1c)
94
2 THE LAPLACE EQUATION
Step 1. First apply the Kelvin transform to map E into BR − {0}. Then
introduce the new unknown function
2 R
2−N
v(y) = |y|
u
y
y=
6 0.
|y|2
With the aid of Theorem 5.2c, verify that the singularity y = 0 is removable. Then, in terms of the new coordinates, (8.1c) becomes
v ∈ C 2 (BR ) ∩ C(B̄R ),
v
∂BR
=R
2−N
∆v = 0 in BR
ϕ ∈ C(∂BR ).
(8.2c)
(8.3c)
Step 2. Solve (8.2c) by means of Poisson formula (3.9). Return to the original coordinates x by inverting the Kelvin transform. In this process use
formula (3.6). To the function so obtained add a radial harmonic function
vanishing for |x| = R and satisfying the last of (8.1c). The solution is
N −2 Z
R
1
R2 − |x|2
u(x) = γ 1 −
−
dσ.
ϕ(y)
|x|
RωN ∂BR
|x − y|N
9c Schauder Estimates up to the Boundary ([222, 223])
Let u ∈ C 2 (E) ∩ C(Ē) be the unique solution of the Dirichlet problem (1.2).
If ∂E and the boundary datum ϕ are of class C 2,η , it is natural to expect that
u ∈ C 2,η (Ē). Prove the following3
Proposition 9.1c Let u ∈ C 2 (RN × R+ ) ∩ C(RN × R+ ) be the unique
bounded solution of the Dirichlet problem (8.1). If ϕ(x) ∈ Co2,η (RN ) then
u ∈ C 2,η (RN × R+ ) and there exists a constant γ depending only upon N , η
and the diameter of the support of ϕ such that
|||u|||2,η;RN ×R+ ≤ γ|||ϕ|||2,η;RN .
Moreover there exist a constant γ depending only upon N and η and independent of the support of ϕ such that
sup
x,y∈RN
xN +1 ,yN +1 ∈R+
|uxi xj (x, xN +1 ) − uxi xj (y, yN +1 )|
|ϕ(x) − ϕ(y)|
.
≤ γ sup
|x − y|η
[|x − y|2 + (xN +1 − yN +1 )2 ]η/2
N
x,y∈R
Proof (Hint:). Apply the same technique of proof of Theorem 9.1 to the Poisson integral (8.1).
The result of Theorem 9.1c is the key step in deriving C 2,η estimates up to the
boundary for solutions of the Dirichlet problem (1.2). The technique consist
of performing a local flattening of ∂E.
3
A version of these estimates is in [101, 150]. Their parabolic counterpart is in
[83, 151].
10c Potential Estimates in Lp (E)
95
10c Potential Estimates in Lp (E)
10.1. Compute the last integral in the proof of Proposition 10.1 by introducing polar coordinates. Prove that for α = 1, the constant γ in (10.3)
is
(p−1)/p
1
p−1
γ = 1/p
diam(E)2−N/p .
2p
−
N
ω (N − 2)
N
10.2. Verify that for N = 2 and α = 1, the constant γ in (10.3) is

1 p
if diam(E) ≤ 1
1
ep−1
γ=
1/p
p−1

(2π)
diam(E)2 p ln diam(E) if diam(E) > 1.
10.3. Prove the following
Corollary 10.1c Let E = BR and N ≥ 3. Then
1/p
Z
p
kAF f k∞,E ≤ γ(N, p, R) − |f (y)| dy
BR
where
p−1
p
p−1
2
.
γ(N, p, R) = 2R
N −2
2p − N
State and prove a similar corollary for the case N = 2.
10.4. In Proposition 10.1 the boundedness of E is essential. For α = 2, give
an example of f ∈ Lp (E) for some p > N/2 and E unbounded for which
AF f is unbounded.
10.5. If E is unbounded, the boundedness of AF f can be recovered by imposing on f a fast decay as |x| → ∞. Prove the following
2N
1/p 2−N/p
Lemma 10.1c Assume f ∈ Lp (E) for some p > N/2, and
|f (x)| ≤ C|x|−(2+ε)
for |x| > Ro
for given positive constants C, Ro , ε. Then AF f ∈ L∞ (E), and there
exists a constant γ depending only on N , C, Ro , ε such that
kAF f k∞,E ≤ γ(1 + kf kp,E ).
10.6. Give an example of AF f such that |∇AF f | ∈ L2 (E) and w2 ∈
/ L∞ (E).
10.1c Integrability of Riesz Potentials
The proof of Proposition 10.1 shows that the constant γ in (10.3) deteriorates
as q → N p/(N − αp)+ . However, (10.3) continues to hold also for the limiting
case
Np
q=
,
provided αp < N.
(N − αp)+
The proof for such a limiting case however, is rather delicate and is based on
Hardy’s inequality ([115], also in [50], Chapter VIII, §18).
96
2 THE LAPLACE EQUATION
10.2c Second Derivatives of Potentials
If f ∈ Coη (E), then AF f is twice continuously differentiable, and formally
Z
Fxi xj (x; y)f (y)dy.
(10.1c)
(AF f )xi xj (x) =
E
The integral is meant in the sense of the improper integral
Z
Z
Fxi xj (x; y)f (y)dy.
Fxi xj (x; y)f (y)dy = lim
E
ε→
E∩[|x−y|>ε]
The limit exists, since f ∈ Coη (E). However, if f ∈ Lp (E) for some p > 1, such
a representation loses its classical meaning. It is natural to ask, in analogy
with Proposition 10.1, whether one may use (10.1c) to define (AF f )xi xj in a
weak sense and whether such weak second derivatives are in Lq (E) for some
q ≥ 1. It turns out that
f ∈ Lp (E) =⇒ (AF f )xi xj ∈ Lp (E)
for 1 < p < ∞.
The proof of this fact cannot be constructed from (10.1) for α = 0, since
the latter would be a divergent integral even if f ∈ Co∞ (E). One has to rely
instead on cancellation properties of the kernel in (10.1c). These estimates are
due to Calderón and Zygmund ([27], see also [246]).
3
BOUNDARY VALUE PROBLEMS BY
DOUBLE LAYER POTENTIALS
1 The Double-Layer Potential
Let Σ be an (N − 1)-dimensional bounded surface of class C 1 in RN for
N ≥ 2, whose boundary ∂Σ is an (N − 2)-dimensional surface of class C 1 .
Fix xo ∈ RN − Σ̄ and consider the cone C(Σ, xo ) generated by half-lines
originating at xo and passing through points of ∂Σ. Let α(xo ) denote the
solid angle spanned by C(Σ, xo ), that is, the area of the portion of the unit
sphere centered at xo , cut by the cone. The double-layer potential generated
in xo by a distribution of dipoles identically equal to 1 on Σ is defined by
Z
Z
∂F (xo ; y)
−1
(xo − y) · n(y)
W (Σ, xo ) = −
dσ =
dσ.
(1.1)
∂n(y)
ω
|xo − y|N
N
Σ
Σ
Here n(·) is the unit normal to Σ exterior to the cone C(Σ, xo ), and F (·; ·) is
the fundamental solution of the Laplacean, introduced in (2.6) of Chapter 2.
...
..................
.... .. .......
.... .... .................
...
....
..
....
..
.
....... .......
.
.
..
.. ..........
....
..... ...
. .
.... ........................... ........
....
.
.
.
.
.
.. ....
..
... . .....
........ ..
....
.... .... .....
.........
............
.... . .
.
....
.
.
.
.
.
.
.
.
.
.. ...
.
.
... ...
.. ...
....
.... ....
....... ...
....
......
....... ....... ..... ... ....
.
.
.
.
.
.
.
.
.
.
... ...
.
......
.... ....
....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.....
...
....
.........
...... .....
.
.
.
.
.
.
.
.
.
.
.
.
.
. ..
.
... .
..
... .. .......
...... ......... ...... ... ..
....
....
.
.... . ..
... .............. ..........
.. ..............
.
.
.
.
.
.
.
.
.
.
.... ..
....................... ............ ...... .............
.
.
.
.
.
.... .
.. .......
... . ........ ....
..
.... ..... . ...... ....................
.......
..
...............
.... ....... ........
..............
... ........... ...... . ... ........
............
.... .... .......... .............
............
.
.
............................. ...........................
.
.
.
.
.
.
.
.
.
.
.
.. .. ...
..... .
.........
.... ...... ..... ............. ......... .....................
.......
.................... ..........
.........................
...........
........................... ..............
..........................................................
.
.
.
.
..................
...................
xo
Fig. 1.1
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_4
97
98
3 BVP BY DOUBLE LAYER POTENTIALS
The same cone is generated by infinitely many surfaces; however, the doublelayer potential depends only on xo and the solid angle α(xo ). This is the
content of the next proposition.
Proposition 1.1 Let Σ1 and Σ2 be any two surfaces generating the same
cone C(Σ, xo ). Then
W (Σ1 , xo ) = W (Σ2 , xo ) = W (Σ, xo ) =
α(xo )
.
ωN
Proof. Let E be the portion of the cone C(Σ1 , xo ) = C(Σ2 , xo ) included by
the surfaces Σ1 and Σ2 . Since xo is outside Ē
Z
Z
1
(xo − y) · n(y)
0=
∆y F (xo ; y)dy =
dσ
ωN ∂E−(Σ1 ∪Σ2 ) |xo − y|N
E
Z
Z
∂F (xo ; y)
∂F (xo ; y)
dσ −
dσ.
+
∂n(y)
∂n(y)
Σ2
Σ1
The first integral on the right-hand side vanishes since (xo − y) is tangent to
the cone and thus normal to n(y). Therefore W (Σ1 , xo ) = W (Σ2 , xo ).
Next, since W (Σ, xo ) is independent of Σ, we replace Σ with the portion
of the sphere ∂BR (xo ) cut by the cone, that is
Σo = ∂BR (xo ) ∩ C(Σ, xo )
for some R > 0.
The normal to Σo exterior to the cone is
n(y) =
y − xo
.
|y − xo |
This in (1.1) gives
W (Σ, xo ) =
1
ωN RN −1
Z
Σo
dσ =
α(xo )
.
ωN
In what follows E, is a bounded open set in RN with boundary ∂E of class
C 1,α for some α ∈ (0, 1]. The double-layer potential generated at a point
xo ∈ RN − ∂E by a continuous distribution of dipoles y → v(y) on ∂E is
Z
∂F (xo ; y)
dσ
v(y)
W (∂E, xo ; v) = −
∂n(y)
∂E
(1.2)
Z
−1
(xo − y) · n(y)
dσ.
=
v(y)
ωN ∂E
|xo − y|N
Proposition 1.2 In (1.2) let v = 1. Then
1 for xo ∈ E
W (∂E, xo ; 1) =
0 for xo ∈ RN − Ē.
(1.3)
2 On the Integral Defining the Double-Layer Potential
99
Proof. The first follows from the Stokes identity (2.3)–(2.4) of Chapter 2,
written for u = 1. If xo is outside Ē, the function y → F (xo ; y) is harmonic
in E. Therefore
Z
Z
∂F (xo ; y)
∆y F (xo ; y)dy =
dσ = 0.
∂n(y)
E
∂E
2 On the Integral Defining the Double-Layer Potential
As xo tends to a point x ∈ ∂E the integrand in (1.2), becomes singular. Such
a singularity however is integrable. This is the content of the next lemma.
Lemma 2.1 There exists a constant C depending only on N , α, and the
structure of ∂E such that
1
|(x − y) · n(x)|
≤C
|x − y|N
|x − y|N −1−α
for all x, y ∈ ∂E.
Proof. It will suffice to prove the lemma for all |x − y| < η for some η > 0.
Fix x ∈ ∂E and assume, after a translation, that it coincides with the origin.
Since ∂E is of class C 1,α , there exists η > 0 such that the portion of ∂E within
the ball Bη , centered at the origin, can be represented, in a local system of
coordinates, as the graph of a function ϕ satisfying
ξN = ϕ(ξ), ξ = (ξ1 , . . . , ξN −1 ), |ξ| < η
ϕ ∈ C 1,α (|ξ| < η), ϕ(0) = 0, |∇ϕ(0)| = 0
|∇ϕ(ξ)| ≤ Cϕ |ξ|α for a given positive constant Cϕ .
(2.1)
.
......
.......
...... ...
.
N
.
.
.
.
.......
........
....
........
...
....
.........
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
......................................................................................................................................................................................................................................................
..
...........
.........
........
η
......
.
.
.
.
.
.
......
......
y = (ξ, ξ )
n(0)...........................
B
ξ
IRN−1
Fig. 2.2
Then n(0) is the unit vector of the ξN -axis, exterior to E, and in the new
coordinates
|ξ| ≤ |y| ≤ (1 + k∇ϕk∞,Bη )|ξ|
and
| − y · n(0)| = |ϕ(ξ)| ≤ Cϕ |ξ|1+α .
Let x ∈ ∂E and for ε > 0 let Sε (x) = ∂E ∩ Bε (x) denote the portion of ∂E
within the ball Bε (x) centered at x and radius ε.
100
3 BVP BY DOUBLE LAYER POTENTIALS
Lemma 2.2 There exist constants C and εo , depending only on N , α, and
the structure of ∂E, such that for every ε ≤ εo
Z
|(z − y) · n(y)|
dσ ≤ C
|z − y|N
Sε (x)
uniformly for all z ∈ Bε (x).
Proof. Fix such a z ∈ Bε (x). Since ∂E is of class C 1+α , we may choose εo
so small that z has a unique projection, say zp , on ∂E. Set δ = |z − zp | and
compute
|z − y|2 = |zp − y|2 + δ 2 − 2(zp − y) · (zp − z).
Since zp ∈ ∂E, by Lemma 2.1
|(zp − y) · (zp − z)| = (zp − y) ·
zp − z
δ
|zp − z|
= |(zp − y) · n(zp )|δ ≤ C|zp − y|1+α δ.
Therefore
|z − y|2 ≥ |zp − y|2 + δ 2 − 2C|zp − y|1+α δ ≥ 41 (|zp − y| + δ)2
provided εo is chosen sufficiently small. Also
|(z − y) · n(y)| ≤ |(zp − y) · n(y)| + δ ≤ C|zp − y|1+α + δ.
Therefore
From this
|zp − y|1+α + δ
|(z − y) · n(y)|
≤C
.
N
|z − y|
(|zp − y| + δ)N
Z
Sε (x)
|(z − y) · n(y)|
dσ ≤
|z − y|N
Z
1
dσ
N −1−α
|z
−
y|
p
Sε (x)
Z
dσ
+ Cδ
.
(|z
−
y| + δ)N
p
Sε (x)
The first integral is convergent, since zp ∈ ∂E, and it is bounded above by
Z
1
dσ.
sup
N −1−α
|x
−
y|
x∈∂E ∂E
To estimate the second integral, first extend the integration to the larger set
S2ε (zp ). Then introduce a local system of coordinates with the origin at zp ,
so that S2ε (zp ) is represented as in (2.1). This gives
Z
Z
dσ
dξ
δ
≤
C
δ
ϕ
N
N
Sε (x) (|x − y| + δ)
|ξ|<2ε (|ξ| + δ)
Z 2ε
dr
≤ Cϕ′ δ
≤ Cϕ′′ .
(r
+
δ)2
0
3 The Jump Condition of W (∂E, xo ; v) Across ∂E
101
3 The Jump Condition of W (∂E, xo; v) Across ∂E
We will compute the limit of W (∂E, xo ; v) as xo → ∂E either from within E or
from outside Ē. Having fixed x ∈ ∂E, denote by {xi } ⊂ E a sequence of points
approaching x from the interior of E. Likewise, denote by {xe } ⊂ RN − Ē a
sequence of points approaching x from the exterior of Ē.
Proposition 3.1 Let v ∈ C(∂E). Then for all x ∈ ∂E
Z
1
1
(x − y) · n(y)
i
lim W (∂E, x ; v) = v(x) −
dσ
v(y)
2
ωN ∂E
|x − y|N
xi →x
Z
1
1
(x − y) · n(y)
lim W (∂E, xe ; v) = − v(x) −
dσ.
v(y)
xe →x
2
ωN ∂E
|x − y|N
Combining these limits gives the jump condition of the potential W (∂E, xo ; v)
across ∂E.
Corollary 3.1 Let v ∈ C(∂E). Then for all x ∈ ∂E
lim W (∂E, xi ; v) − lim
W (∂E, xe ; v) = v(x).
e
xi →x
x →x
Proof (of Proposition 3.1). For ε > 0, let Sε (x) = ∂E ∩ Bε (x) and write
Z
Z
(xi − y) · n(y)
(xi − y) · n(y)
dσ
=
v(x)
dσ
v(y)
i
N
|x − y|
|xi − y|N
Sε (x)
∂E
Z
(xi − y) · n(y)
+
[v(y) − v(x)]
dσ
|xi − y|N
Sε (x)
Z
(xi − y) · n(y)
dσ.
+
v(y)
|xi − y|N
∂E−Sε (x)
Choose ε ≤ εo , where εo is the number claimed by Lemma 2.2, and set
Z
(xi − y) · n(y)
dσ = α(ε, xi )
−
|xi − y|N
Sε (x)
where by Proposition 1.1, α(ε, xi ) is the solid angle of the cone generated by
the lines through xi and the points of ∂Sε (x). By Lemma 2.2
Z
(xi − y) · n(y)
[v(y) − v(x)]
dσ
|xi − y|N
Sε (x)
Z
(xi − y) · n(y)
dσ
≤ sup |v(y) − v(x)|
|xi − y|N
y∈Sε (x)
Sε (x)
≤ C sup |v(y) − v(x)|.
y∈Sε (x)
Therefore
102
3 BVP BY DOUBLE LAYER POTENTIALS
Z
∂E
v(y)
(xi − y) · n(y)
dσ = − v(x)α(ε, xi ) + O(ε)
|xi − y|N
Z
(xi − y) · n(y)
+
v(y)
dσ.
|xi − y|N
∂E−Sε (x)
Now let xi → x to obtain
α(ε; x)
− O(ε)
lim W (∂E, xi ; v) = v(x)
i
ωN
x →x
Z
(x − y) · n(y)
−
v(y)
dσ
|x − y|N
∂E−Sε (x)
(3.1)
where α(ε; x) is the solid angle of the cone generated by lines through x and
points of ∂Sε (x). As ε → 0, α(ε; x) → 12 ωN . To prove the proposition, we let
ε → 0 in (3.1) with the aid of Lemma 2.1.
IRN
E
..
..
..
...........
..
..........
................ ......................
..........
......
........
. .......
...... .
......
.....
. ..
.... . .
....
. . ..... .........
.... ..... .. . .
. . ... ...
.... ..
. . .. ....
....... .. . .
. . ...
. . ............
. .
.....
. . .. .
.......... . .
. . ... .... ........
..... ... .. . .
.
. . .
.... ............ . .
.
.
. . .... .... ........
.
.
.....
. . . ..
.. .. ...... . .
.....
.
.
.
.
. .... . ......
.....
.
.
.
..
. .. . . .
.....
....
..... . . .... .
. ..... . . .........
..... ...... ........
..... . . . .. ..... . .
. . . . . ......
......... ..........
.
. ...
.......
. .
. .
.
..
.. . . . . . ................................................................ . . . . ...
. ..
. .
. . . . ..... . . . .
. .
.. .
... . . . . . . . . . . . . . . . . . . . . .
.
. .
. ..
. .
. . . . . . . . . . .
. .
. .. . . . . . . . . . . . . . . . . . . ..... .
. ..... . .
. . . . . . . . . . .
. . ... .
. . . . . . . . . . . . . . . . . . . . ... .
. ..... . .
. . . . . . . . . . .
. .
. .
. .. . . . . . . . . . . . . . . . . . . ..... .
. . ... . .
. . . . . . . . . . .
. .
. .
. . .. . . . . . . . . . . . . . . . . ...... .
. . ... . .
. . . . . . . . . . .
. .
. .
. . ..... . . . . . . . . . . . . . . . . ..... . .
.
. .
.
. . . . . . . . . . .
.
. .
. . . ....... . . . . . . . . . . . . . . ....... . . .
. .
. .
. . . . . . . . . . .
.. .
. .
. . . ......... . . . . . . . . . . . . ....... . . . .
. .
. .
. . . . . . . . . . .
. .
. .
. . . . . ........... . . . . . . . . ............ . . . .
. .
. .
. .. . . . . . . . .. .
. .
. .
. . . . . . ..................... . . ..................... . . . . . .
. .
. .
. . . . .................... . . . .
. .
. .
. . . . . . . . . . . . . . . . . . . . . .
. .
. .
. . . . . . . . . . .
. .
. .
"; x)
x
(
E
B"
Fig. 3.3
Corollary 3.2 For all x ∈ ∂E
Z
1
1
(x − y) · n(y)
dσ = .
−
ωN ∂E |x − y|N
2
Proof. Apply (3.1) with v = 1 and use (1.3).
Remark 3.1 Combining this with Proposition 1.2, we conclude that the
double-layer potential x → W (∂E; x) generated by a constant distribution
of dipoles on ∂E, at points x ∈ RN , is a function that is discontinuous across
∂Ω, and its values are, up to a multiplicative constant

 1 for x ∈ E
(3.2)
W (∂E; x) = 21 for x ∈ ∂E

0 for x ∈ RN − Ē.
4 More on the Jump Condition Across ∂E
103
4 More on the Jump Condition Across ∂E
Fix x ∈ ∂E and denote by n(x) the outward unit normal at x. For xo ∈
RN − ∂E, set
Z
1
(xo − y) · n(x)
W̃ (∂E, xo ; v) = −
dσ
v(y)
ωN ∂E
|xo − y|N
Z
(4.1)
= ∇z
v(y)F (z; y)dσ · n(x)
.
∂E
z=xo
As xo → x the behavior of W̃ (∂E, xo ; v) is similar to that of the double-layer
potential W (∂E, xo ; v), provided xo approaches x along the normal n(x). Let
{xi } and {xe } denote sequences approaching x from the inside and respectively
outside of E, say for example
xi = x − δi n(x),
xe = x + δe n(x)
where {δi } and {δe } are sequences of positive numbers decreasing to zero.
Proposition 4.1 Let v ∈ C(∂E). Then for all x ∈ ∂E
Z
1
(x − y) · n(x)
1
dσ
v(y)
lim W̃ (∂E, xi ; v) = v(x) −
δi →0
2
ωN ∂E
|x − y|N
Z
1
(x − y) · n(x)
1
dσ.
v(y)
lim W̃ (∂E, xe ; v) = − v(x) −
δe →0
2
ωN ∂E
|x − y|N
Corollary 4.1 Let v ∈ C(∂E). Then for all x ∈ ∂E
lim W̃ (∂E, xi ; v) − lim
W̃ (∂E, xe ; v) = v(x).
e
xi →x
x →x
Proof (of Proposition 4.1). We prove only the first statement. Write
Z
1
(xi − y) · (n(y) − n(x))
lim W̃ (∂E, xi ; v) = lim
dσ
v(y)
δi →0
δi →0 ωN ∂E
|xi − y|N
Z
(xi − y) · n(y)
−1
dσ.
v(y)
+ lim
δi →0 ωN ∂E
|xi − y|N
The second limit is computed by means of Proposition 3.1, and equals
Z
1
1
(x − y) · n(y)
dσ.
v(x) −
v(y)
2
ωN ∂E
|x − y|N
To compute the first limit, set S2ε = ∂E ∩ B2ε (x) for 0 < ε ≪ 1, and write
Z
(xi − y) · (n(y) − n(x))
dσ
v(y)
|xi − y|N
∂E
Z
(xi − y) · (n(y) − n(x))
v(y)
=
dσ
|xi − y|N
∂E−B2ε (x)
Z
(xi − y) · (n(y) − n(x))
dσ.
v(y)
+
|xi − y|N
S2ε (x)
104
3 BVP BY DOUBLE LAYER POTENTIALS
Without loss of generality, we may assume that the sequence {xi } is contained
in Bε (x). Then for ε fixed
Z
(xi − y) · (n(y) − n(x))
lim
v(y)
dσ
δi →0 ∂E−B2ε (x)
|xi − y|N
Z
(x − y) · (n(y) − n(x))
=
v(y)
dσ.
|x − y|N
∂E−B2ε (x)
It remains to prove that
Z
(xi − y) · (n(y) − n(x))
dσ
lim
v(y)
δi →0 S (x)
|xi − y|N
2ε
Z
(x − y) · (n(y) − n(x))
=
v(y)
dσ.
|x − y|N
S2ε (x)
Compute
|xi − y|2 = |x − y|2 + δi2 + 2δi (x − y) · n(x).
By Lemma 2.1, since |x − y| < ε
|(x − y) · n(x)| ≤ const|x − y|1+α ≤ εα const|x − y|.
Therefore, if ε is chosen so small that εα const ≤ 1
|xi − y|2 ≥ |x − y|2 + δi2 − 2εα const|x − y|δi ≥
1
(|x − y| + δi )2 .
4
Next estimate
|(xi − y) · (n(y) − n(x))|
|x − y|α
|x − y|α
≤γ i
≤γ
i
N
N
−1
|x − y|
|x − y|
(|x − y| + δi )N −1
γ
≤
|x − y|N −1−α
for a constant γ depending only upon N and the structure of ∂E. Therefore
to compute the last limit, it suffices to pass to the limit under the integral
with the aid of the Lebesgue dominated convergence theorem.
5 The Dirichlet Problem by Integral Equations ([192])
Let E be a bounded domain in RN with boundary ∂E of class C 1,α for some
α ∈ (0, 1], and consider the Dirichlet problem (1.2) of Chapter 2. Seek a
solution of such a problem in the form of a double-layer potential
Z
−1
(x − y) · n(y)
u(x) =
dσ
(5.1)
v(y)
ωN ∂E
|x − y|N
6 The Neumann Problem by Integral Equations ([192])
105
for some unknown density v ∈ C(∂E). To impose the boundary data u = ϕ
on ∂Ω, let x tend to points of ∂E. Using Proposition 3.1, we conclude that
v(·) must satisfy the integral equation
Z
1
v=
KD (·; y)v(y)dσ + ϕ
(5.2)
2
∂E
where KD (·; ·) is the Dirichlet kernel
KD (x; y) =
1 (x − y) · n(y)
.
ωN |x − y|N
(5.3)
Proposition 5.1 Suppose that (5.2) has a solution v ∈ C(∂E). Then (5.1)
for such a v defines a solution of the Dirichlet problem (1.2) of Chapter 2.
Proof. The function u defined by (5.1) is harmonic in E, and by Proposition 3.1, it takes the boundary values ϕ on ∂E.
For v ∈ L∞ (∂E), set
AD v =
Z
KD (·; y)v(y)dσ.
(5.4)
∂E
Proposition 5.2 The function x → AD v(x) is continuous in ∂E.
Proof. Let {xn } be a sequence of points in ∂E converging to some xo ∈ ∂E.
First observe that {KD (xn ; ·)} → KD (xo ; ·) a.e. in ∂E. Then write
Z
lim |AD v(xn ) − AD v(xo )| ≤ lim
|KD (xn ; y) − KD (xo ; y)||v(y)|dy.
∂E
Pass to the limit under integral, with the aid of Lemma 2.1 (5.2. of the
Complements).
Corollary 5.1 Let {vn } be equi-bounded in L∞ (∂E). Then {AD vn } is equibounded and equi-continuous in ∂E (5.3. of the Problems and Complements).
6 The Neumann Problem by Integral Equations ([192])
Consider the Neumann problem (1.3) of Chapter 2 for a datum ψ ∈ C(∂E).
For x ∈ ∂E, such a datum is taken in the sense
lim ∇u(x − δn(x)) · n(x) = ψ(x).
δ→0
(6.1)
Seek solutions of the Neumann problem in the form of a single-layer potential
Z
F (·; y)v(y)dσ
(6.2)
u=
∂E
106
3 BVP BY DOUBLE LAYER POTENTIALS
where F (·; ·) is the fundamental solution of the Laplacean, introduced in (2.6)
of Chapter 2, and v(·) is an unknown surface density. To compute v, impose
the boundary condition in the sense of (6.1). First, for xi ∈ E and x ∈ ∂E,
compute
Z
1
(xi − y) · n(x)
∇u(xi ) · n(x) = −
dσ.
v(y)
ωN ∂E
|xi − y|N
Then take xi = x − δn(x) and take the limit as δ → 0 of the integral on the
right-hand side by making use of Proposition 4.1. We conclude that v(·) must
satisfy the integral equation
Z
1
v=
KN (·; y)v(y)dσ + ψ
(6.3)
2
∂E
where KN (·; ·) is the Neumann kernel
KN (x; y) =
1 (x − y) · n(x)
.
ωN |x − y|N
(6.4)
Proposition 6.1 Suppose that (6.3) has a solution v ∈ C(∂E). Then (6.2),
for such a v, defines a solution to the Neumann problem (1.3) of Chapter 2.
Proof. The function defined by (6.2) is harmonic in E and by the previous
calculations, satisfies the Neumann data on ∂E.
Theorem 6.1. A necessary and sufficient condition of solvability of (6.3) is
that ψ be of zero-average over ∂E, i.e., that (1.4) of Chapter 2 holds.
Proof (Necessity). Integrate (6.3) over ∂E in dσ(x). By Corollary 3.1
Z
Z
1
KD (y; x)dσ(x) = .
KN (x; y)dσ(x) = −
2
∂E
∂E
The sufficient part of the theorem will be proved in the next Chapter.
For v ∈ L∞ (∂E) set
Z
AN v =
KN (·; y)v(y)dσ.
(6.5)
∂E
Proposition 6.2 The function x → AN v(x) is continuous in ∂E.
Corollary 6.1 Let {vn } be equi-bounded in L∞ (∂E). Then {AN vn } is equibounded and equi-continuous in ∂E.
7 The Green’s Function for the Neumann Problem
107
7 The Green’s Function for the Neumann Problem
Consider the family of Neumann problems
N (x; ·) ∈ C 2 (E) ∩ C 1 (Ē) for all x ∈ E, ∆y N (x; y) = k in E
∂
∂
N (x; y) =
F (x; y), y ∈ ∂E
∂n(y)
∂n(y)
(7.1)
where F is the fundamental solution of the Laplacean, introduced in (2.6) of
Chapter 2, and k is a constant. Integrating the equation by parts and using
(1.3) gives
Z
∂F (x; y)
dσ = −1.
k |E| =
∂E ∂n(y)
Therefore, a necessary condition of solvability is k = −|E|−1 . Assuming for
the moment that (7.1) has a solution set
(x, y) → G(x; y) = F (x; y) − N (x; y) for x 6= y.
This is called Green’s function for the Neumann problem, and it satisfies
∆y G = −k in E,
∂
G(x; y) = 0 on ∂E;
∂n(y)
x 6= y.
(7.2)
Green’s function is not unique; indeed if G(x; ·), is a Green’s function, then
G(x; ·) + v(x)
for all v ∈ C 2 (E)
is still a Green’s function for the Neumann problem. Having determined one
such a function, say for example G1 (x; y), we let
Z
def
v = − − G1 (·; y)dy and G(x; y) = G1 (x; y) + v(x).
E
In this way, among all the possible Green’s functions for the Neumann problem, we have selected the one with zero-average for all x ∈ E. Such a selection
implies that G(·; ·) is symmetric. This is a particular case of the following
Lemma 7.1 Let G(·; ·) be a Green’s function for the Neumann problem satisfying
Z
x→
G(x; y)dy = const.
E
Then G(x; y) = G(y; x).
The proof is the same as in Lemma 3.1 of Chapter 2. From now on, we will
select G satisfying the zero-average property. Therefore, by symmetry
∆y G = ∆x G = −k
in E for x 6= y.
108
3 BVP BY DOUBLE LAYER POTENTIALS
Let u be a solution of the Neumann problem (1.3) of Chapter 2. From the
Stokes identity (2.3)–(2.4) of Chapter 2, subtract the Green’s identity (2.2),
of the same Chapter, written for u and N (x; ·). This gives
Z
Z
u dy.
ψG(·; y)dσ − k
u=
E
∂E
If u is a solution of the Neumann problem, then u + C are also solutions of
the same problem, for all constants C. Choosing
Z
C = −− udy
E
we select, among all solutions of the Neumann problem, the one with the
zero-average property and satisfying the representation
Z
(7.3)
ψG(·; y)dσ.
u=
∂E
This representation is a candidate for a solution of the Neumann problem. By
the symmetry of G(·; ·)
Z
Z
ψ∆x G(·; y) dσ = −k
∆x u =
ψ dσ = 0.
∂E
∂E
Thus the condition that ψ be of zero-average over ∂E is necessary for (7.3) to
define a harmonic function. It would remain to establish that the boundary
datum is taken in the sense of (6.1). This verification could be carried out
if one had an explicit expression for G(·; ·). This would be analogous to the
Dirichlet problem for the ball, where a verification of the boundary data was
possible via the explicit Poisson representation of Theorem 3.1 of Chapter 2.
Even though the method is elegant, the actual calculation of the Green’s
function G(·; ·) can be effected explicitly only for domains with a simple geometry such as balls or cubes (see Section 7c of the Complements).
7.1 Finding G(·; ·)
One might look for G(·; ·) of the form
G(x; y) = F (x; y) − γo |y|2 + h(x; y)
up to the addition of a function x → v(x). Here γo is a constant to be determined, and
h(x; ·) ∈ C 2 (E) ∩ C 1 (Ē) is harmonic for all x ∈ E.
Such a G(·; ·) satisfies the first of (7.2) for the choice 2N γo = k. Imposing the
boundary conditions on G(x; ·) implies that h(x; ·) must satisfy
8 Eigenvalue Problems for the Laplacean
∆h(x; ·) = 0 in E,
109
∂h(x; y)
∂F (x; y)
= 2γo y · n(y) −
on ∂E.
∂n(y)
∂n(y)
This family of Neumann problems can be solved by the method of integral
equations outlined in the previous section. Specifically, one looks for h(x; ·) in
the form of a single-layer potential
Z
h(x; y) =
v(x; η)F (y; η) dσ(η), x, y ∈ E
∂E
where the unknown x-dependent density distribution v(x; ·) satisfies the integral equation
Z
1
∂F (x; y)
1
(y − η) · n(y)
v(x; y) = 2γo y · n(y) −
+
dσ(η).
v(x; η)
2
∂n(y)
ωN ∂E
|y − η|N
This integral equation is solvable if and only if
Z
∂h(x; y)
dσ = 0 for all x ∈ E.
∂E ∂n(y)
This is part of the existence theory for such integral equations that will be
developed in the next chapter. To verify the zero-average condition, compute
Z
∂F (x; y)
dσ = −W (∂E, x; 1) = −1
∂E ∂n(y)
by Proposition 1.2. On the other hand, by the divergence theorem and the
indicated choices of γo and k
Z
Z
2γo
y · n(y)dσ = γo
div ∇|y|2 dx = 2N γo |E| = −1.
∂E
E
8 Eigenvalue Problems for the Laplacean
Consider the problem of finding λ ∈ R − {0} and a nontrivial u ∈ C 2 (E) ∩
C η (Ē), for some η ∈ (0, 1) satisfying
∆u = −λu in E,
and
u = 0 on ∂E.
(8.1)
This is the eigenvalue problem for the Laplacean with homogeneous Dirichlet
data on ∂E. If (8.1) has a nontrivial solution, by the results of Section 12 of
Chapter 2, such a solution u satisfies
Z
u=λ
G(·; y)u dy
(8.2)
E
where G(·; ·) is the Green’s function for the Laplacean in E. The nontrivial pair
(λ, u) represents an eigenvalue and an eigenfunction for the integral equation
(8.2). Conversely, if (8.2) has a nontrivial solution pair (λ, u), such that u ∈
C η (Ē), for some η ∈ (0, 1), then by the same procedure of Section 12 of
Chapter 2, such a u is also a solution of (8.1). We summarize
110
3 BVP BY DOUBLE LAYER POTENTIALS
Lemma 8.1 A nontrivial pair (λ, u) is a solution of (8.1) if and only if it
solves (8.2).
8.1 Compact Kernels Generated by Green’s Function
Let E be a bounded open set in RN with boundary ∂E of class C 1 and let
G(·; ·) be the Green’s function for the Laplacean in E. Set
Z
p
L (E) ∋ f → AG f =
(8.3)
G(·; y)f (y)dy for some p ≥ 1
E
provided the right-hand side defines a function in Lq (E) for some q ≥ 1.
Theorem 8.1. AG is a compact mapping in Lp (E) for all 1 ≤ p ≤ ∞, i.e.,
it maps bounded sets in Lp (E), into pre-compact sets in Lp (E).
Green’s function G is defined in (3.3) of Chapter 2, where F (·; ·) is the fundamental solution of the Laplace equation in RN , for N ≥ 2, defined in (2.6)
of Chapter 2, and Φ is introduced in (3.1) of the same Chapter. Setting
Z


F (·; y)f (y)dy
A
f
=


 F
E
Lp (E) ∋ f →
for some p ≥ 1
(8.4)
Z



A f =
Φ(·; y)f (y)dy
Φ
E
the proof reduces to showing that both AF and AΦ are compact in Lp (E).
9 Compactness of AF in Lp (E) for 1 ≤ p ≤ ∞
The operator AF maps bounded sets in Lp (E) into bounded sets of Lp (E), for
all 1 ≤ p ≤ ∞. This is content of Proposition 10.1 of Chapter 2. For p > N , the
compactness of AF follows from Corollary 10.1 of Chapter 2. Indeed, in such
a case, AF maps bounded sequences {fn } ⊂ Lp (E) into sequences {AF fn } of
equi-Lipschitz continuous functions in Ē. Therefore compactness follows from
the Ascoli–Arzelà theorem.
To establish compactness in Lp (E) for 1 ≤ p ≤ N , for a fixed vector
h ∈ RN , introduce the translation operator
v(· + h) if · +h ∈ E
p
L (E) ∋ v → Th v =
0
otherwise.
Also for δ > 0 set
Eδ = {x ∈ E dist(x, ∂E) > δ}.
(9.1)
The proof uses the following characterization of pre-compact subsets of Lp (E)
([50] Chapter 5, Section 22).
9 Compactness of AF in Lp (E) for 1 ≤ p ≤ ∞
111
Theorem 9.1. A bounded subset K ⊂ Lp (E), for 1 ≤ p < ∞ is pre-compact
in Lp (E) if and only if for every ε > 0 there exists δ > 0 such that for all
vectors h ∈ RN of length |h| < δ, and for all v ∈ K
kTh v − vkp,E < ε
kvkp,E−Eδ < ε.
and
(9.2)
To verify the assumptions of the theorem, set v = AF f and use Proposition 10.1 of Chapter 2 and (10.3) of the same Chapter, for α = 1. Fix δ > 0
so small that E − Eδ is not empty and let h ∈ RN be such that |h| < δ. Then
Z
Eδ
|v(x + h) − v(x)|dx ≤
Z
Eδ
≤ |h|
Z
Z
1
0
1
0
Z
d
v(x + th) dtdx
dt
Eδ
|∇v(x + th)|dxdt
1
≤ |h||E|1− q k∇AF f kq,E ≤ γ|h|kf kp,E
where p and q are as in the indicated proposition, including possibly the
limiting cases. Therefore for all σ ∈ (0, p1 )
Z
Eδ
|Th v − v|p dx =
≤
Z
Eδ
Z
|Th v − v|pσ+p(1−σ) dx
Eδ
pσ Z
|Th v − v|dx
Eδ
|Th v − v|
p(1−σ)
1−pσ
1−pσ
.
dx
Choose σ from
p(1 − σ)
=q∈
1 − pσ
p,
Np
(N − p)+
,
i.e.,
σ=
q−p
.
p(q − 1)
One verifies that such a choice is possible if q is in the range (10.2) of Chapter 2, for α = 1. Therefore
Z
p(1−σ)
|Th v − v|p dx ≤ γδ pσ kf kp,E .
Eδ
On the other hand
Z
E−Eδ
p
p
|Th v − v| dx ≤ 2 |E − Eδ |
≤ γ|E − Eδ |
q−p
q
q−p
q
Z
E
p/q
|v| dx
q
kf kpp,E .
Since ∂E is of class C 1 , there exists a constant γ depending on N and the
structure of ∂E such that |E − Eδ | ≤ γδ. Combining these estimates
kTh v − vkp,E = kTh AF f − AF f kp,E ≤ γδ
q−p
pq
(1 + kf kp,E ).
112
3 BVP BY DOUBLE LAYER POTENTIALS
10 Compactness of AΦ in Lp (E) for 1 ≤ p < ∞
Since Φ(x; ·) is harmonic in E and equals F (x; ·) on ∂E, possible singularities
of Φ(x; y) occur for x ∈ ∂E. Therefore, for δ > 0 fixed, Φ(·; y) ∈ C ∞ (Eδ )
uniformly in y ∈ Ē. If {fn } is a bounded sequence in Lp (E), then the sequence
{AΦ fn } is equi-bounded and equi-continuous in Ēδ . By the Ascoli–Arzelà
theorem, a subsequence can be selected, and relabeled with n, such that for
every ε > 0, there exists a positive integer n(ε) such that
kAΦ fn − AΦ fm k∞,Eδ ≤ ε
for n, m ≥ n(ε).
(10.1)
The selection of the subsequence depends on δ. Let {δj } be a sequence of
positive numbers decreasing to zero, and let {AΦ fnj } be the subsequence, out
of {AΦ fn }, for which (10.1) holds within Eδj . By diagonalization, one may
select a subsequence, and relabel it with n, such that {AΦ fn } is a Cauchy
sequence in C(Ēδj ) for each fixed j = 1, 2, . . . . We claim that {AΦ fn } is a
Cauchy sequence in Lp (E). Fix ε > 0 arbitrarily small and j ∈ N arbitrarily
large. There exists a positive integer m(ε, j) such that
kAΦ fn − AΦ fm k∞,Eδj ≤ ε
Next, for n, m ≥ m(ε, j)
Z
p
kAΦ fn − AΦ fm kp,E =
Eδj
for n, m ≥ m(ε, j).
p
|AΦ fn − AΦ fm | dx +
≤ εp |E| +
Z
E−Eδj
Z
E
Z
E−Eδj
|AΦ fn − AΦ fm |p dx
p
Φ(x; y)|fn (y) − fm (y)|dy dx.
Let us assume that N ≥ 3, the proof for N = 2 being similar. By Proposition 12.1 of Chapter 2 the last integral is majorized by
Z
E
Z
E
p/q
q
F (x; y)|fn (y) − fm (y)|dy dx
|E − Eδj |1−p/q
for some q > p.
11 Compactness of AΦ in L∞ (E)
Lemma 11.1 Let f ∈ L∞ (E). Then AΦ f is Hölder continuous in Ē. Namely,
there exist constants γ > 1 and 0 < η ≤ 1, that can be determined a priori in
terms of N and the structure of ∂E only such that
|AΦ f (x1 ) − AΦ f (x2 )| ≤ γkf k∞,E |x1 − x2 |η
for all x1 , x2 ∈ Ē.
11 Compactness of AΦ in L∞ (E)
113
Proof. The points xi ∈ Ē being fixed, set δ = |x1 − x2 |α , where α > 0 is to be
chosen, and denote by Bδ (x̄) the ball of radius δ centered at x̄ = 21 (x1 + x2 ).
For such a δ, let Eδ be defined as in (9.1), where without loss of generality we
may assume that δ is so small that E4δ 6= ∅. Assume first that Bδ (x̄) ⊂ E2δ .
Then
Z
AΦ f (x1 )−AΦ f (x2 ) =
[Φ(x1 ; y) − Φ(x2 ; y)]f (y)dy
E
Z Z 1
d
Φ(sx1 + (1 − s)x2 ; y)ds f (y)dy
=
E
0 ds
Z
Z 1
= (x1 − x2 ) ·
∇x
Φ(sx1 + (1 − s)x2 ; y)f (y)dy ds.
0
E
The function AΦ f is harmonic in B2δ (x̄), and sx1 + (1 − s)x2 ∈ Bδ (x̄) for all
s ∈ [0, 1]. Therefore by Theorem 5.2 of Chapter 2
Z
Z
γ
∇x
Φ(x; y)f (y)dy ≤
sup
Φ(x; y)f (y)dy
δ x∈B2δ (x̄) E
E
Z
γ
γ
F (x; y)dy ≤ kf k∞,E .
≤ kf k∞,E sup
δ
δ
x∈E E
Combining these estimates
|x1 − x2 |
δ
provided Bδ (x̄) ⊂ E2δ . Assume now that Bδ (x̄) ∩ (E − E2δ ) 6= ∅, and write
Z
AΦ f (x1 ) − AΦ f (x2 ) =
[Φ(x1 ; y) − Φ(x2 ; y)]f (y)dy
ZE
[Φ(x1 ; y) − Φ(x2 ; y)]f (y)dy
=
E4δ
Z
[Φ(x1 ; y) − Φ(x2 ; y)]f (y)dy = I1 + I2 .
+
|AΦ f (x1 ) − AΦ f (x2 )| ≤ γ̃kf k∞,E
E−E4δ
I1 = (x1 − x2 ) ·
= (x1 − x2 ) ·
Z
Z
1
∇x
0
1
0
Z
E4δ
Z
E4δ
Φ(sx1 + (1 − s)x2 ; y)f (y)dy ds
∇y Φ(sx1 + (1 − s)x2 ; y)f (y)dy ds
by symmetry of Φ(·; ·). Since y → Φ(sx1 + (1 − s)x2 ; y) is harmonic in E4δ , by
Theorem 5.2 of Chapter 2
γ
|∇Φ(sx1 + (1 − s)x2 ; y)| ≤
sup Φ(sx1 + (1 − s)x2 ; z)
δ z∈Bδ (y)
≤
γ
δ
sup F (sx1 + (1 − s)x2 ; z) ≤
z∈Bδ (y)
γ̂
δ N −1
.
114
3 BVP BY DOUBLE LAYER POTENTIALS
Therefore
|I1 | ≤ γ̃kf k∞,E
Next
|I2 | ≤
≤
Z
|x1 − x2 |
.
δ N −1
[Φ(x1 ; y) + Φ(x2 ; y)]|f (y)|dy
E−E4δ
Z
[F (x1 ; y) + F (x2 ; y)]|f (y)|dy
E−E4δ
≤ γkf k∞,E |E − E4δ |1/N ≤ γkf k∞,E δ 1/N .
Therefore if Bδ (x̄) ∩ (E − E2δ ) 6= ∅
|AΦ f (x1 ) − AΦ f (x2 )| ≤ γ̃
|x1 − x2 |
1/N
.
+
δ
δ N −1
The remaining cases are treated by inserting between x1 and x2 , finitely many
points {ξ1 , . . . , ξn }, so that each of the pairs (x1 , ξ1 ), (ξj , ξj+1 ), and (ξn , x2 )
falls in one of the previous cases. The number n will depend on the structure
of ∂E.
Corollary 11.1 AG is compact in L∞ (E).
Corollary 11.2 Let u ∈ L2 (E) be a solution of (8.2). Then u ∈ C η (Ē) for
some η > 0.
Proof. Applying Proposition 10.1 of Chapter 2 a finite number of times implies that u ∈ L∞ (E). Then the conclusion follows from Lemma 11.1 and
Corollary 10.1 of Chapter 2.
Remark 11.1 The corollary provides the necessary regularity for the eigenvalue problems (8.1) and (8.2) to be equivalent.
Problems and Complements
2c On the Integral Defining the Double-Layer Potential
2.1. Prove a sharper version of Lemma 2.1. In particular, find the optimal
conditions on ∂E to ensure that
(x − y) · n(y)
∈ L1 (∂E).
|x − y|N
5c The Dirichlet Problem by Integral Equations
115
2.2. Consider the integral in (2.1). As xo → x ∈ ∂E, the integrand tends to
(x − y) · n(y)
v(y)
|x − y|N
for a.e. y ∈ ∂E.
Moreover, such a function is in L1+ε (∂E) for some ε > 0. This follows
from Lemma 2.1. However, the limit cannot be carried under the integral.
Explain.
5c The Dirichlet Problem by Integral Equations
5.1. Let E ⊂ RN be open and bounded and with boundary ∂E of class C 1 .
Formulate the following exterior Dirichlet problem in terms of an integral
equation:
u ∈ C 2 (RN − Ē) ∩ C(RN − E), ∆u = 0 in RN − E
u ∂E = ϕ ∈ C(∂E),
lim u(x) = 0.
|x|→∞
Compare with Section 8.1c of the Problems and Complements of Chapter 2.
5.2. In the proof of Proposition 5.2, justify the passage of the limit under
the integral. Hint:
Z
|KD (xn ; y) − KD (xo ; y)||v(y)|dσ
∂E
Z
=
|KD (xn ; y) − KD (xo ; y)||v(y)|dσ
∂E∩[|y−xo |<ε]
Z
+
|KD (xn ; y) − KD (xo ; y)||v(y)|dσ.
∂E∩[|y−xo |≥ε]
5.3. Prove a stronger version of Proposition 5.2. Namely, if v ∈ L∞ (∂E),
then {AD v} is Hölder continuous with exponent α/N , where α is the
constant appearing in Lemma 2.1. Hint : For x1 , x2 ∈ ∂E
|AD v(x1 )−AD v(x2 )|
≤
kvk∞,∂E
ωN
Z
∂E
(x1 − y) · n(y) (x2 − y) · n(y)
dσ.
−
|x1 − y|N
|x2 − y|N
May assume that |x1 − x2 | < 1 and set
∂i = {y ∈ ∂E |xi − y| < |x1 − x2 |1/N }.
Divide the integral into one extended over ∂1 ∩∂2 and another one extended
over the complement.
116
3 BVP BY DOUBLE LAYER POTENTIALS
6c The Neumann Problem by Integral Equations
6.1. Let E ⊂ RN be open and bounded and with boundary ∂E of class C 1 .
Formulate the following exterior Neumann problem in terms of an integral
equation:
u ∈ C 2 (RN − Ē) ∩ C 1 (RN − E),
∆u = 0 in RN − E
∂
u
= ϕ ∈ C(∂E),
lim u(x) = 0.
∂n ∂E
|x|→∞
6.2. Prove a stronger version of Proposition 6.2. Namely, if v ∈ L∞ (∂E),
then {AN v} is Hölder continuous with exponent α/N , where α is the
constant appearing in Lemma 2.1.
7c The Green’s Function for the Neumann Problem
7.1c Constructing G(·; ·) for a Ball in R2 and R3
Attempt finding G(·; ·) of the form
G(x; y) = F (x; y) + Φ(x; y) + h(x; y) + γ|y|2
|
{z
}
−N (x;y)
where F and Φ are defined in (2.6) and (3.7) of Chapter 2 respectively, γ is
a constant, and h(x; ·) is a suitable harmonic function in BR to be chosen to
satisfy the last of (7.2). As in Section 7.1 one computes 2N γo |E| = −1. Using
the explicit expression of Φ
∂
1
(F + Φ + γ|y|2 ) =
+ 2γo R.
∂n(y)
ωN R|x − y|N −2
Therefore the harmonic function h(x; ·) has to be chosen to satisfy
−1
∂
h(x; y) =
− 2N γo R.
∂n(y)
ωN R|x − y|N −2
7.1.1c The Case N = 2
Choose h = 0 and γo = −1/4πR2 to conclude that Green’s function for the
Neumann problem for the disc DR ⊂ R2 is
−1
|x|
R2
1
2
G(x; y) =
ln |ξ − y|
+ ln |x − y| −
|y|
,
ξ
=
x
2π
R
4πR2
|x|2
up to an arbitrary additive smooth function of x.
8c Eigenvalue Problems
117
7.1.2c The Case N = 3
The function h(x; ·) is given by
h(x; y) =
1
ln H,
4π
H=
One computes
(ξ − y) · x
+ |ξ − y|,
|x|
−1
4π∇h =
H
and
x
ξ−y
+
|x| |ξ − y|
ξ=
R2
x.
|x|2
(∗)
x
2
1
ξ−y
x
ξ−y
·
− 2
+
+
H|ξ − y| H
|x| |ξ − y|
|x| |ξ − y|
2
x
ξ−y
2
−
·
= 0.
1+
=
H|ξ − y| H2
|ξ − y| |x|
4π∆h =
From (∗), taking into account that ξ = R2 x/|x|2
4π
∂h
y
= 4π∇h ·
∂n(y)
R
−1 x · y (ξ − y) · (y − ξ) (ξ − y) · ξ
=
+
+
RH |x|
|ξ − y|
|ξ − y|
x
x
(ξ − y) · ξ
1
(ξ − y) ·
+ |ξ − y| −
·ξ−
=
RH
|x|
|x|
|ξ − y|
i
h


x
+ |ξ − y| R2
(ξ − y) · |x|
1
1 
= 1 −
H−
.
=
RH
|ξ − y|
|x|
R |x − y|
The last equality follows from (3.6) of Chapter 2, since y ∈ ∂BR . We conclude
that Green’s function for the Neumann problem for the ball BR ⊂ R3 is
1
1
1
R
G(x; y) =
+
4π |x − y| |x| |ξ − y|
1
x
1
+
ln (ξ − y) ·
+ |ξ − y| −
|y|2
4π
|x|
8πR3
up to an arbitrary additive smooth function of x.
8c Eigenvalue Problems
8.1. Formulate the homogeneous Neumann eigenvalue problem
∆u = −λu in E,
∂
∂n u
= 0 on ∂E.
(8.1c)
8.2. Find eigenvalues and eigenfunctions of (8.1) and (8.1c), when E is a
parallelepiped in R3 of sides a, b, c. Do the same for a disc DR ⊂ R2 .
8.3. Let AG be defined as in (8.3) with G replaced by G. Prove the analogue
of Theorem 8.1.
4
INTEGRAL EQUATIONS AND
EIGENVALUE PROBLEMS
1 Kernels in L2 (E)
Let E be a bounded open set in RN with boundary ∂E of class C 1 . For
complex-valued f and g in L2 (E), set
Z
hf, gi =
f ḡ dx and kf k2 = hf, f i
E
and say that f and g are orthogonal if hf, gi = 0. A complex-valued dx × dxmeasurable function K(·; ·) defined in E × E is a kernel acting in L2 (E) if the
two operators
Z
Z
Af =
K(·; y)f (y)dy,
A∗ f =
K̄(y; ·)f (y)dy
E
E
map bounded subsets of L2 (E) into bounded subsets of L2 (E), equivalently,
if there is a constant γ depending only upon N and E such that
kAf k ≤ γkf k
and
kA∗ f k ≤ γkf k,
for all f ∈ L2 (E).
It would be sufficient to require only one of these, since any of them implies
the other. Indeed, assuming that the first holds
Z Z
|hA∗ f, gi| =
K̄(y; x)f (y)dy ḡ(x)dx
E
=
Z
E
E
Z
f (y)
K(y; x)g(x)dx dy = |hf, Agi| ≤ γkf kkgk.
E
∗
The operators A and A are adjoint in the sense that
hAf, gi = hf, A∗ gi
for all f, g ∈ L2 (E).
Their norm is defined by
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_5
119
120
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
kA∗ k = sup kA∗ f k.
kAk = sup kAf k,
kf k=1
kf k=1
By the characterization of the L2 (E)-norm ([50], Chapter V, Section 4)
Z Z
kAk = sup sup
K(x; y)f (y)dy ḡ(x)dx
kf k=1 kgk=1
= sup sup
kf k=1 kgk=1
E
Z
E
≤ sup sup kf k
kf k=1 kgk=1
Z
= sup
kgk=1
E
Z
E
Z
f (y)
K̄(x; y)g(x)dx dy
E
Z
E
Z
2
K̄(x; y)g(x)dx dy
E
2
K̄(x; y)g(x)dx dy
E
!1/2
!1/2
= kA∗ k.
A similar calculation gives kA∗ k ≤ kAk. Thus kAk = kA∗ k.
A kernel K(·; ·) in L2 (E) is compact if the resulting operators A and A∗
are compact in L2 (E), i.e., if they map bounded subsets of L2 (E) into precompact subsets of L2 (E). If A is compact, A∗ is also compact.
A kernel K(·; ·) in L2 (E) is symmetric if K(x; y) = K(y, x) for a.e. (x, y) ∈
E × E. If it is symmetric and real-valued, then A = A∗ .
1.1 Examples of Kernels in L2 (E)
Given two n-tuples {ϕ1 , . . . , ϕn } and {ψ1 , . . . , ψn } of linearly independent,
complex-valued functions in L2 (E), set
K(x; y) =
n
P
ϕi (x)ψ̄i (y)
i=1
for a.e. (x, y) ∈ E × E.
(1.1)
A kernel of this kind is called separable, or of finite rank, or degenerate. For
such a kernel for any f ∈ L2 (E)
Z
n
P
ϕi hf, ψi i.
K(·; y)f (y)dy =
E
i=1
Thus separable kernels are compact, but need not be symmetric. The Green
function G(·; ·) for the Laplacean with homogeneous Dirichlet data is a realvalued, symmetric, compact kernel in L2 (E) (see Theorem 8.1 of Chapter 3).
This last example shows that a kernel K(·; ·) in L2 (E) need not be in L2 (E ×
E).
1.1.1 Kernels in L2 (∂E)
If ∂E is of class C 1,α for some α > 0, one might consider complex-valued
kernels defined and measurable in ∂E × ∂E and introduce in a similar manner
2 Integral Equations in L2 (E)
121
integral operators A and A∗ for such kernels, where the integrals are over
∂E for the Lebesgue surface measure on it. Examples of such kernels are
the Dirichlet kernel KD introduced in (5.3) and the Neumann kernel KN
introduced in (6.4) of Chapter 3. The corresponding operators AD and AN
are introduced in (5.4) and (6.5) of the same Chapter. By Corollary 5.1 and
Corollary 6.1 they map L2 (∂E) into L2 (∂E). The kernels KD and KN are
real-valued but not symmetric.
2 Integral Equations in L2 (E)
A Fredholm integral equation of the second kind in L2 (E) is an expression of
the form ([80])
Z
K(·; y)u(y)dy + f
(2.1)
u=λ
E
where λ is a complex parameter, f is a complex-valued function in L2 (E), and
K(·; ·) is a complex-valued kernel in L2 (E). A solution of (2.1) is a complexvalued function u ∈ L2 (E) for which (2.1) holds a.e. in E.
To the integral equation (2.1) associate the homogeneous, and the adjoint
homogeneous, equations
Z
Z
V=λ
K(x; y)U(y)dy,
U =λ
K̄(y; x)V(y)dy.
(2.2)
E
E
The general solution of (2.1) is the sum of a particular solution and a solution
of the associated homogeneous equation.
Lemma 2.1 The integral equation (2.1) has at most one solution if and only
if U = 0 is the only solution of the associated homogeneous equation (2.2).
Denoting by I the identity operator, (2.1) can be rewritten in the operator
form
(I − λA)u = f.
(2.3)
2.1 Existence of Solutions for Small |λ|
Theorem 2.1. Let λ and K(·; ·) satisfy |λ|kAk < 1. Then for every f ∈
L2 (E), there exists a solution to the integral equation (2.1). The solution is
unique if the associated homogeneous equation (2.1) admits only the trivial
solution.
Proof. If |λ| is small, a first approximation to a possible solution u is uo = f .
Then progressively improve the approximation by setting
un = λAun−1 + f
n = 1, 2, . . . .
Set Ao = I and An = AAn−1 for n ∈ N, and estimate
(2.4)
122
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
kAn k = sup kAn f k = sup kA(An−1 f )k
kf k=1
n−1
kf k=1
kA(A
f )k n−1
kA
f k ≤ kAkkAn−1 k.
n−1 f k
kA
kf k=1
= sup
By iteration, kAn k ≤ kAkn for all n ∈ N. With this symbolism, the approximating solutions un take the explicit form
un =
n
P
λi Ai f.
(2.5)
i=0
From this, for every pair of positive integers n > m
kun − um k ≤ kf k
n
P
i=m+1
|λ|i kAki → 0
as m, n → ∞.
Therefore {un } is a Cauchy sequence in L2 (E) and we let u denote its limit.
Also
kAun − Auk ≤ kAkkun − uk → 0 as n → ∞.
Therefore {Aun } → Au in L2 (E). To prove the theorem, we let n → ∞ in
(2.4), in the sense of L2 (E).
P i
Motivated by the convergence of the series
|λ| kAki and by the formal
symbolism of (2.5), write
u=
P
def
λi Ai f = (I − λA)
−1
f.
(2.6)
The operator (I − λA)−1 : L2 (E) → L2 (E) is called the resolvent, and it
satisfies
(I − λA)(I − λA)−1 = (I − λA)−1 (I − λA) = I.
Since hAn f, gi = hf, A∗n gi for all f, g ∈ L2 (E), there also hold
h(I − λA)−1 f, gi = hf, (I − λ̄A∗ )−1 gi.
(2.7)
3 Separable Kernels
If the kernel K(·; ·) is of finite rank, as in (1.1), rewrite (2.1) in the form
Z
Z
n
n
P
P
ϕi (u − f )ψ̄i dy + λ
ϕi
u−f = λ
f ψ̄i dy.
(3.1)
i=1
E
i=1
E
The associated homogeneous and adjoint homogeneous equations are
Z
Z
n
n
P
P
ϕi
ψi
U =λ
U ψ̄i dy,
V = λ̄
V ϕ̄i dy.
i=1
E
i=1
E
(3.2)
3 Separable Kernels
123
3.1 Solving the Homogeneous Equations
Solutions of (3.2) are of the form U =
n
P
i=1
wi ϕi and V =
n
P
w̃i ψi where the
i=1
numbers wi = hU, ψi i and w̃i = hV, ϕi i are to be determined. Putting this
form of U into the first of (3.2) gives
Z n
n
n
P
P
P
ϕi
wi ϕi = λ
wj ϕj (y) ψ̄i (y)dy
i=1
i=1
=λ
n
P
E
n
P
i=1 j=1
j=1
wj ϕi
Z
ϕj ψ̄i dy = λ
n
P
i=1
E
n
P
aij wj ϕi
j=1
where aij = hϕj , ψi i. Since the set of functions {ϕi }1n is linearly independent,
this leads to the linear system
w = (w1 , . . . , wn )t .
[I − λ(aij )]w = 0,
(3.3)
Analogously, putting the form of V into the second of (3.2) and taking into
account that the set {ψi }n1 is linearly independent leads to the linear system
w̃ = (w̃1 , . . . , w̃n )t .
[I − λ̄(āji )]w̃ = 0,
(3.3)∗
Let r be the rank of [I − λ(aij )]. If r = n then det[I − λ(aij )] 6= 0 and (3.3)–
(3.3)∗ have only the trivial solution. Otherwise, the systems have n−r linearly
independent solutions, say wj and w̃j for j = 1, . . . , n − r, and (3.2) have,
respectively, the n − r solutions
Uj =
n
P
Vj =
wij ϕi ,
i=1
n
P
w̃ij ψi ,
i=1
j = 1, . . . , n − r.
3.2 Solving the Inhomogeneous Equation
Solutions to (3.1) are of the form u − f =
n
P
vi ϕi where the complex numbers
i=1
vi are to be determined from (3.1). Putting this in (3.1), and setting fi =
hf, ψi i, yields the linear system
[I − λ(aij )]v = λf .
If det[I − λ(aij )] 6= 0, then for every f ∈ RN , this system admits a unique
solution. Otherwise, the system is solvable if and only if f is orthogonal to
the (n − r)-dimensional subspace spanned by the solutions of (3.3)∗ , that is,
if and only if
Z
Z
n
n
P
P
f
w̃ij ψi dy
0 = f · w̃j =
f w̃ij ψ̄i dy =
i=1 E
E i=1
Z
=
f V̄j dy = hf, Vj i,
j = 1, . . . , n − r.
E
124
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Theorem 3.1. Let K(·; ·) be separable. Then the integral equation (2.1) is
solvable if and only if f is orthogonal to all the solutions of the adjoint homogeneous equation (2.2). In particular, if λ is not an eigenvalue of the matrix
(aij ), then (2.1) is uniquely solvable for every f ∈ L2 (E).
More generally, one might consider separable kernels of the type
Ksep (·; ·) =
where
λ → aij (λ) =
n
P
i=1
Z
ϕi (·; λ)ψ i (·; λ)
ϕi (y; λ)ψ̄j (y; λ)dy
(3.4)
(3.5)
E
are analytic functions of λ in the complex plane C. Then det[I − λ aij (λ) ]
can vanish only at isolated points of C. Therefore, for such kernels, (2.1) is
uniquely solvable for every f ∈ L2 (E) except for isolated values of λ in C.
Remark 3.1 The theorem, due to Fredholm, discriminates between those
values of λ that are eigenvalues of (aij ) and the remaining ones. For this
reason it is referred to as the Fredholm alternative ([79, 80], see also Mikhlin
[183] and Tricomi [260]).
4 Small Perturbations of Separable Kernels
Consider the integral equation (2.1) for kernels of the form
K(·; ·) = Ksep (·; ·) + Ko (·; ·)
(4.1)
where Ksep (·; ·) is separable and Ko (·; ·) is a kernel in L2 (E). Setting
Z
Z
Ko (·; y)f (y)dy,
A∗o f =
Ao f =
K̄o (y; ·)f (y)dy
E
E
the perturbation Ko (·; ·) is said to be small if
|λ|kAo k < 1
and
|λ|kA∗o k < 1.
(4.2)
This implies that the resolvent (I − λAo )−1 is well defined in the sense of (2.6)
and permits one to rewrite (2.1) in the form
Z
Ksep (·; y)u dy + f
(I − λAo )u = λ
E
Z
n
P
−1
ψ̄i (I − λAo ) (I − λAo )u dy + f.
=λ
ϕi
i=1
E
Set z = (I − λAo )u and observe that by (2.7)
4 Small Perturbations of Separable Kernels
Z
E
ψ̄i (I − λAo )−1 (I − λAo )u dy =
Z
−1
(I − λ̄A∗o )
E
ψi z dy.
Therefore, solving the integral equation is equivalent to solving
Z
n
P
−1
(I − λ̄A∗o ) ψi z dy + f.
ϕi
z=λ
i=1
125
(4.3)
E
This, in turn, has the associated adjoint homogeneous equation
Z
n
P
−1
(I − λ̄A∗o ) ψi
V = λ̄
ϕ̄i V dy
i=1
(4.4)
E
which can be rewritten as
(I − λ̄A∗o )V = λ̄
or equivalently
V = λ̄
n
P
i=1
ψi
Z
E
ϕ̄i V dy + λ̄
Z
E
n
P
i=1
ψi
Z
E
ϕ̄i V dy
K̄o (y; ·)V dy = λ̄
Z
E
K̄(y; ·)V dy.
This is precisely the adjoint homogeneous equation in (2.2) associated with
the original kernel K(·; ·). Thus V ∈ L2 (E) is a solution of (4.4) if and only if
it is a solution of the adjoint homogeneous equation
Z
∗
V = λ̄A V,
Af =
K(·; y)f dy
(4.5)
E
associated with the original kernel.
4.1 Existence and Uniqueness of Solutions
By Theorem 3.1, the integral equation (4.3) is solvable if and only if f is
orthogonal to all the solutions of the adjoint homogeneous equation (4.4), that
is, if and only if f is orthogonal to all the solutions of the adjoint homogeneous
equation (4.5). The solution of (4.3) is unique if the associated homogeneous
equation
Z
n
P
ϕi
v=λ
v ψ̄i dy + λAo v
(4.6)
i=1
E
admits only the trivial solution. We may regard this as an integral equation
with separable kernel and with forcing term f = λAo v. By Theorem 3.1, such
an equation has at most one solution if λ is not an eigenvalue of the matrix
(aij ). In such a case, since v = 0 solves (4.6), it must be the only solution.
Recalling that λ is restricted by the condition (4.2), we conclude that if λ is
not an eigenvalue of (aij ) satisfying |λ| < kAo k−1 , then (4.6) has only the
trivial solution, and consequently (4.3) has at most one solution.
126
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Analogously, if λ is not an eigenvalue of (aij ) satisfying |λ| < kAo k−1 ,
then (4.5) has only the trivial solution. Therefore any f ∈ L2 (E) would be
orthogonal to all the solutions of the adjoint homogeneous equation (4.5), and
therefore (4.3) is uniquely solvable for every f ∈ L2 (E).
Theorem 4.1. If the kernel K(·; ·) is a small perturbation of a separable kernel, in the sense of (4.1)–(4.2), the integral equation (2.1) is solvable if and
only if f is orthogonal to all the solutions of the adjoint homogeneous equation
(4.5). In particular, if λ is not an eigenvalue of (aij ) such that |λ| < kAo k−1 ,
then (2.1) is uniquely solvable for every f ∈ L2 (E).
More generally, one might consider kernels that are small perturbations of
a separable kernel of the type of (3.4), with the corresponding functions in
(3.5) analytic in the disc D = {|λ| < kAo k−1 } ⊂ C. Then det[I − λ aij (λ) ]
can vanish only at isolated points of D. Therefore, for such kernels, (2.1) is
uniquely solvable for every f ∈ L2 (E) except for isolated values of λ in D.
5 Almost Separable Kernels and Compactness
A kernel K(·; ·) in L2 (E) is almost separable if for all ε > 0 it can be decomposed as in (4.1) with Ksep (·; ·) separable and a small perturbation Ko (·; ·)
such that
Z
kAo k = sup
kf k=1
E
Ko (·; y)f dy ≤ ε.
(5.1)
The perturbation kernel Ko (·; ·) is not required to be compact. It turns out
however that the original kernel K(·; ·) is compact.
Proposition 5.1 (F. Riesz [213]) An almost separable kernel K(·; ·) in
L2 (E) is compact.
Proof. Let A be the operator generated by K(·; ·). It will suffice to prove that
every bounded sequence {fn } ⊂ L2 (E) contains a subsequence {fn′ } ⊂ {fn }
such that {Afn′ } is Cauchy in L2 (E). For j ∈ N, let Asep,j and Ao,j be the
operators corresponding to the decomposition (4.1) and (5.1) for the choice
ε = 1/j. Since Asep,j are compact, select by diagonalization, a subsequence
{fn′ } ⊂ {fn } such that {Asep,j fn′ } is convergent for all j ∈ N. Then
kAfn′ − Afm′ k < kAsep,j fn′ − Asep,j fm′ k + 2kAo,j k sup kfn k.
n
Corollary 5.1 Let A : L2 (E) → L2 (E) be such that for all ε > 0 it can be
decomposed into the sum of a compact operator and a “small perturbation” Aε
of norm kAε k < ε. Then A is compact.
5 Almost Separable Kernels and Compactness
127
5.1 Solving Integral Equations for Almost Separable Kernels
It follows from Theorem 4.1, and the remarks following it, that (2.1) for an
almost separable kernel can always be solved for every f ∈ L2 (E) except at
most finitely many values of λ within the disc [|z| < ε−1 ] ⊂ C. Since ε > 0 is
arbitrary, we conclude that solvability is ensured for all f ∈ L2 (E), except for
countably many complex numbers {λn }. Every disc |z| < ε−1 of the complex
plane contains at most finitely many such exceptional values of λ. Therefore
if the sequence {λn } is infinite, then {|λn |} → ∞. Equivalently, for all ε > 0,
the adjoint homogeneous equation associated to such almost separable kernels
has only the trivial solution, except for finitely many values of λ within the
disc [|z| < ε−1 ] ⊂ C. These are the eigenvalues of the adjoint homogeneous
equation. Since ε > 0 is arbitrary, we conclude:
Theorem 5.1. The integral equation (2.1) with almost separable kernel K(·; ·)
is solvable if and only if f is orthogonal to all the solutions of the associated
adjoint homogeneous equation in (2.2). Moreover, it is always uniquely solvable for every f ∈ L2 (E) except for countably many values of λ. These are
the eigenvalues of the associated adjoint homogeneous equation (2.2). If the
sequence {λn } is infinite, then {|λn |} → ∞.
5.2 Potential Kernels Are Almost Separable
A potential kernel is a measurable function K(·; ·) : E × E → R such that
|K(x; y)| ≤ C|x − y|−N +α
for almost all (x, y) ∈ E × E
(5.2)
for some positive constants C and α. For δ > 0 let
 −N +α
if K(x, y) ≥ δ −N +α
 δ
Kδ (x; y) =
K(x; y)
if |K(x; y)| < δ −N +α
 −N +α
−δ
if K(x, y) ≤ −δ −N +α .
Since Kδ (·; ·) is uniformly continuous in Ē× Ē, by the Weierstrass theorem, for
each ε > 0 there exists a polynomial Pn (x; y) in the 2N variables (x, y) ∈ Ē×Ē
such that
ε
kKδ − Pn k∞,E×E ≤ p .
2 |E|
Writing K = Pn + (K − Pn ), the perturbation Ko = K − Pn satisfies (5.1).
Indeed, for all f ∈ L2 (E) of norm kf k = 1
Z
Z
Z
|K(·; y) − Kδ (·; y)||f |dy
|Kδ (·; y) − Pn (·; y)||f |dy +
|Ko (·; y)f |dy ≤
E
E
E
Z
ε
|x − y|−N +α |f |dy.
≤ +C
1
2
|x−y|<C N −α δ
It remains to choose δ so small that the L2 (E) norm of the last integral is less
than ε/2. This is possible by virtue of Proposition 10.1 of Chapter 2.
128
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Remark 5.1 Analogous considerations can be carried almost verbatim for
integral equations set on ∂E such as (5.2) and (6.3) of Chapter 3. By virtue
of Lemma 2.1 of that Chapter, the Dirichlet and Neumann kernels KD and
KN , introduced in (5.3) and (6.4) of Chapter 3 respectively, are potential,
almost separable kernels in ∂E×∂E. For these integral equations, Theorem 5.1
continues to hold with the proper modifications.
The idea of approximating potential kernels by separable ones, appears in E.
Schmidt ([227]) and J. Radon ([210]).
6 Applications to the Neumann Problem
Solving the Neumann problem (1.3) of Chapter 2 is equivalent to solving the
integral equation (6.3) of Chapter 3. The latter is in turn solvable if and only
if the Neumann datum ψ is orthogonal, in the sense of L2 (∂E), to all the
solutions of the adjoint, homogeneous equation
Z
Z
1
1
(x − y) · n(y)
V(x) =
KN (y; x)V dσ = −
V dσ
2
ω
|x − y|N
N
∂E
∂E
(6.1)
Z
Z
∂F (x; y)
V dσ.
KD (x; y)V dσ = −
=−
∂n
∂E
∂E
By Corollary 3.2 of Chapter 3, this is solved by V = const. If the constants
are the only solutions, then the zero-average condition (1.4) of Chapter 2 on
the Neumann datum ψ would imply that such a ψ is orthogonal to all the
solutions of the homogeneous, adjoint equation (6.1) and thus would provide
the characteristic condition of solvability of the Neumann problem. Therefore
the sufficient part of Theorem 6.1 of Chapter 3 will be a consequence of the
following
Proposition 6.1 The integral equation (6.1) admits only the constant solutions.
Proof. Solutions V ∈ L2 (∂E) of (6.1) are continuous in ∂E. Indeed, applying
Corollary 10.2 of Chapter 2 a finite number of times, one finds V ∈ L∞ (∂E).
Then V ∈ C(∂E), by Proposition 5.2 of Chapter 3. Let v be the harmonic
extension of V in E and denote by ϕ the resulting normal derivative of such
an extension on ∂E, that is
∆v = 0 in E,
v = V,
and
def ∂v
ϕ =
∂n
on ∂E.
By the Stokes representation formula (2.7) of Chapter 2
Z
Z
∂F (x; y)
V
E ∋ x → v(x) = −
ϕF (x; y) dσ.
dσ +
∂n
∂E
∂E
7 The Eigenvalue Problem
129
Letting x → ∂E and using Proposition 3.1 of Chapter 3, gives
Z
Z
1
V= V−
KD (·; y)V dσ +
ϕF (·; y) dσ.
2
∂E
∂E
Therefore if V is a solution of (6.1), then
Z
ϕF (x; y) dσ = 0 for all x ∈ ∂E.
(6.2)
∂E
To prove the proposition it suffices to establish that (6.2) implies ϕ = 0.
Indeed, in such a case, v would be harmonic in E and with zero flux on ∂E;
thus v = V = const. Assume N ≥ 3 and consider the function
Z
H=
ϕF (·; y)dσ.
∂E
It is harmonic in E and it vanishes on ∂E. Therefore it vanishes identically
in E. Likewise, it is harmonic in RN − E, it vanishes on ∂E, and |H(x)| → 0
as |x| → ∞. Therefore H = 0 in RN − E. Fix x ∈ ∂E and let {xi } and {xe }
be sequences of points approaching x respectively from the interior and the
exterior of E. Let also W̃ (∂E, xo ; ϕ) be defined as in (4.1) of Chapter 3. By
the previous remarks
W̃ (∂E, xo ; ϕ) = −∇H(xo ) · n(x) = 0
in xo ∈ RN − ∂E.
On the other hand by the jump condition of Corollary 4.1 of Chapter 3
0 = lim
W̃ (∂E, xi ; ϕ) − lim
W̃ (∂E, xe ; ϕ) = ϕ(x).
e
i
x →x
x →x
For N = 2 see Section 6c of the Complements.
7 The Eigenvalue Problem
In what follows we will assume that A is generated by a real-valued, compact,
symmetric, almost separable kernel K(·; ·) in L2 (E), so that A = A∗ . Consider
the problem of finding nontrivial pairs (λ, u), solutions of
u = λAu,
λ ∈ C,
u ∈ L2 (E).
(7.1)
The numbers λ are called the eigenvalues of the operator A, and the functions
u are its corresponding eigenfunctions.
Proposition 7.1 Any two distinct eigenfunctions corresponding to two distinct eigenvalues are orthogonal in L2 (E). Moreover, the eigenvalues of A are
real and the eigenfunctions of A are real-valued.
130
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Remark 7.1 If (λ, u) is a solution pair to (7.1), then also (λ, µu), for all µ ∈
C, are solution pairs. Therefore a more precise statement of the proposition
would be that the eigenvalues of A are real, and the eigenfunctions can be
taken to be real-valued.
Proof (of Proposition 7.1). Let (λi , ui ), i = 1, 2, be distinct solution pairs, say
u1 = λ1 Au1 ,
u2 = λ2 Au2 ,
λ1 6= λ2 .
Multiply the first by u2 , and integrate over E to obtain
Z
Z
Z
Z
1
1
u1 Au2 dy =
u1 u2 dx =
Au1 u2 dy =
u1 u2 dy.
λ1 E
λ2 E
E
E
Therefore hu1 , ū2 i = 0. Let now (λ, u) be a nontrivial solution of (7.1). Then
ū = λAu = λ̄Aū.
Therefore the pair (λ̄, ū) is also a solution of (7.1). If λ 6= λ̄, then hu, ui =
kuk2 = 0. Thus λ = λ̄. Since both u and ū are eigenfunctions for the same
eigenvalue λ, the functions
u + ū
= Re(u),
2
u − ū
= Im(u)
2i
are also eigenfunctions for the same eigenvalue λ. Thus u can be taken to be
real.
Proposition 7.2 The operator A admits at most countably many distinct
eigenvalues {λn }. If the sequence {λn } is infinite, then {|λn |} → ∞. Moreover, to each eigenvalue λ there correspond finitely many, linearly independent
eigenfunctions {uλ,1 , . . . , uλ,nλ }, for some nλ ∈ N.
Proof. Regard (7.1) as an integral equation of the type of (2.1) with f = 0.
According to Theorem 5.1 this is uniquely solvable except for countably many
numbers {λn }. If λ 6= λn , then u = 0 is the only solution. Therefore nontrivial pairs (λ, u) occur for at most countably many values of λ. To prove
the second statement, let λ be a fixed eigenvalue of (7.1). Since K(·; ·) is realvalued and almost separable, it can be written as in (4.1) with |λ|kAo k < 1,
and the integral equation (7.1) can be rewritten as an analogue of (4.3) for
z = (I − λAo )u, that is
Z
n
P
ϕi (I − λA∗o )−1 ψi z dy.
z=λ
(7.2)
i=1
E
Solutions of this are of the form
z=λ
n
P
i=1
zi ϕi ,
where
zi =
Z
E
(I − λA∗o )−1 ψi z dy.
8 Finding a First Eigenvalue and Its Eigenfunctions
131
Multiply (7.2) by (I−λAo∗ )−1 ψj and integrate over E to arrive at the algebraic
system
Z
zj = λaij zi ,
aij =
E
ϕi (I − λA∗o )−1 ψj dy.
This has at most n linearly independent vector solutions (z1,j , . . . , zn,j ) for
j = 1, . . . , n. Accordingly, (7.2) has finitely many solutions, and (7.1) has
finitely many linearly independent solutions u = (I − λAo )z for a given λ.
An eigenvalue λ of A is simple if to λ there corresponds only one eigenfunction u up to a multiplicative constant µ. The eigenvalues of (7.1) need not
be simple. To an eigenvalue λ there corresponds a maximal set of linearly
independent eigenfunctions {vλ,1 , . . . , vλ,nλ }. Any linear combination of these
is an eigenfunction for the same eigenvalue λ. We let
Eλ = {the linear span of the eigenvectors of λ}.
By the Gram–Schmidt orthonormalization procedure we may arrange for Eλ
to be spanned by an orthonormal system of eigenvectors {uλ,1 , . . . , uλ,nλ }.
Corollary 7.1 The set of eigenfunctions of (7.1) can be chosen to be orthonormal in L2 (E).
8 Finding a First Eigenvalue and Its Eigenfunctions
Let S1 be the unit sphere of L2 (E). If (λ, u) is a a nontrivial solution pair
of (7.1), by possibly replacing u with ukuk−1, one may assume that u ∈ S1 .
Thus
kAϕk2 ≤ kAk2 for all ϕ ∈ S1 and kAuk2 = λ−2 .
This suggests that the eigenvalue λ and the eigenfunction u satisfy
sup kAϕk2 = kAuk2 = λ−2
(8.1)
ϕ∈S1
and that they can be found by solving such an extremal problem.1
Theorem 8.1. The eigenvalue problem (7.1) admits a nontrivial solution.
Proof. Select ϕ1 ∈ S1 . If kAϕ1 k ≥ kAϕk for all ϕ ∈ S1 , then the supremum in
(8.1) is achieved at ϕ1 . If not, there exists ϕ2 ∈ S1 such that kAϕ2 k > kAϕ1 k.
Proceeding in this manner, we generate a maximizing sequence {ϕn } ⊂ S1 .
Since A is compact, one may select out of {ϕn } a subsequence, relabeled with
n, such that {ϕn } → u weakly in L2 (E) and {Aϕn } → w, strongly in L2 (E),
and in addition
def
lim kAϕn k2 = sup kAϕk2 = λ−2 .
n→∞
1
ϕ∈S1
This idea, due to Hilbert [119], applies to general, linear, symmetric, compact
operators in L2 (E); also in F. Reillich [216].
132
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Lemma 8.1 u = λ2 A2 u.
Proof. Observe first A2 ϕn → Aw strongly in L2 (E). Indeed
kA2 ϕn − Awk = kA[Aϕn − w]k ≤ kAkkAϕn − wk.
For the assertion, it will suffice to show that
kϕn − λ2 A2 ϕn k → 0
as n → ∞.
(*)
Indeed, since {A2 ϕn } converges to Aw strongly in L2 (E), this would imply
that {ϕn } → u strongly in L2 (E), and as a consequence w = Au and Aw =
A2 u. To prove (*) write
kϕn − λ2 A2 ϕn k2 = kϕn k2 − 2λ2 hA2 ϕn , ϕ̄n i + λ4 kA2 ϕn k2 .
Since A is symmetric hA2 ϕn , ϕ̄n i = kAϕn k2 . Moreover kAϕn k2 ≤ λ−2 , and
kA2 ϕn k2 =
kA(Aϕn )k2
1
kAϕn k2 ≤ sup kAϕk4 ≤ 4 .
kAϕn k2
λ
ϕ∈S1
Combining these estimates
kϕn − λ2 A2 ϕn k2 ≤ 2 − 2λ2 kAϕn k2 → 0.
To prove the theorem, rewrite the conclusion of Lemma 8.1 as
(I − λ2 A2 )u = (I + λA)(I − λA)u = 0.
If (I − λA)u = 0, then the pair (λ, u) is a nontrivial solution of the eigenvalue
problem (7.1). Otherwise, setting (I − λA)u = ψ, the pair (−λ, ψ) solves
(I − (−λ)A)ψ = 0, and therefore is a nontrivial solution of (7.1).
9 The Sequence of Eigenvalues
Let λ1 be the eigenvalue claimed by Theorem 8.1, denote by E1 the linear span
of all the eigenvectors of λ1 , and let E1⊥ be its orthogonal complement. Motivated by the previous maximization procedure, we construct another eigenvalue λ2 by the formula
sup
ϕ∈S1 ∩E1⊥
kAϕk2 = λ−2
2 .
(9.1)
If Aϕ = 0 for all ϕ ∈ S1 ∩ E1⊥ , then λ2 = ∞. Otherwise, we proceed as
before and find a nontrivial pair (λ2 , u2 ) such that λ21 < λ22 and u2 ⊥ E1 .
Having constructed the first n eigenvalues {λ1 , . . . , λn } and the corresponding
eigenspaces {E1 , . . . , En }, construct λn+1 by the maximization problem
9 The Sequence of Eigenvalues
sup
ϕ∈S1 ∩[E1 ∪···∪En ]⊥
kAϕk = λ−2
n+1 .
133
(9.2)
If for some n ∈ N, Aϕ = 0 for all ϕ ∈ S1 ∩ [E1 ∪ · · · ∪ En ]⊥ , then λn+1 = ∞
and the process terminates. Otherwise, proceeding inductively, we construct
2
, and a sequence {um } of
a sequence {λn } of eigenvalues such that λn2 < λn+1
eigenfunctions that can be chosen to form an orthonormal sequence in L2 (E).
Such a sequence in general need not be complete. Necessary and sufficient
conditions on the kernel K(·; ·) to ensure completeness will be given in the
next section.
9.1 An Alternative Construction Procedure of the Sequence of
Eigenvalues
Having determined λ1 , let {uλ1 ,1 , . . . , uλ1 ,n1 } be a set of real-valued linearly
independent orthonormal eigenfunctions spanning the eigenspace E1 , and set
K1 (x; y) =
n1
1 P
uλ ,i (x)uλ1 ,i (y).
λ1 i=1 1
The kernel K1 (·; ·) is symmetric, and the corresponding operator
Z
n1
1 P
K1 (·; y)f (y)dy =
L2 (E) ∋ f → A1 f =
uλ ,i huλ1 ,i , f¯i
λ1 i=1 1
E
is compact and symmetric. Then compute an eigenvalue, λ2 , for the problem
u = λ(A − A1 )u
(9.3)
by the previous procedure, that is
sup k(A − A1 )ϕk2 = λ−2
2 .
ϕ∈S1
Unlike the maximization in (9.1) and (9.2), here the supremum is taken over
the entire unit sphere S1 of L2 (E). However, the operator (A − A1 ) is, roughly
speaking, inactive on the eigenspace E1 . In particular, if A = A1 , then λ2 = ∞,
and the process terminates.
Lemma 9.1 A pair (λ, u) different from (λ1 , uλ1 ,i ) for all i = 1, . . . , n1 , is a
solution of the eigenvalue problem (7.1) if and only if it is a solution of the
eigenvalue problem (9.3).
Proof. If (λ, u) solves (7.1) and λ 6= λ1 , then u ⊥ E1 , and therefore it is a
solution of (9.3). Now let (λ, u) solve (9.3). Multiply both sides by uλ1 ,i and
integrate over E. Since A − A1 is symmetric
hu, uλ1 ,i i = λh(A − A1 )u, uλ1 ,i i = λhu, (A − A1 )uλ1 ,i i
hu, uλ1 ,i i
=0
= λ hu, Auλ1 ,i i −
λ1
134
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
since uλ1 ,i is an eigenfunction for A corresponding to λ1 . This implies that
u = λ(A − A1 )u = λAu − λ
n1
P
i=1
uλ1 ,i huλ1 ,i , ui = λAu.
Proceeding inductively, construct λn+1 from
sup (A −
ϕ∈S1
n
P
Aj )ϕ
j=1
2
= λ−2
n+1
(9.4)
where Aj is the compact symmetric operator in L2 (E) corresponding to the
kernel
nj
1 P
Kj (x; y) =
uλ ,i (x)uλj ,i (y)
λj i=1 j
and {uλj ,1 , . . . , uλj ,nj } are real-valued linearly independent orthonormal eigenfunctions corresponding to the eigenvalue λj . If, for some n ∈ N
K(x; y) =
n
P
Kj (x; y) =
j=1
nj
n 1 P
P
uλj ,i (x)uλj ,i (y)
j=1 λj i=1
set λn+1 = ∞, and the process terminates. If not, recall that {λ2n } → ∞ and
deduce from (9.4) that
lim sup
n→∞
A−
n
P
j=1
Aj f = 0
for all f ∈ L2 (E).
Therefore, letting n → ∞ in (9.4) gives a formal expansion of the kernel
K(·; ·), in terms of its eigenvalues and eigenfunctions. Since the right-hand
side of (9.4) tends to zero as n → ∞, such an expansion can be rigorously
interpreted in the sense of
hK(x; ·), f i =
for all f ∈ L2 (E). Equivalently
n
P
j=1
nj
∞ 1 P
P
uλj ,i (x)huλj ,i , f i
j=1 λj i=1
Aj f → Af
in
L2 (E).
(9.5)
(9.6)
10 Questions of Completeness and the Hilbert–Schmidt
Theorem
Let {λn } and {uλj ,i } be the sequences of eigenvalues and corresponding eigenfunctions of A. If {λn } is finite, say {λ1 , . . . , λm } for some m ∈ N, set λj = ∞
for j > m and uλj ,i = 0 for i = 1, . . . , nj for all j > m. Reorder {uλj ,i } into a
10 Questions of Completeness and the Hilbert–Schmidt Theorem
135
sequence {vn } of real-valued linearly independent orthonormal eigenfunctions,
and rewrite (9.5) in the form
hK(x; ·), f i =
∞ 1
P
vi (x)hvi , f i
i=1 λi
(10.1)
where λi remains the same as {vi }, for i = 1, . . . , nλi spans the eigenspace
Ej . The system {vn } is complete if it spans the whole of L2 (E). Equivalently
if [span{vn }]⊥ = {0}, that is if hf, vi i = 0 for all i implies f = 0. If {vn } is
complete then every f ∈ L2 (E) can be represented as
f=
∞
P
i=1
hf, vi ivi
in the sense
f−
n
P
i=1
hf, vi ivi → 0 as n → ∞.
The series on the right-hand side is the Fourier series of f .
Proposition 10.1 Af = 0 ⇐⇒ hf, vi i = 0 for all i ∈ N.
Proof. For fixed (λi , vi ), hf, vi i = λi hAf, vi i. This proves the implication =⇒.
The converse statement follows from (10.1).
Corollary 10.1 The orthonormal system {vn } is complete in L2 (E) if and
only if f 6= 0 implies Af 6= 0.
Remark 10.1 If the kernel K(·; ·) is of finite rank, then {vn } cannot be
complete in L2 (E).
The corollary gives a simple criterion to check the completeness of {vn }. We
will apply it to the case when K(·; ·) is the Green’s function G(·; ·) for the
Laplacean with homogeneous Dirichlet data on ∂Ω.
R
Proposition 10.2 L2 (E) ∋ f 6= 0 =⇒ E G(·; y)f dy 6= 0.
Proof. Let ϕ ∈ Co∞ (E) and recall the representation of Corollary 3.1 of Chapter 2. For f ∈ L2 (E)
Z Z
G(x; y)f (y)∆ϕ̄(x) dydx
hAf, ∆ϕi =
E E
Z
Z
G(x; y)∆ϕ̄(x) dxdy
=
f (y)
E
E
Z
f ϕ̄dy = −hf, ϕi.
=−
E
10.1 The Case of K(x; ·) ∈ L2 (E) Uniformly in x
Assume that K(·; ·) is a real-valued compact symmetric kernel acting on L2 (E)
that generates an orthonormal system {vn } complete in L2 (E). It is natural
to ask whether, or under what conditions, the Fourier series of a function f ∈
136
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
L2 (E) converges to f in some stronger topology, for example in the topology
of the uniform convergence in Ē. This requires more stringent assumptions
on f and on the kernel K(·; ·).
Assume that kK(x; ·)k ≤ C for some positive constant C uniformly in x.
In such a case, in (10.1), we may take f (x; y) = K(x; y) to obtain
∞ 1
P
2
2
2 vi (x) ≤ C
λ
i=1 i
for all x ∈ E.
(10.2)
Theorem 10.1 (Hilbert–Schmidt). Let K(·; ·) be a real-valued compact
symmetric kernel in L2 (E), which generates an orthonormal system {vn } complete in L2 (E) and such that kK(x; ·)k ≤ C for some C > 0, uniformly in x.
Then every function f ∈ L2 (E) that can be represented as
Z
f=
K(·; y)g dy for some g ∈ L2 (E)
(10.3)
E
P∞
has a Fourier seriesP i=1 hf, vi ivi , absolutely and uniformly convergent to f
n
in E, that is, f − i=1 hf, vi ivi ∞ → 0 as n → ∞.
P∞
Proof. It suffices to show that
i=m |hf, vi i||vi | ∞ → 0 as m → ∞. This is
a consequence of (10.2) and the representation (10.3). Indeed, for all x ∈ E
∞
P
i=m
|hf, vi i||vi (x)|
2
2
|vi (x)|
|λi |
i=m
∞
∞
P 1 2
P
2
|hg, vi i|
≤
2 vi (x) .
i=m λi
i=m
=
∞
P
|hg, vi i|
Remark 10.2 The Green’s function G(·; ·) satisfies the assumptions of the
Hilbert–Schmidt theorem only for N = 2, 3.
Corollary 10.2 Let N = 2, 3. Then a function f ∈ Co1 (E) ∩ C 2 (Ē) has a
Fourier series that converges absolutely and uniformly to f in E.
Proof. It follows from the Hilbert–Schmidt theorem and the representation of
Corollary 3.1 of Chapter 2.
11 The Eigenvalue Problem for the Laplacean
Eingenvalues and eigenfunctions for the Laplacean with homogeneous Dirichlet data are those related to (8.1)–(8.2) of the previous Chapter, which were
shown to be equivalent.
11 The Eigenvalue Problem for the Laplacean
137
Theorem 11.1. The eigenvalues of the Laplacean on a bounded open set
E ⊂ RN , with homogeneous Dirichlet data on ∂E, are positive and form
a monotone increasing sequence {λn } → ∞ as n → ∞. Moreover, the corresponding orthonormal system of eigenfunctions {vn } is complete in L2 (E).
Finally, the first eigenfunction can be taken to be positive, and λ1 is simple.
Proof. If the pair (λ, u) solves (8.1) of Chapter 3, is nontrivial, and λ < 0,
then u cannot take a positive maximum in E. Indeed, if a positive maximum
were taken at some xo ∈ E
−∆u(xo ) = λu(xo ) < 0.
By the same argument, u cannot attain a negative minimum in E. Therefore
u = 0. The statement about completeness follows from Proposition 10.2.
The maximization process (8.1) implies that if the supremum is achieved
for some u1 then it is achieved also for |u1 |. Indeed, since G(·; ·) ≥ 0
λ1−2 = kAu1 k2 =
Z
2
G(·; y)u1 dy
E
≤
Z
E
2
G(·; y)|u1 |dy
= λ−2
1 .
Thus u1 and |u1 | are both eigenfunctions for the same eigenvalue λ1 . In particular
−∆|u1 | = λ1 |u1 |.
This in turn implies that the function
E × R ∋ (x, t) → w(x, t) = |u1 (x)|e
√
λ1 t
is a non-negative harmonic function in the (N + 1)-dimensional strip E × R.
By the Harnack estimate of Corollary 5.1 of Chapter 2, |u1 | > 0 in E, and
therefore u1 = |u1 |. We conclude that all the eigenfunctions corresponding to
λ1 can be taken to be positive. In particular, no two of them can be orthogonal.
Thus λ1 is simple.
11.1 An Expansion of the Green’s Function
Formula (9.5) provides an expansion of the Green’s function G(·; ·) in terms
of its eigenvalues and eigenfunctions. Namely, for all f ∈ L2 (E) and for a.e.
x∈E
hG(x; ·), f i =
nj
∞ 1 P
P
1
uλ1 (x)huλ1 , f i +
uλj ,i (x)huλj ,i , f i.
λ1
j=2 λj i=1
138
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
Problems and Complements
2c Integral Equations
2.1c Integral Equations of the First Kind
Equation (2.1) was also called an integral equation of the second kind. An
integral equation of the first kind is of the form
Z
K(·; y)u dy = f.
(2.1c)
E
Here f ∈ L2 (E) is given, K(·; ·) is a kernel in L2 (E), and u is the unknown
function. Below we give an example of an integral equation of the first kind.
2.2c Abel Equations ([2, 3])
A particle constrained on a vertical plane falls from rest under the action
of gravity along a trajectory γ. On the vertical plane introduce a Cartesian
system originating at ground level, and with j directed along the ascending
vertical. If the particle is initially at level x from the ground, we seek the
trajectory γ such that it will hit the ground after a time t = f¯(x), where f is
a given function. Parametrize γ by the angle θ that the tangent line at points
of γ forms with the horizontal axis, taken counterclockwise starting from the
positive direction of the horizontal axis.
p
The speed of the falling particle at level y ∈ [0, x] is 2g(x − y), where g
is the acceleration of gravity. The velocity along j is
p
dy
= − 2g(x − y) sin θ
dt
or, separating the variables
dy
p
= −dt.
2g(x − y) sin θ
Integrate the left-hand side from the initial level x to the final level 0, and the
right-hand side from the initial time 0 to the final time f¯(x). This gives the
Abel integral equation of the first kind
Z x
p
1
v(y)dy
√
= f (x) where v(y) =
and f = − 2gf¯.
(2.2c)
x−y
sin θ
0
When f =const, this is the problem of the tautochrone trajectory. More generally, an Abel integral equation takes the form
2c Integral Equations
Z
x
0
v(y)dy
= f (x)
(x − y)α
139
(2.3c)
for some α > 0 and f ∈ C 1 [0, ∞). This can be recast in the form (2.1c) as
follows. First limit x not to exceed some fixed positive number a. Then set
(x − y)−α if 0 ≤ y < x
K(x; y) =
0
if x ≤ y ≤ a.
and rewrite (2.3c) as
Z
a
K(·; y)vdy = f.
0
Kernels of this kind are said to be of Volterra type ([266, 267]).
2.3c Solving Abel Integral Equations
In (2.3c) replace x by a running variable η, multiply both sides by (x − η)α−1
and integrate in dη over (0, x). Interchanging the order of integration gives
Z x
Z x
Z x
f (η)
dη
dy =
dη.
v(y)
1−α (η − y)α
(x
−
η)
(x
−
η)1−α
0
0
y
Compute the integral in braces by the change of variables
η=y+
1
(x − y),
s+1
s ∈ [0, ∞).
This gives
Z
x
y
dη
dη =
(x − η)1−α (η − y)α
Z
0
∞
ds
π
=
s1−α (1 + s)
sin απ
where the last integral has been computed by the method of residues ([31]
page 107). Combining these calculations
Z x
Z x
π
f (y)
dy
v(y)dy =
sin απ 0
(x
−
y)1−α
0
Z
1 x
xα
f (0) +
(x − y)α f ′ (y)dy.
=
α
α 0
Taking the derivative gives an explicit representation, of the solution of the
Abel integral equation (2.3c), in the form
Z x
π
f (0)
f ′ (y)
v(x) =
+
dy
.
(2.4c)
1−α
sin απ x1−α
0 (x − y)
140
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
2.4c The Cycloid ([3])
To find the parametric equations of the tautochrone trajectory, in (2.4c) take
α = 12 and f = C. Using (2.2c)
√
π x
C
.
sin θ =
v(x) = √ ,
C
π x
Denoting by x = x(θ) the vertical component of the parametrization of γ
x(θ) =
C2
C2
2
sin
θ
=
(1 − cos 2θ).
π2
2π 2
(2.5c)
Let y = y(θ) denote the horizontal component of the parametrization of γ,
and let γ have the local representation y = y(x). Then
dy =
and by integration
dx
C 2 2 sin θ cos θ
C2
= 2
dθ = 2 (1 + cos 2θ)dθ
tan θ
π
tan θ
π
1
C2
y(θ) = 2 θ + cos 2θ + co
π
2
(2.6c)
for a constant co . The equations (2.5c) and (2.6c) are the parametric equations
of a cycloid.
2.5c Volterra Integral Equations ([266, 267])
Let f be bounded and continuous in R+ , and consider the Volterra equation
Z x
u(x) = λ
K(x; y)u(y)dy + f (x)
0
where K(·; ·) is bounded and continuous in R+ ×R+ . Assuming that K(x; y) =
0 for y > x, rewrite this as
Z ∞
u=λ
K(·; y)u(y)dy + f.
0
Prove that for all x ∈ R+
K nxn
|(A f )| ≤ sup kf k
,
n!
R+
n
where (Af )(x) =
Z
x
K(x; y)f (y)dy
0
and where K is an upper bound for |K(·; ·)|. Conclude that a solution must
be continuous and locally bounded in R+ .
2.1. Say in what sense the Dirichlet and Neumann problems for the Laplacean
in a bounded domain are mutually adjoint. Hint: See Sections 5–6 of
Chapter 3 and the arguments of Section 6 of this Chapter.
3c Separable Kernels
141
2.2. Find A2 if K(x; y) = e|x−y| and E = (0, 1).
2.3. Find A2 and A3 if K(x; y) = x − y and E = (0, 1).
One might ask whether these integral equations set in R+ , have a solution
if K(·; ·) does not vanish for y > x. It turns out that some decay has to
imposed on K(·; ·). For kernels of the type K(x; y) = K(x − y) and K(s) → 0
exponentially fast as s → ∞ a theory is developed by N. Wiener and E. Hopf
([279]). See also G. Talenti [250].
3c Separable Kernels
3.1c Hammerstein Integral Equations ([114])
Consider the nonlinear integral equation of Hammerstein type
Z
u=
K(x; y)f (y, u(y))dy.
E
If the kernel is separable, set
Z
γi =
ψi f (y, u(y))dy
K(·; ·) =
E
n
P
ϕi ψi
i=1
where the numbers γi are to be determined from
Z
n
n
n
P
P
P
γi ϕi (y) dy.
ϕi
γi ϕi =
ψi f y,
u=
i=1
i=1
E
i=1
Therefore, the numbers γi are the possible solutions (real or complex) of the
system
Z
n
P
γi ϕi (y) dy.
γi =
ψi f y,
i=1
E
3.1. Solve the Hammerstein equations
Z
Z 1
2
u(x) = λ
u(x) = λ
xyu (y)dy,
0
1
−1
|x − y|
dy.
1 + u2 (y)
3.2. Let ϕ ∈ L2 [0, 1] be non-negative. Show that if kϕk2,[0,1] > 1, there are
no real-valued solutions of the Hammerstein equation
u(x) =
1
2
Z
0
1
ϕ(x)ϕ(y) 1 + u2 (y) dy.
142
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
6c Applications to the Neumann Problem
Prove Proposition 6.1 for N = 2 by the following steps.
Step 1: Consider the double-layer potential
R2 − ∂E ∋ x → W (∂E, x; ϕ) =
1
2π
Z
ϕ(y)
∂E
∂ ln |x − y|
dσ.
∂n(y)
By the first part of the proof of Proposition 6.1, such a function is identically zero in E. Prove that it vanishes on ∂E. Thus W (∂E, ·; ϕ) is harmonic
in R2 − E, vanishes as |x| → ∞ and has zero normal derivative on ∂E.
Step 2: Prove that ∀N ≥ 2, there exists at most one solution to the problem
u ∈ C 2 (RN − Ē) ∩ C 1 (RN − E), ∆u = 0 in RN − Ē
∂u
= 0 on ∂E, and
lim u(x) = 0.
∂n
|x|→∞
Prove that positive maxima or negative minima cannot occur on ∂E.
9c The Sequence of Eigenvalues
9.1. Let A be generated by the Green’s function for the Laplacean with
homogeneous Dirichlet data. Prove that the maximization process (8.1) is
formally equivalent to
min k∇uk2 = λ2
ϕ∈Co ∩S1
where Co = {u ∈ Co (E) |∇u| ∈ L2 (E)}.
9.2. Let f ∈ L2 (E). Prove that the minimum
min
{f1 ,...,fn }∈Cn }
f−
n
P
fi ui
i=1
is achieved for fi = hf, ui i. Hint: Compute the derivatives
n
P
∂
2
f−
fi ui .
∂fi
i=1
Pn
2
9.3. Prove Bessel’s inequality Pi=1 fi ≤ kf k2.
∞
9.4. Prove Parseval’s identity i=1 fi2 = kf k2 .
10c Questions of Completeness
10.1. If K(x; ·) ∈ L2 (E) uniformly in x ∈ E, then (10.2) gives another proof
that to each eigenvalue λi there correspond only finitely many linearly independent eigenfunctions. Hint: If ni is the number of linearly independent
eigenfunctions corresponding to the eigenvalue λi , then ni ≤ C|λi |E|.
10c Questions of Completeness
10.2. Prove that
2
nq
2
L
sin nπ
L x
o
143
is a complete orthonormal system in the
space L (0, L). Hint: Compute the eigenvalues and eigenfunctions of the
Laplacean in one dimension, over (0, L) with homogeneous Dirichlet data
on x = 0 and x = L.
m
10.3. Let m be an even positive integer, and let Codd
(0, L) denote the space
m
of functions in C (0, L) whose even order derivatives vanish at x = 0 and
x = L, i.e.,
∂j
∂j
+
ϕ(0
)
=
ϕ(L− ) = 0
∂xj
∂xj
for all even integers
0 ≤ j ≤ m.
Denoting by {vn } the complete orthonormal system in L2 (0, L), of the
m
previous problem prove that if ϕ ∈ Codd
(0, L)
∂j
2Lj
ϕ
(nπ)j ∂xj
|hϕ, vn i| ≤
∞,[0,L]
for all 0 ≤ j ≤ m.
As a consequence
ϕ−
j−1
P
hϕ, vn ivn
n=1
∞,[0,L]
≤
const ∂ j
ϕ
j m ∂xj
∞,[0,L]
.
10.1c Periodic Functions in RN
A function f : RN → R is periodic of period 1 if f (x + n) = f (x) for all
x ∈ RN and every N -tuple of integers n ∈ ZN .
Let Q = (0, 1)N denote the unit cube in RN . Every f ∈ L2 (Q) can be
regarded as the restriction to Q of a periodic function in RN of period 1. If
f is periodic of period 1, there exists a constant γ such that f + γ is periodic
of period 1 and has zero average over Q. Consider the space
Z
2
2
Lp (Q) = f ∈ L (Q)
f dx = 0
Q
where the subscript p denotes “periodic function”. An orthogonal basis for
L2p (Q) is found by solving the eigenvalue problem
−∆u = λu in Q,
and
∂u
= 0 on ∂Q.
∂n
Verify that
N
Q
j=1
cos nj πxj ,
N
Q
j=1
sin nj πxj ;
n = (n1 , . . . , nN ) ∈ ZN , |n|2 =
N
P
j=1
n2j
are eigenfunctions for the eigenvalues λ = (π|n|)2 . Any complex linear combination of these is still an eigenfunction. Prove that the system {eiπhn,xi } for
n ∈ ZN is a complete orthogonal basis for L2p (Q).
144
4 INTEGRAL EQUATIONS AND EIGENVALUE PROBLEMS
10.2c The Poisson Equation with Periodic Boundary Conditions
Consider the Neumann problem
u ∈ C 2 (Q) ∩ C 1 (Q̄),
∆u = f ∈ C 1 (Q̄),
∂u
= 0 on ∂Q.
∂n
The necessary and sufficient condition for solvability is that f has zero average
over Q, which we assume. Write
P ˆ iπhn,xi
fn e
, where fˆn = hf, eiπhn,xi i
f=
n∈ZN
and seek a solution of the type
P
ûn eiπhn,xi ,
u=
n∈ZN
Prove that
ûn =
fˆn
−π|n|2
for all
ûn = hu, eiπhn,xi i.
n ∈ ZN − {0}.
11c The Eigenvalue Problem for the Laplacean
11.1. A linear operator A : L2 (E) → L2 (E) is positive if
hAf − Ag, f − gi ≥ 0
for all f, g ∈ L2 (E).
The operator A generated by the Green’s function for the Laplacean with
homogeneous Dirichlet data on ∂E, is positive in the sense that hAf, f i >
0 for all f ∈ L2 (E), f 6= 0. Assume first that f ∈ Coη (E) for some
η ∈ (0, 1). Then the function Af is the unique solution of the problem
u ∈ C 2 (E) ∩ C(Ē),
−∆u = f,
in E,
u
∂E
= 0.
Therefore
hAf, f i = hu, −∆ui = k∇uk2 .
Prove the positivity of A for general f ∈ L2 (E).
11.2. Prove that if A is a symmetric, positive, compact operator in L2 (E),
then its eigenvalues are positive.
5
THE HEAT EQUATION
1 Preliminaries
Consider a material homogeneous body occupying a region E ⊂ RN with
boundary ∂E of class C 1 and outward unit normal n. Identify the body with
E and denote by k > 0 its dimensionless conductivity. The temperature distribution (x, t) → u(x, t) satisfies the second-order parabolic equation
ut = k∆u for x ∈ E and t ∈ (t1 , t2 )
(1.1)
where (t1 , t2 ) ⊂ R is some time interval of observation. By changing the time
scale, we may assume that k = 1. Set formally
H(·) =
∂
− ∆,
∂t
H ∗ (·) =
∂
+ ∆.
∂t
The formal operators H(·) and H ∗ (·) are called the heat operator and the
adjoint heat operator respectively. If 0 < T < ∞ denote by ET the cylindrical
domain E × (0, T ], and if E = RN , let ST denote the strip RN × (0, T ]. The
heat operator and its adjoint are well defined for functions in the class
H(ET ) = {u : ET → R ut , uxi xj ∈ C(ET ), i, j = 1, . . . , N }.
Information on the thermal status of the body is gathered at the boundary of
E over an interval of time (0, T ]. That is, one might be given the temperature
or the heat flux at ∂E × (0, T ). Physically relevant problems consist in finding
the temperature distribution in E for t ≥ 0, from information on ∂E × (0, T )
and the knowledge of the temperature x → uo (x) at time t = 0. This leads to
the following boundary value problems:
1.1 The Dirichlet Problem
Find u ∈ H(ET ) ∩ C(ĒT ) satisfying
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_6
145
146
5 THE HEAT EQUATION
H(u) = 0
u
∂E×[0,T ]
in ET
= g ∈ C(∂E × (0, T ])
(1.2)
u(·, 0) = uo ∈ C(Ē).
1.2 The Neumann Problem
Find u ∈ H(ET ) ∩ C 1 (ĒT ) satisfying
H(u) = 0 in ET
Du · n = g ∈ C(∂E × (0, T ))
u(·, 0) = uo ∈ C(Ē).
(1.3)
where D denotes the gradient with respect to the space variables only.
1.3 The Characteristic Cauchy Problem
Find u ∈ H(ST ) ∩ C(S̄T ) satisfying
H(u) = 0
in ST
u(·, 0) = uo ∈ C(RN ) ∩ L∞ (RN ).
(1.4)
The initial datum in (1.4) is taken in the topology of the uniform convergence
over compact sets K ⊂ RN , that is, ku(·, t) − uo k∞,K → 0, as t → 0, for all
such K. In (1.4) the data are assigned on the characteristic surface t = 0. The
Cauchy–Kowalewski theorem fails to hold in such a circumstance. Even if uo is
analytic, a solution of (1.4) near t = 0, that is for small positive and negative
times, in general cannot be found. Indeed, changing t into −t does not preserve
(1.1) and the PDE distinguishes between solutions forward and backward in
time. This corresponds to the physical fact that heat conduction is, in general,
irreversible, i.e., given x → uo (x), we may predict future temperatures, but we
cannot in general determine the thermal status that generated that particular
temperature distribution.
2 The Cauchy Problem by Similarity Solutions
The PDE H(u) = 0 is invariant by linear transformations x̄ = hx, t̄ = h2 t
for h 6= 0. These are transformations that leave invariant the ratio ξ = |x|2 /t.
This suggests looking for solutions u that are “separable” in the variables t
and ξ, that is, solutions of the form u(x, t) = h(t)f (ξ). Substituting this in
the PDE H(u) = 0 gives
th′ f − 2N hf ′ = hξ[4f ′′ + f ′ ].
2 The Cauchy Problem by Similarity Solutions
147
Setting each side equal to zero yields
f (ξ) = exp(−ξ/4),
h(t) = t−N/2
up to multiplicative constants. These remarks imply that a solution of H(u) =
0 in RN × (0, ∞) is given by
Γ (x, t) =
2
1
e−|x| /4t
(4πt)N/2
(2.1)
where the multiplicative constant (4π)−N/2 has been chosen to satisfy the
normalization (Section 2.1c of the Complements)
Z
|x−y|2
1
(2.2)
e− 4(t−s) dy = 1
N/2
[4π(t − s)]
RN
for all x ∈ RN and all s < t.
Remark 2.1 The function Γ is called the heat kernel or the fundamental
solution of the heat equation. It satisfies
(x, t) → Γ (x, t) ∈ C ∞ (RN × R+ )
x → Γ (x, t) is analytic for t > 0.
∗
Let H(η,τ ) and H(η,τ
) denote respectively the heat operator and its adjoint
with respect to the variables η ∈ RN and τ ∈ R. By direct calculation
H(x,t) Γ (x − y; t − s) = 0
∗
H(y,s)
Γ (x − y; t − s) = 0
for s < t < ∞.
(2.3)
Assume that u ∈ H(ST ) is a solution of the Cauchy problem (1.4) satisfying
Z
|u(x, t)|dx < ∞ for all 0 ≤ t ≤ T
(2.4)
RN
and the asymptotic decay
Z
y
lim sup
Γ (x − y; t)Du · dσ = 0
r
r→∞
|y|=r
lim sup
r→∞
Z
y
uDΓ (x − y; t) · dσ = 0
r
|y|=r
for all 0 ≤ t < T
(2.5)
where dσ denotes the surface measure on the sphere |y| = r. Multiply the
first of (1.4) viewed in the variables (y, s) by Γ (x − y; t − s), and integrate by
parts in dyds over the cylindrical domain Br × (0, t − ε) for ε ∈ (0, t). Letting
r → ∞ with the aid of (2.4) and (2.5), we arrive at
148
5 THE HEAT EQUATION
Z
RN
2
u(y, t − ε)ε
−N/2 − |x−y|
4ε
e
dy =
Z
uo (y)t−N/2 e−
|x−y|2
4t
dy.
(2.6)
RN
We let ε → 0 as follows. Fix σ > 0 and write
Z
Z
|x−y|2
|x−y|2
u(y, t − ε)ε−N/2 e− 4ε dy =
u(y, t − ε)ε−N/2 e− 4ε dy
RN
|y−x|>σ
Z
|x−y|2
+
u(y, t − ε)ε−N/2 e− 4ε dy
|y−x|≤σ
=
Iε(1)
+ Iε(2) .
As ε → 0, the first integral on the right-hand side tends to zero. We rewrite
the second integral as
Z
|x−y|2
(2)
[u(y, t − ε) − u(x, t)]ε−N/2 e− 4ε dy
Iε =
|y−x|<σ
Z
|x−y|2
+ u(x, t)
ε−N/2 e− 4ε dy
|y−x|<σ
Z
|y|2
= [u(x, t) + O(σ + ε)]
ε−N/2 e− 4ε dy
|y|<σ
where O(σ + ε) denotes a quantity that tends to zero as (σ + ε) → 0. Write
Z
Z
Z
2
2
|y|2
−N/2
−N/2
− |y|
−N/2
− |y|
4ε
4ε
dy = ε
dy − ε
e
e− 4ε dy.
ε
e
RN
|y|<σ
|y|>σ
The first integral can be computed from (2.2) with x = 0 and (t − s) = ε, i.e.,
Z
|y|2
−N/2
ε
e− 4ε dy = (4π)N/2 .
RN
√
To estimate the second integral, introduce the change of variables y = 2 εη,
whose Jacobian is (4ε)N/2 . This gives
Z
Z
|y|2
2
ε−N/2
e− 4ε dy =
e−|η| dη.
σ
|η|> 2√
ε
|y|>σ
This integrals tends to zero as ε → 0, for σ > 0 fixed. Combine these calculations in (2.6), and let ε → 0, while σ > 0 remains fixed, to obtain
Z
|x−y|2
(4π)N/2 u(x, t) = t−N/2
uo (y)e− 4t dy + O(σ).
RN
Letting σ → 0 gives the representation formula
Z
Z
2
1
− |x−y|
4t
uo (y)dy =
Γ (x − y; t)uo (y)dy.
u(x, t) =
e
(4πt)N/2 RN
RN
(2.7)
2 The Cauchy Problem by Similarity Solutions
149
Therefore every solution of the Cauchy problem satisfying the decay conditions
(2.4)–(2.5) must be represented as in (2.7). Now consider (2.7), regardless of
its derivation process. If uo ∈ C(RN )∩L∞ (RN ), the integral on the right-hand
side is convergent and defines a function u that satisfies the decay conditions
(2.4)–(2.5). Moreover, by Remark 2.1, u(x, t) ∈ C ∞ (ST ) and
x → u(x, t) is locally analytic in RN for all 0 < t ≤ T.
(2.8)
Theorem 2.1. Let uo ∈ C(RN ) ∩ L∞ (RN ). Then u defined by (2.7) is a
solution to the Cauchy problem (1.4). Moreover, u is bounded in RN × R+ ,
and it is the only bounded solution to the Cauchy problem (1.4).
Proof (existence). By construction H(u) = 0 in ST . Moreover
Z
ku(·, t)k∞,RN ≤ kuo k∞,RN
Γ (x − y; t)dy = kuo k∞,RN .
(2.9)
RN
Therefore u defined by (2.7) is bounded in RN × R+ . It remains to show
that the initial datum is taken in the topology of uniform convergence over
compact subsets of RN . Fix a compact set K ⊂ RN , recall the normalization
(2.2), and write for x ∈ K
Z
|x−y|2
1
u(x, t) − uo (x) =
[uo (y) − uo (x)]e− 4t dy.
N/2
(4πt)
RN
Divide the domain of integration on the right-hand side into |x − y| < σ and
|x − y| ≥ σ where σ > 0 is arbitrary but fixed. As t → 0, the integral extended
over |x − y| > σ tends to zero and the one extended over |x − y| < σ is
majorized by
Z
sup |uo (y) − uo (x)|
x∈K
|x−y|<σ
RN
Γ (x − y; t)dy.
Therefore, for arbitrary σ > 0
lim ku(x, t) − uo (x)k∞,K ≤
t→0
sup |uo (y) − uo (x)|.
x∈K
|x−y|<σ
The proof of uniqueness will make use of the maximum principle discussed
in the next sections. A first form of such a principle can be read from (2.9),
that is, the supremum of |u(·, t)| at all instants t > 0 is no larger than the
supremum of |uo |.
Remark 2.2 Suppose that in (2.7), uo is non-negative, not identically zero,
and supported in the ball Bε of radius ε > 0 centered at some point in RN .
Then u(x, t) is strictly positive for all (x, t) ∈ ST . In particular, the initial
disturbance, confined in Bε , for however small ε, is felt by the solution at
any |x| however large, and any positive t, however small. Thus the initial
disturbance propagates with infinite speed.
150
5 THE HEAT EQUATION
2.1 The Backward Cauchy Problem
Let S T = RN × (−T, 0), and consider the problem of finding u ∈ H(S T ) ∩
C(S̄ T ) satisfying
H(u) = 0 in S T
(2.10)
u(·, 0) = uo ∈ C(RN ) ∩ L∞ (RN ).
The backward problem (2.10) is ill posed in the sense that unlike the forward
problem (1.4), it is not solvable in general within the class of bounded solutions. Indeed, if a bounded, continuous solution did exist for every choice of
data uo ∈ C(RN ) ∩ L∞ (RN ), we would have by (2.7) and Theorem 2.1
Z
Γ (x − y; T )u(y, −T ) dy
(2.11)
uo (x) =
RN
and this would contradict (2.8), if for example, uo is merely continuous.
3 The Maximum Principle and Uniqueness (Bounded
Domains)
Let E be a bounded open subset of RN and let ∂∗ ET = ∂ET − E × {T } denote
the parabolic boundary of ET .
Theorem 3.1. Let u ∈ H(ET ) ∩ C(ĒT ) satisfy H(u) ≤ 0(≥ 0) in ET . Then
inf u = inf u .
sup u = sup u
ET
∂∗ ET
ET
∂∗ ET
Proof. We prove the statement only for H(u) ≤ 0. Let ε ∈ (0, T ) be arbitrary
but fixed, and consider the function
ĒT −ε ∋ (x, t) → v(x, t) = u(x, t) − εt
which satisfies H(v) < −ε < 0 in ĒT −ε . Since v is continuous in ĒT −ε , it
achieves its maximum at some (xo , to ) ∈ ĒT −ε . If (xo , to ) ∈
/ ∂∗ ET −ε , then
H(v)(xo , to ) ≥ 0, contradicting H(v) < 0. Thus (xo , to ) ∈ ∂∗ ET −ε and
u(x, t) ≤ 2εT + sup u
∂∗ ET
for all (x, t) ∈ ĒT −ε for all ε > 0.
Corollary 3.1 Let u ∈ H(ET ) ∩ C(ĒT ) satisfy H(u) = 0 in ET . Then
kuk∞,ET = kuk∞,∂∗ET .
Remark 3.1 Theorem 3.1 is a weak maximum principle since it does not
exclude that u might obtain its extremal values also at some other points in
ĒT . For example, u could be identically constant in ET . A strong maximum
principle would assert that this is the only other possibility.
3 The Maximum Principle and Uniqueness (Bounded Domains)
151
3.1 A Priori Estimates
Denote by λ the diameter of E. After a rotation and translation, we may, if
necessary, arrange the coordinate axes so that
x1o − λ ≤ x1 ≤ x1o
for all x = (x1 , . . . , xN ) ∈ Ē,
o
) ∈ ∂E. This is possible since the heat operafor some xo = (xo1 , . . . , xN
tor is invariant under rotations and translations of the space variables. Let
u ∈ H(ET ) ∩ C(ĒT ) be such that kH(u)k∞,ET < ∞ and construct the two
functions
o
w± (x, t) = kuk∞,∂∗ET + eλ [1 − e(x1 −x1 ) ]kH(u)k∞,ET ± u.
One verifies that H(w± ) ≥ 0 in ET and that
w±
∂∗ ET
≥ kuk∞,∂∗ ET ± u
∂∗ ET
.
Therefore w± ≥ 0, by Theorem 3.1. This gives the following a priori estimate.
Corollary 3.2 Let u ∈ H(ET ) ∩ C(ĒT ). Then
kuk∞,ET ≤ kuk∞,∂∗ ET + (ediam(E) − 1)kH(u)k∞,ET .
3.2 Ill Posed Problems
A boundary value problem for H(u) = 0 with data prescribed on the whole
boundary of ET in general is not well posed. For example, consider the
rectangle R = [0 < x < 1] × [0 < t < 1], and let ϕ ∈ C(∂R) be nonconstant and such that it takes an absolute maximum on the open line segment
[0 < x < 1] × [t = 1]. Then the problem
u ∈ H(R),
ut − uxx = 0 in R,
u
∂R
=ϕ
cannot have a solution, for it would violate Theorem 3.1.
3.3 Uniqueness (Bounded Domains)
Corollary 3.3 There exists at most one solution u ∈ H(ET ) ∩ C(ĒT ) of the
boundary value problem
ut − ∆u = f ∈ C(ĒT ),
u
∂∗ ET
= g ∈ C(∂∗ ET ).
Proof. If u and v are solutions, w = u − v solves
wt − ∆w = 0 in ET ,
and hence w ≡ 0 by Theorem 3.1.
w
∂∗ ET
=0
152
5 THE HEAT EQUATION
4 The Maximum Principle in RN
Results analogous to Theorem 3.1 are possible in RN if one imposes some
conditions on the behavior of x → u(x, t) as |x| → ∞. Such conditions are
dictated by the solution formula (2.7). For such a formula to have a meaning,
uo does not have to be regular or bounded. It would suffice to require the
convergence of the integral on the right-hand side for 0 < t ≤ T . The next
proposition gives some sufficient conditions for this to occur.
Proposition 4.1 Assume that uo ∈ L1loc (RN ) and satisfies the growth condition
there exist positive constants Co , αo , ro such that
2
(4.1)
|uo (x)| ≤ Co eαo |x| for almost all |x| ≥ ro .
Then (2.7) defines a function u ∈ C ∞ (ST ) for every T ∈ 0, 4α1 o . Moreover,
H(u) = 0 in ST , and for every ε ∈ 0, 4α1 o , there exists positive constants α,
C, and r depending upon αo , Co , ro , N , and ε such that
2
ε2 kuo k1,Bro + Ceα|x|
2N
(4.2)
1
for all |x| > r and for all 0 < t <
− ε.
4αo
Proof. Fix ε ∈ 0, 4α1 o and |x| > ro + ε, and write the integral in (2.7) as
|u(x, t)| ≤ Γ ε;
Z
|y|≤ro
Γ (x − y; t)uo (y)dy +
Z
|y|>ro
Γ (x − y; t)uo (y)dy = J1 + J2 .
For |x − y| > ε
|J1 | ≤ sup Γ (ε; t)kuo k1,Bro = Γ ε;
t≥0
ε2 kuo k1,Bro .
2N
√
In estimating J2 we perform the change of variables y − x = 2 tη, of Jacobian
(4t)N/2 , and use (4.1) to estimate |uo (y)| from above. This gives
Z
√ 2
2
|J2 | ≤ Co π −N/2
e−|η| eαo |x+2 tη| dη.
|y|>ro
By the Schwarz inequality, for all δ > 0
√
1
|x + 2 tη|2 ≤ 1 + |x|2 + 4(1 + δ)t|η|2 .
δ
Therefore, for all |x| > r = ro + ε
|J2 | ≤ Co π
−N/2 αo (1+1/δ)|x|2
e
Z
RN
2
e−(1−4αo (1+δ)t)|η| dη.
4 The Maximum Principle in RN
153
The integral on the right-hand side is convergent if
t<
1
1
=
− ε.
4αo (1 + δ)
4αo
2
This defines the choice of δ. Therefore |J2 | ≤ Ceα|x| , where
Z
2
1
−N/2
C = Co π
e−(1−4αo (1+δ)t)|η| dη and α = αo 1 + .
δ
N
R
In deriving a maximum principle for solutions of the heat equation in ST , we
require that such solutions satisfy a behavior of the type (4.2) as |x| → ∞,
but we make no further reference to the representation formula (2.7).
Theorem 4.1. Let u ∈ H(ST )∩C(S̄T ) satisfy H(u) ≥ 0 in ST and u(·, 0) ≥ 0.
Assume moreover that
there exist positive constants C, α, r such that
2
(4.3)
u(x, t) ≥ −Ceα|x| for all |x| ≥ r and all 0 ≤ t ≤ T.
Then u ≥ 0 in ST .
Proof. Choose β > α so large that T >
in the strip ST1 . The function
v(x, t) =
1
8β
def
= T1 . We first prove that u ≥ 0
2
1
eβ|x| /(1−4βt)
(1 − 4βt)N/2
2
satisfies H(v) = 0, and v(x, t) ≥ eβ|x| , in ST1 . Let ε > 0 be arbitrary but
fixed and set w = u + εv. In view of the arbitrariness of ε, it will suffice to
show that w ≥ 0 in ST1 . The function w satisfies H(w) ≥ 0 in ST1 , w(·, 0) ≥ 0,
and
lim inf w(x, t) ≥ 0, uniformly in t ∈ [0, T1 ].
|x|→∞
Therefore, having fixed (xo , to ) ∈ ST1 and σ > 0, there exists ρ > |xo | such
that w(x, t) ≥ −σ for |x| ≥ ρ for all t ∈ [0, T1 ]. On the (bounded) cylinder
Q = [|x| < ρ] × (0, T1 ) the function w̄ = w + σ satisfies H(w̄) ≥ 0 in Q
and w̄ ≥ 0 on the parabolic boundary ∂∗ Q of Q. Therefore w̄ ≥ 0 in Q, by
Theorem 3.1. In particular, w(xo , to ) ≥ −σ for all σ > 0. Therefore w ≥ 0
in ST1 , since (xo , to ) ∈ ST1 is arbitrary. To conclude the proof we repeat the
1
argument in adjacent non-overlapping strips of width not exceeding 8β
, up to
cover the whole of ST .
Theorem 4.2. Let u ∈ H(ST ) ∩ C(S̄T ) satisfy H(u) ≤ 0 in ST and
there exist positive constants C, α, r such that
2
u(x, t) ≤ Ceα|x| for all |x| ≥ r and all 0 ≤ t ≤ T.
Then
u(x, t) ≤ sup u(·, 0)
RN
for all (x, t) ∈ ST .
(4.4)
154
5 THE HEAT EQUATION
Proof. We may assume that u(·, 0) ∈ L∞ (RN ); otherwise, the statement
is trivial. Assume first that T is so small that 4αT < 1 and consider the
(bounded) cylinder Q = [|x| < ρ] × (0, T ). The function
w =u−
2
ε
e|x| /4(T −t) ,
[4π(T − t)]N/2
ε>0
satisfies H(w) ≤ 0 in ST , and w(x, 0) ≤ supRN u(·, 0) for |x| < ρ. Moreover,
for |x| = ρ,
2
2
w |x|=ρ ≤ Ceαρ − ε(4πT )−N/2 eρ /4T .
Therefore, since 4T < 1/α, having fixed ε > 0, the parameter ρ can be chosen
so large that w||x|=ρ ≤ 0. The conclusion now follows from Theorem 3.1 and
the arbitrariness of ε. If 4αT ≥ 1, subdivide ST into finitely many strips of
1
width less than 4α
.
4.1 A Priori Estimates
Proposition 4.2 Let u ∈ H(ST ) ∩ C(S̄T ) satisfy (4.3) and (4.4). Then
kuk∞,ST ≤ ku(·, 0)k∞,RN + T kH(u)k∞,ST .
Proof. Assume that kuo k∞,RN and kH(u)k∞,ST are finite; otherwise, the
statement is trivial. The functions
w± = ku(·, 0)k∞,RN + tkH(u)k∞,ST ± u
satisfy H(w± ) ≥ 0 in ST and w± (·, 0) ≥ 0. Moreover, both w̄± satisfy the
asymptotic behavior (4.3). Therefore w± ≥ 0 in ST , by Theorem 4.1.
Remark 4.1 The functional dependence of this estimate is optimal. Indeed,
the estimate holds with equality for the function u = 1 + t.
4.2 About the Growth Conditions (4.3) and (4.4)
The conclusion of Theorem 4.1 fails if (4.3) is replaced by
u(x, t) ≥ −keβ|x|
2+ε
for any ε > 0.
However Theorem 4.2 continues to hold for a growth slightly faster than (4.4).
Precisely (S. Tacklind [249])
u(x, t) ≤ Ceα|x|h(|x|)
as |x| → ∞
where h(·) is positive nondecreasing and satisfies the optimal condition
Z ∞
ds
= +∞.
h(s)
5 Uniqueness of Solutions to the Cauchy Problem
155
5 Uniqueness of Solutions to the Cauchy Problem
Consider the class of functions w ∈ H(ST ) satisfying the growth condition
there exist positive constants C, α, r such that
2
(5.1)
|u(x, t)| ≤ Ceα|x| for all |x| ≥ r and all 0 ≤ t ≤ T.
Let u, v ∈ H(ST ) ∩ C(S̄T ) be solutions of the Cauchy problem (1.4) with
initial data uo , vo ∈ C(RN ) ∩ L∞ (RN ). If both u and v satisfy (5.1), then by
Proposition 4.2,
ku − vk∞,ST ≤ kuo − vo k∞,RN .
This inequality represents both a uniqueness and a stability result. Namely:
(i). Uniqueness: solutions of the Cauchy problem (1.4) are unique within
the class (5.1).
(ii). Stability: within such a class, small variations on the data, measured
in the norm of L∞ (RN ), yield small variations on the solution measured
in the same norm.
Proof (of Theorem 2.1 (Uniqueness)). If uo ∈ L∞ (RN ) ∩ C(RN ), the function
u defined by the representation formula (2.7) is bounded by virtue of (2.9).
It solves the heat equation in ST , and it satisfies (5.1) by virtue of Proposition 4.1. Therefore, it is the only bounded solution of the Cauchy problem
(1.4).
5.1 A Counterexample of Tychonov ([263])
The growth condition (5.1) is essential for uniqueness, as shown by the following counterexample due to Tychonov.
Proposition 5.1 There exists a non-identically zero solution to the Cauchy
problem
ut = uxx in R × (0, ∞), u(x, 0) = 0.
Proof. For z ∈ C, let
ϕ(z) =
and define
e−1/z
0
2
for z 6= 0
for z = 0

∞ dn
x2n
P
ϕ(t)
u(x, t) = n=0 dtn
(2n)!

0
Proceeding formally
for t > 0
for t = 0.
(5.2)
156
5 THE HEAT EQUATION
∞ dn
P
x2n
ϕ(t)
=0
n
t=0 (2n)!
n=0 dt
∞ dn
P
∂2u
x2n−2
=
ϕ(t)2n(2n − 1)
2
n
∂x
(2n)!
n=0 dt
n
2(n−1)
∞
P d
x
ϕ(t)
=
n
dt
(2(n
− 1))!
n=1
2n
n+1
∞
P d
x
∂u
ϕ(t)
=
.
=
n+1
dt
(2n)!
∂t
n=0
lim u(x, t) =
t→0
(i)
(ii)
These calculations become rigorous after we prove the following
Lemma 5.1 The series in (5.2) and (i)–(ii) are uniformly convergent in a
neighborhood of every point of R × R+ .
Proof. The function z → ϕ(z) is holomorphic in C − {0}. We identify the
t-axis as the real axis of the complex plane. If t > 0 is fixed, the circle
t
γ = z ∈ C z = t + eiθ ,
2
0 < θ ≤ 2π
does not meet the origin, and by the Cauchy formula ([31] page 72)
Z
n!
dn
ϕ(z)
ϕ(t)
=
dz for all n ∈ N.
dtn
2πi γ (z − t)n+1
From this
n!
dn
ϕ(t) ≤
dtn
2π
Z
γ
−2
n!
e−Re(z )
|dz| =
|z − t|n+1
2π
For z ∈ γ
2
1
z = t 1 + eiθ
2
2
2
and
From this Re(z −2 ) ≥ (2t)−2 and
dn
ϕ(t) ≤ n!
dtn
n Z 2π
−2
2
e−Re(z ) dθ.
t
0
1
1 1 + 41 e−2iθ + e−iθ
= 2
2 2 .
z2
t
1 + 21 eiθ
n
2
2
e−1/4t ,
t
n ∈ N.
Fix a > 0. For all |x| < a, the series in (5.2) is majorized, term by term, by
the uniformly convergent series
n 2 n
∞
2
2
2 P
(a )
1
= e−1/4t ea /t .
e−1/4t
t
n!
n=0
Here we have used the Stirling inequality
2n n!
1
≤ .
(2n)!
n!
1
6 Initial Data in Lloc
(RN )
157
Remark 5.1 The function in (5.2) can also be defined for t < 0. Therefore
the backward Cauchy problem
ut − ∆u = 0 in RN × (−∞, 0),
u(x, 0) = 0
fails in general to have a unique solution.
6 Initial Data in L1loc (RN )
The Cauchy problem for the heat equation can be solved uniquely for rather
1
(RN ), provided they satisfy the
coarse initial data, for example uo ∈ Lloc
growth condition (4.1). The solution will exist only within the strip ST for
0 < T < 4α1 o , and the initial datum is taken in the sense of L1loc (RN ), i.e.,
ku(·, t) − uo k1,K → 0 as t → 0, for all compact K ⊂ RN .
(6.1)
Theorem 6.1. Let uo ∈ L1loc (RN ) satisfy (4.1). Then (2.7) defines a solution
of the Cauchy problem
1
4αo
in the sense of L1loc (RN ).
H(u) = 0
in ST for 0 < T <
u(·, 0) = uo
(6.2)
Such a solution is unique within the class (5.1).
Proof. Fix ρ ≥ ro , where ro is the constant in the growth condition (4.1). For
almost all x ∈ Bρ , write
Z
Γ (x − y; t)|uo (y) − uo (x)|dy.
|u(x, t) − uo (x)| ≤
RN
Integrating in dx over Bρ
Z
Z
|u(x, t) − uo (x)|dx ≤
Bρ
Bρ
=
Z
Bρ
+
Z
Z
RN
Z
Γ (x − y; t)|uo (y) − uo (x)|dydx
|x−y|≤σ
Bρ
Z
Γ (x − y; t)|uo (y) − uo (x)|dydx
|x−y|>σ
Γ (x − y; t)|uo (y) − uo (x)|dydx
= I1 + I2 .
Let h be a vector in RN of size |h| ≤ σ. Then the first integral is estimated
by
Z
I1 ≤ sup
|uo (x + h) − uo (x)|dx.
h∈RN
|h|≤σ
Bρ
158
5 THE HEAT EQUATION
The second integral is estimated by
Z
Z
−|η|2
I2 ≤ π −N/2
e
√
|η|>σ/2 t
Bρ
√
|uo (x + 2 tη) − uo (x)|dxdη.
√
such that 2 t|η| < 2ρ, estimate
For all t > 0 and η ∈ RN
Z
√
|uo (x + 2 tη) − uo (x)|dx ≤ 2kuo k1,B2ρ .
Bρ
√
If 2 t|η| ≥ 2ρ, by the growth condition (4.1)
Z
√
2
2
|uo (x + 2 tη) − uo (x)|dx ≤ kuo k1,B2ρ + C|Bρ | sup e2αo |x| +8αo t|η| .
x∈Bρ
Bρ
Therefore for ρ > ro fixed
ku(·, t) − uo k1,Bρ ≤ const(ρ)
Z
2
√
|η|>σ/2 t
e−(1−8αo t)|η| dη
+ sup kuo (x + h) − uo (x)k1,Bρ .
h∈RN
|h|≤σ
The proof is concluded by recalling that the translation Th uo = uo (· + h) is
continuous in L1loc (RN ) ([50], Chapter IV, Section 20).
6.1 Initial Data in the Sense of L1loc (RN )
Part of the definition of a solution to the Cauchy problem (1.4) or (6.2) is
to make precise in what sense the initial data are taken. The notion (6.1) is
the weakest unambiguous requirement for data uo ∈ L1loc (RN ). If (6.1) holds,
then there exists a sequence of times {tn } → 0 such that
u(x, tn ) → uo (x)
for almost all x ∈ RN .
and one might be tempted to take this as the sense in which u(·, t) takes its
datum at t = 0. Such a definition might, however, generate ambiguity. Indeed,
the uniqueness may be lost, as shown by the following examples. The function
Γ satisfies the heat equation in S∞ and the growth condition (5.1). Moreover,
Γ (·, t) → 0 a.e. in RN , as t → 0, and yet Γ 6≡ 0. For such a “solution” the
identically zero initial datum is not taken in the sense of L1loc (RN ). Indeed,
for all ρ > 0
Z
Z
−N/2
−|η|2
Γ (x, t)dx = π
dη → 1 as t → 0.
√ e
Bρ
|η|<ρ/2 t
Even more striking is the following example in one space dimension. The
function
7 Remarks on the Cauchy Problem
159
x
x2
v(x, t) = √ 3/2 e− 4t
4 πt
is a solution of the heat equation in R×R+ satisfies all the previous properties,
and in addition, v(x, t) → 0 as t → 0 for all x ∈ R. And yet v 6≡ 0. One checks
that for all ρ > 0
Z ρ
√
1
t
as t → 0
|v(x, t)|dx → √
π
−ρ
that is, the initial datum uo = 0 is not taken in the sense of L1loc (RN ).
7 Remarks on the Cauchy Problem
7.1 About Regularity
Let uo be locally analytic in RN and assume that it satisfies the growth
condition (4.1). Then formula (2.7) defines the unique solution, within the
class (5.1), to the Cauchy problem (1.4) in ST for 0 < T < 4α1 o . Such a solution
is locally analytic in the space variables. It is also analytic in the time variable
within RN × (ε, T ) for all ε ∈ (0, T ). Having in mind the Cauchy–Kowalewski
theorem, it is natural to ask whether u is analytic in the x and t variables up
to t = 0. This is in general false, as shown by the following argument.
If u were analytic in t up to t = 0, then u, ut , ∆u would have, in a right
neighborhood of the origin, the absolutely convergent series representations
∞
P
u(x, t) =
ut (x, t) =
∆u(x, t) =
n=0
∞
P
n=1
∞
P
ϕn (x)tn
(7.1)
ϕn (x)ntn−1
∆ϕn (x)tn
n=0
with analytic coefficients ϕn . This in the equation H(u) = 0 gives
ϕn+1 =
1
∆ϕn ,
(n + 1)
n = 0, 1, . . .
From this, by iteration, starting from ϕo = uo
ϕn =
∆n uo
,
n!
n = 0, 1, 2, . . .
where ∆0 = I and ∆n = ∆n−1 ∆, for n ∈ N. Putting this in (7.1) gives a
representation of u in the form
u(x, t) =
∞ ∆n u
P
o n
t .
n!
n=0
160
5 THE HEAT EQUATION
From the uniform convergence, it follows that
∆n uo (x) n
t →0
n!
as n → ∞
(7.2)
for (x, t) fixed in the domain of uniform convergence. Let Dn uo denote the
generic derivative of uo , of order n . Expanding uo about x within a ball of
radius t we must have
|Dn uo (x)| n
t →0
n!
as n → ∞.
(7.3)
Now there exist locally analytic initial data uo satisfying (7.3) but not (7.2).1
7.2 Instability of the Backward Problem
We have already remarked that the backward Cauchy problem (2.10) in general does not have a solution. If it does, the datum uo must by analytic by
Remark 2.1 and the representation formula (2.11). Stability however might
be lost, as shown by the following example, due to Hadamard ([111]):
x
2
u(x, t) = εe−t/ε sin
, ε>0
ε
solves (2.10) with N = 1 and uo (x) = ε sin(x/ε). As ε → 0, uo → 0 in the
L∞ (R)-norm. Yet for all t < 0, for all intervals (−ρ, ρ), and for all 0 < p ≤ ∞
ku(·, t)kp,(−ρ,ρ) → ∞
as ε → 0.
8 Estimates Near t = 0
N
Let uo ∈ L∞
loc (R ) satisfy the growth condition (4.1), and let u be defined by
(2.7) in the strip ST for 0 < T < 4α1 o . We will study the behavior of u(·, t) and
|Du(·, t)| as t → 0. Since uo is locally bounded, in (4.1) we may take ro = 0,
by possibly modifying the constant Co . We will also investigate the behavior
of ut (·, t) and uxi xj (·, t) as t → 0, under the more stringent assumption that
δ
uo ∈ Cloc
(RN ) for some δ ∈ (0, 1).
N
Proposition 8.1 Let uo ∈ L∞
loc (R ) satisfy (4.1) with ro = 0. For all ρ > 0,
there exist constants Aℓ , for ℓ = 0, 1, depending only on ρ, N , αo , and Co ,
such that
|u(x, t)| ≤ Ao ,
|Du(x, t)| ≤ A1 t−1/2
(8.1)
for (x, t) ∈ Bρ × (0, T ].
1
2
Give examples of such functions in R. Hint: Attempt ex or ln(1 + x2 ), or a
variant of these.
8 Estimates Near t = 0
161
δ
Proposition 8.2 Let uo ∈ Cloc
(RN ), for some δ > 0, satisfy (4.1) with
ro = 0. For all ρ > 0 there exists a constant A depending only on ρ, N ,
αo , Co , δ, and the Hölder constant of uo over Bρ such that
|ut (x, t)| + |uxi xj (x, t)| ≤ Atδ/2−1
(8.2)
for (x, t) ∈ Bρ × (0, T ] and for all i, j = 1, . . . , N .
Proof (Proposition 8.1). Both estimates will follow from estimating
Jℓ =
Z
RN
|x − y|
2t
ℓ
Γ (x − y; t)|uo (y)|dy
for ℓ = 0, 1.
√
The change of variable y − x = 2 tη yields
Z
√
2
1
|η|ℓ e−|η| |uo (x + 2 tη)|dη
Jℓ = N/2 √ ℓ
π
( t) RN
2αo |x|2 Z
2
Co e
≤ N/2 √ ℓ
|η|ℓ e−(1−8αo t)|η| dη.
π
( t) RN
Thus if |x| < ρ and t is so small that (1 − 8αo t) ≥ 21
Z
const(Co , N, αo ) ∞ N −1+ℓ − 1 r2
Jℓ ≤
r
e 2 dr.
tℓ/2
0
Proof (Proposition 8.2). First one computes
Z
Z
∂2
∂2
Γ (x − y; t)dy
0=
Γ (x − y; t)dy =
∂xi ∂xj RN
RN ∂xi ∂xj
Z
∂2
=
Γ (x − y; t)dy.
RN ∂yi ∂yj
Then for |x| < ρ and 0 < t ≤ T
|ut (x, t)| + |uxi xj (x, t)| ≤ 2
N
P
h,k=1
Z
RN
∂2
Γ (x − y; t)(uo (y) − uo (x))dy
∂yh ∂yk
1
|x − y|
Γ (x − y; t)|uo (y) − uo (x)|dy
+
≤ 2N
4t2
2t
RN
Z
|x − y|2
1
=N
Γ (x − y; t)|uo (y) − uo (x)|dy
+
2t2
t
|y|<2ρ
Z
|x − y|2
1
+N
Γ (x − y; t)|uo (y) − uo (x)|dy
+
2t2
t
|y|>2ρ
Z
= H1 + H2 .
δ
Since uo ∈ Cloc
(RN )
2
162
5 THE HEAT EQUATION
H1 ≤ 2N ho (4πt)−N/2
Z
|y|<2ρ
|x − y|2+δ
|x − y|δ − |x−y|2
4t
+
e
dy
4t2
2t
where ho √
is the Hölder constant of uo over B2ρ . Perform the change of variables
y − x = 2 tη, and majorize the resulting integral by extending it to the whole
of RN to get
H1 ≤
Ã
t1−δ/2
Z
∞
0
2
|η|N −1 (|η|2+δ + |η|δ )e−|η| d|η|
where à = 4N ho π −N/2 ωN 2δ . To estimate H2 , perform the same change of
variables to get
Z
(|η|2 + 1) −(1−8αo t)|η|2
−N/2 2αo |x|2
H2 ≤ 4Co N π
e
e
dη.
√
t
|η|>ρ/2 t
If t is so small that (1 − 8αo t) ≥ 12 , this gives H2 ≤ Âtδ/2−1 , where
 =
sup
4Co N π
−N/2 2αo ρ2
e
t∈(0,1/4αo )
Z
√
|η|>ρ/2 t
(|η|2 + 1) − 1 |η|2
e 2 dη.
tδ/2
9 The Inhomogeneous Cauchy Problem
Consider the problem of finding u ∈ H(ST ) satisfying
H(u) = f in ST ,
u(·, 0) = uo .
(9.1)
Assume that uo is in L1loc (RN ) and satisfies the growth condition (4.1). The
initial datum in (9.1) is taken in the sense of L1loc (RN ). On the forcing term
f , we assume
δ
f (·, t) ∈ Cloc
(RN ) for some δ > 0 uniformly in t > 0.
(9.2)
Moreover f (·, t) is required to satisfy the same growth condition as (4.1),
uniformly in t, i.e.,
|f (x, t)| ≤ Co eαo |x|
2
for |x| > ro uniformly in t ≥ 0.
(9.3)
Theorem 9.1. Let (9.2) and (9.3) hold. Then there exists a solution to the
inhomogeneous Cauchy problem (9.1) in the strip ST for 0 < T < 4α1 o . Moreover, the solution is unique within the class (5.1) and is represented by
u(x, t) =
Z
RN
Γ (x − y; t)uo (y)dy +
Z tZ
0
RN
Γ (x − y; t − s)f (y, s)dyds. (9.4)
10 Problems in Bounded Domains
163
Proof. Since the heat operator is linear, u can be constructed as the sum of
the solution of the homogeneous Cauchy problem (f = 0) and the solution
of (9.1) with uo = 0. Thus it suffices to take uo = 0 in (9.1). The family of
homogeneous Cauchy problems
(x, t; s) → v(x, t; s) ∈ H(RN × [0 < s < t ≤ T ])
vt − ∆v = 0 in RN × (s, T )
v(·, s; s) = f (·, s)
has, for all 0 < s < t ≤ T , the unique bounded solution
Z
v(x, t; s) =
Γ (x − y; t − s)f (y, s)dy
RN
valid for 0 < t − s < T . We claim that the function
Z t
v(x, t; s)ds
u(x, t) =
0
solves (9.1), with uo = 0. To show this, first observe that by virtue of the
estimates of the previous section and assumption (9.2) and (9.2), the integrals
Z t
Z t
Z t
vt (x, t; s)ds,
v(x, t; s)dx,
vxi xj (x, t; s)ds
(9.5)
0
0
0
are uniformly convergent over compact subsets of RN . The convergence of
the first integral implies that u(·, t) → 0 as t → 0, in the sense of L1loc (RN ).
Moreover, by direct calculation
Z t
ut = v(x, t; t) +
vt (x, t; s)ds
0
Z t
∆v(x, t; s)ds = f (x, t) + ∆u(x, t)
= f (x, t) +
0
where the calculation of the derivatives under the integral is justified by the
uniform convergence of the integrals in (9.5). Thus u is a solution of (9.1) with
uo = 0. Such a solution is unique in view of (9.3).
This method is a particular case of the Duhamel principle. See Section 3.1c
of the Complements of Chapter 6.
10 Problems in Bounded Domains
Let E be a bounded region of RN with boundary ∂E of class C 1 and consider
the Dirichlet problem (1.2). If g = 0, the problem is referred to as the homogeneous Dirichlet problem. We may solve such a homogeneous problem by separation of variables, i.e., by seeking solutions of the form u(x, t) = X(x)T (t).
Using the PDE, we find
164
5 THE HEAT EQUATION
T ′ (t) = −λT (t),
∆X = −λX,
t > 0;
X
∂E
= 0.
(10.1)
The second of these is solved by an infinite sequence of pairs (λn , vn ), where
{λn } is an increasing sequence of positive numbers and {vn } is a sequence
of functions that form a complete orthonormal set in L2 (E) (Section 11 of
Chapter 4). In particular, the initial datum uo , regarded as an element of
L2 (E), can be expanded as
uo (x) =
∞
P
i=1
huo , vi ivi (x)
with
kuo k22,E =
P
|huo , vi i|2 .
Then with λn determined by (10.1), one has Tn (t) = To,n e−λn t , where To,n are
selected to satisfy the initial condition uo . This gives approximate solutions
of the form
n
P
To,i e−λi t vi (x), To,i = huo , vi i.
un (x, t) =
i=1
Lemma 10.1 The sequence {un (·, t)}, is Cauchy in L2 (E), uniformly in t.
Proof. Fix ε > 0 and let no = no (ε) be such that
∞
P
i>no
|huo , vi i|2 < ε.
(10.2)
Next for all m > n > no and all 0 ≤ t ≤ T
2
≤
kum (·, t) − un (·, t)k2,E
m
P
i=n
huo , vi ie−λi t vi (x)
2
2,E
≤
∞
P
i>no
|huo , vi i|2 < ε.
Thus, formally, a solution to the homogeneous Dirichlet problem (1.2) is
u(x, t) =
∞
P
i=1
huo , vi ie−λi t vi (x)
(10.3)
where the convergence of the series is meant in the sense of L2 (E), uniformly
in t ∈ [0, T ]. It remains to interpret in what sense the PDE is satisfied and in
what sense u takes the boundary data.
Lemma 10.2 Let u be defined by (10.3). Then t → u(·, t) is continuous in
L2 (E), Moreover, u(·, t) takes the initial datum uo in the sense of L2 (E)
ku(·, t) − uo k2,E → 0
as t → 0.
(10.4)
Finally, u(·, t) satisfies the decay estimate
ku(·, t)k2,E ≤ e−λ1 t kuo k2,E
where λ1 is the first eigenvalue of the Laplacean in E.
(10.5)
10 Problems in Bounded Domains
165
Proof. From the definitions
u(x, t) − uo (x) =
∞
P
i=1
To,i (e−λi t − 1)vi (x).
Fix ε > 0 and choose no as in (10.2). Then
2
=
ku(·, t) − uo k2,E
≤
∞
P
i=1
no
P
i=1
|huo , vi i|2 (e−λi t − 1)2
|huo , vi i|2 (e−λi t − 1)2 + ε
2
+ ε.
≤ (1 − e−λno t )2 kuo k2,E
Therefore, letting t → 0 gives
lim sup ku(·, t) − uo k2,E ≤
√
ε.
t→0
This proves (10.4) and also that t → u(·, t) is continuous at t = 0, in the
topology of L2 (E). The continuity at every t ∈ [0, T ] is proved in a similar
fashion. The decay estimate (10.5) follows from the representation (10.3),
Parseval’s identity, and the fact that {λn } is an increasing sequence.
Remark 10.1 This construction procedure as well as Lemmas 10.1 and 10.2
require only that the initial datum uo be in L2 (E).
10.1 The Strong Solution
Assume that N ≤ 3 and that the initial datum uo is in C 2 (Ē) and satisfies
uo = 0 and Duo = 0 on ∂E. Then, by Corollary 10.2 of Chapter 4, the series
in (10.3) is absolutely and uniformly convergent. This implies that u satisfies
the homogeneous boundary data on ∂E, in the sense of continuous functions.
Also, the series
∞
P
i=0
huo , vi i
d −λi t
e
vi (x)
dt
and
∞
P
i=0
huo , vi ie−λi t ∆vi (x)
are absolutely and uniformly convergent. Therefore, the heat operator H(·)
can be applied term by term in (10.3) to give
H(u) =
∞
P
i=0
huo , vi iH[e−λi t vi (x)] = 0.
We conclude that if 1 ≤ N ≤ 3, and if uo satisfies the indicated regularity properties, then u as defined by (10.3) is a solution of the homogeneous
Dirichlet problem (1.2).
166
5 THE HEAT EQUATION
10.2 The Weak Solution and Energy Inequalities
If N > 3, or if uo ∈ L2 (E), we will interpret the PDE in a weak sense. By
construction, each un satisfies
un,t − ∆un = 0 in ET
=0
n
P
huo , vi ivi .
un (·, 0) =
un (·, t)
(10.6)
∂E
i=0
2
Let ϕ ∈ C (ĒT ) vanish on ∂E for all t. Multiply the PDE satisfied by un by
ϕ and integrate by parts over Et to obtain
Z tZ
Z
Z
[un ϕt + un ∆ϕ] dx dt =
(un ϕ)(t)dx −
(uo,n ϕ)(x, 0)dx.
E
0
E
E
Letting n → ∞ gives
Z
Z
Z tZ
uH ∗ (ϕ) dx dt =
(uϕ)(t)dx −
uo ϕ(x, 0)dx.
0
E
E
(10.7)
E
In this limiting process we use Lemma 10.1, which is valid for all N ≥ 1. We
regard (10.7), as a weak notion of a solution of the homogeneous Dirichlet
problem (1.2), and we call u a weak solution.
Lemma 10.3 Weak solutions in the sense of (10.5), (10.7) are unique.
Proof. The difference w = u1 − u2 of any two solutions satisfies (10.3), and in
particular
Z tZ
Z
(10.8)
(wϕ)(t)dx −
w∆ϕ dx dτ = 0
0
E
E
for all ϕ ∈ Co1 (E) independent of t. Since w ∈ L2 (ET ), it must have for a.e.
t ∈ (0, T ) a representation in terms of the eigenfunctions {vn }, i.e.,
w(x, t) = lim
n
P
n→∞ i=0
ai (t)vi (x)
In (10.8) choose ϕ = vi to get
Z t
ai (t) + λi
ai (s)ds = 0
0
Thus ai (·) = 0 for all i ∈ N, and w = 0.
for a.e. t ∈ (0, T ).
for all i ∈ N.
Remark 10.2 The choice ϕ = vi is admissible if vi ∈ Co1 (E). By Corollary 11.2 of Chapter 3, the eigenfunctions vi are Hölder continuous in Ē. By
the Schauder estimates of Section 9 of Chapter 2, vi ∈ C 2+η (E), and by a
bootstrap argument, vi ∈ C ∞ (E). Actually vi are of class Co1+η up to ∂E.
Such an estimate up to the boundary, has been indicated in Section 9c of the
Complements of Chapter 2.
11 Energy and Logarithmic Convexity
167
Remark 10.3 If uo ∈ Co1 (E), and u is smooth enough, we may take u = ϕ
in (10.7) to obtain the energy identity
Z tZ
1
1
ku(t)k22,E − kuo k22,E +
|Du|2 dx dt = 0.
2
2
0
E
This identity also contains a statement of uniqueness since the PDE is linear.
Indeed, uo = 0 implies u(·, t) = 0.
11 Energy and Logarithmic Convexity
Let E be a bounded open set in RN with boundary ∂E of class C 1 and let
u ∈ H(ET )∩C(ĒT ) be a solution of the homogeneous Dirichlet problem (1.2).
The quantity
E(t) = ku(·, t)k22,E
is the thermal energy of the body E at time t.
Proposition 11.1 For every 0 ≤ t1 < t < t2 ≤ T
t2 −t
t−t1
E(t) ≤ [E(t1 )] t2 −t1 [E(t2 )] t2 −t1 .
(11.1)
Proof. Assume first that u is sufficiently regular as to justify the formal calculations below. Multiply the first of (1.2) by u(·, t) and integrate by parts
over E, taking into account that u(·, t) vanishes on ∂E. This gives
Z
Z
′
E =2
u∆u dx = −2
Du · Du dx.
E
From this
′′
E = −4
Z
E
E
Dut · Du dx = 4
Z
ut ∆u dx = 4
Z
E
E
From this and Hölder’s inequality
Z
2
′2
uut dx ≤ EE ′′ .
E = 2
u2t dx.
E
First assume that E(t) > 0 for all t ∈ [t1 , t2 ]. Then the function t → ln E(t) is
well defined and convex in such an interval, since
2
d2
E ′′ E − E ′
ln E =
≥ 0.
2
dt
E2
Therefore, for all t1 < t < t2
ln E(t) ≤
t2 − t
t − t1
ln E(t1 ) +
ln E(t2 ).
t2 − t1
t2 − t1
If E(t) ≥ 0, replacing it with Eε = E + ε for ε > 0 proves (11.1) for Eε .
Then let ε → 0. These calculations can be made rigorous by working with the
approximate solutions {un } of (10.5) and then by letting n → ∞.
168
5 THE HEAT EQUATION
Remark 11.1 The energy E(·) can also be defined for solutions of the homogeneous Neumann problem (1.2), and (11.1) holds for it ([200]).
11.1 Uniqueness for Some Ill Posed Problems
Corollary 11.1 There exists at most one solution to the homogeneous backward Dirichlet problem
u ∈ H(ET ) ∩ C(ĒT ), H(u) = 0 in ET
u(·, T ) = uT ∈ C(Ē), u(·, t) ∂E = 0.
Proof. It suffices to show that uT = 0 implies u(·, t) = 0. This follows from
(11.1) with t2 = T .
12 Local Solutions
We have observed that solutions of the Cauchy problem representable by (2.7)
are analytic in the space variables and C ∞ in time for t > 0. It turns out that
this is also the case for every local solution of the heat equation in a spacetime cylindrical domain ET . Let Qρ = Bρ × (−ρ2 , 0) denote the cylinder with
“vertex” at the origin, height ρ2 , and transversal cross section the ball Bρ .
For (xo , to ) ∈ RN +1 , we let (xo , to ) + Qρ denote the box congruent to Qρ and
with “vertex” at (xo , to ), i.e.
(xo , to ) + Qρ = [|x − xo | < ρ] × [to − ρ2 , to ].
If (xo , to ) ∈ ET , we let ρ > 0 be so small that (xo , to ) + Q4ρ ⊂ ET . We also
denote the integral average of |u| over (xo , to ) + Q4ρ by
Z
Z
1
|u| dy ds.
|u| dy ds =
−
|Q4ρ | (xo ,to )+Q4ρ
(xo ,to )+Q4ρ
Proposition 12.1 (Gevrey [97]) Let u ∈ H(ET ) be a solution of the heat
equation in ET . There exist constants γ and C depending only on N such that
for every box (xo , to ) + Q4ρ ⊂ ET
sup
(xo ,to )+Qρ
Z
C |α| |α|!
D u ≤γ
|u| dy ds
−
ρ|α| (xo ,to )+Q4ρ
α
(12.1)
for all multi-indices α. Moreover
sup
(xo ,to )+Qρ
Z
∂ku
C 2k (2k)!
≤γ
|u| dy ds
−
∂tk
ρ2k
(xo ,to )+Q4ρ
for all positive integers k.
(12.2)
12 Local Solutions
169
Proof. It suffices to prove only (12.1), since
∂k
u = ∆k u.
∂tk
After a translation, we may assume that (xo , to ) coincides with the origin and
u is a solution of the heat equation in Q4ρ . Construct a non-negative smooth
cutoff function ζ in Q4ρ satisfying
|Dζ| ≤
ζ = 1 in Q2ρ ,
1
,
ρ2
0 ≤ ζt ≤
The function
w=
1
,
4ρ
|ζyi yj | ≤
1
i, j = 1, . . . , N
ρ2
ζ(y, s) = 0 for |y| ≥ 4ρ and s ≤ −(4ρ)2 .
uζ in [|y| ≤ 4ρ] × (−(4ρ)2 , 0)
0 otherwise
coincides with u within Q2ρ and satisfies
def
H(w) = uH(ζ) − 2Du · Dζ = f
in RN × (−(4ρ)2 , 0].
Therefore, it can be viewed as the unique solution of the inhomogeneous
Cauchy problem
H(w) = f in RN × (−(4ρ)2 , 0],
w(·, −(4ρ)2 ) = 0.
By Theorem 9.1
w(x, t) =
Z
t
−(4ρ)2
Z
RN
Γ (x − y; t − s)f (y, s) dy ds.
From this, after an integration by parts
Z t
Z
w(x, t) =
u(y, s) Γ (x − y; t − s)H ∗ (ζ)
−(4ρ)2
=
Z
−(2ρ)2
−(4ρ)2
+
RN
Z
+2
t
+ 2DΓ (x − y; t − s) · Dζ dy ds
Z
Γ (x − y; t − s)ζt u dy ds
|y|<4ρ
−(4ρ)2
Z t
Z
−(4ρ)2
2ρ<|y|<4ρ
Z
Γ (x − y; t − s)∆ζ u dy ds
2ρ<|y|<4ρ
DΓ (x − y; t − s) · Dζ u dy ds.
Observe that in these integrals, if |x| < ρ and −ρ2 ≤ t ≤ 0, the kernel is not
singular. Take the space derivatives of any order of both sides, for x and t in
such a range, and use the properties of the cutoff function ζ to obtain
170
5 THE HEAT EQUATION
1
sup |D u| ≤ 2
ρ
Qρ
α
Z
−(2ρ)2
−(4ρ)2
Z
|y|<4ρ
|Dα Γ (x − y; t − s)||u| dy ds
Z t
Z
1
|Dα Γ (x − y; t − s)||u| dy ds
ρ2 −(4ρ)2 2ρ<|y|<4ρ
Z
Z
n
P
1 t
|Dα Γxi (x − y; t − s)||u| dy ds
+
ρ −(4ρ)2 2ρ<|y|<4ρ i=1
+
= J1 + J2 + J3 .
In the estimates to follow we denote by C and γ generic positive constants that
can be different in different contexts. These may be quantitatively determined
a priori only in terms of N and are independent of the multi-index α.
Lemma 12.1 There exists a positive constant C such that
|α|
ρ
|α|!
|Dα Γ (x − y; t − s)| ≤ C |α|
+ |α| Γ (x − y; t − s)
t−s
ρ
for all (x, t) ∈ Qρ and (y, s) ∈ Q4ρ , and for every multi-index α.
Assuming the lemma for the moment, we proceed to estimate Ji . In estimating J1 observe that within the domain of integration t − s > ρ2 . Therefore
|Dα Γ (x − y; t − s)| ≤ γ
and
J1 ≤ γ
C |α| |α|! 1
ρ|α| |Q4ρ |
Z
Q4ρ
C |α| |α|!
ρ|α|+N
|u| dy ds.
The estimation of J2 and J3 hinges on the supremum of the function
g(τ ) =
1 −A/τ
e
,
τm
τ >0
where A and m are given positive constants. The supremum of g is achieved
for τ = A/m, and
m m
e−m for all τ ≥ 0.
|g(τ )| ≤
A
Within the domain of integration of J2 and J3 one has |x − y| > ρ, provided
|x| < ρ. Therefore
2
ρ|α|
e−ρ /4(t−s)
[4(t − s)]|α|+N/2
2
1
|α|!
e−ρ /4(t−s)
+ γC |α| |α|
N/2
ρ [4(t − s)]
|Dα Γ (x − y; t − s)| ≤ γC |α|
≤γ
C |α|
ρ
|α||α| e−|α| + γ
|α|+N
C |α| |α|!
.
ρ|α|+N
12 Local Solutions
171
By Stirling’s inequality mm e−m ≤ γm!. Therefore by modifying the constants
C and γ
C |α| |α|! 1
|Dα Γ (x − y; t − s)| ≤ γ
ρ|α| ρN
for (x, t) ∈ Qρ and (y, s) ∈ Q4ρ − Q2ρ . With this estimate in hand, we deduce
that for (x, t) ∈ Qρ and all multi-indices α
C |α| |α|! 1
J2 ≤ γ
ρ|α| |Q4ρ |
Z
|u| dy ds.
Q4ρ
As for J3 , the previous calculations give
C |α|+1 |α|!(|α| + 1) 1
J3 ≤ γ
|Q4ρ |
ρ|α|
Z
Q4ρ
|u| dy ds.
Now the constant C can be further modified so that
C |α|+1 (|α| + 1) ≤ C̄ |α|
for all multi-indices α
and the theorem follows.
Proof (of Lemma 12.1). Fix a multi-index α of size |α| = n and let β be a
multi-index of size |β| = n + 1. Then
Dβ Γ = Dα Γxi = Dα
(x − y)i
Γ
2(t − s)
for some i = 1, . . . , N . From this
n−1
ρ
β
|Dα Γ | +
|Dᾱ Γ |
2|D Γ | ≤
t−s
(t − s)
where ᾱ is a multi-index of size |ᾱ| = n − 1. The lemma holds for n = 1. By
induction, assuming that it does hold for multi-indices α of size |α| ≤ n, we
show that it continues to hold for multi-indices β of size |β| = n + 1. Using
the induction hypothesis
2 β
|D Γ | ≤ C n
Γ
ρ
t−s
n+1
+
ρ
t−s
2n!
+
ρn
ρ
t−s
n
n−1
.
Cρ
By Young’s inequality
n+1
n+1
n+1
2n!
ρ
2 n n (n!) n
1
ρ
+
≤
t − s ρn
n+1 t−s
n + 1 ρn+1
n
n+1
n−1
ρ
(n − 1)n+1
1
ρ
n
+
.
≤
t−s
Cρ
n+1 t−s
n + 1 C n+1 ρn+1
172
5 THE HEAT EQUATION
Using Stirling’s inequality and choosing C sufficiently large
(n − 1)n+1 1
≤ (n + 1)!.
n + 1 C n+1
This in turn implies
n+1 |Dβ Γ | C n
ρ
4n
1/n (n + 1)!
2
.
≤
+ 1+
(n!)
Γ
2
t−s
(n + 1)2
ρn+1
The number (n!)1/n is the geometric mean of the first n integers, which is
majorized by its arithmetic mean. Therefore
Pn
4n
2n
4n
1/n
i=1 i
(n!)
≤
=
.
(n + 1)2
(n + 1)2 n
n+1
These remarks in the previous inequality yield
3
|Dβ Γ |
≤ Cn
Γ
2
ρ
t−s
n+1
(n + 1)!
.
+
ρn+1
It remains to choose C so that 32 C n ≤ C n+1 .
12.1 Variable Cylinders
To simplify the symbolism, let us assume that (xo , to ) coincides with the origin.
The estimates of Theorem 12.1 give information on Dα u on the cylinder Qρ
in terms of the L1 -norm of u over the larger box Q2ρ . The proof could be
repeated with minor variations to derive a similar statement for any pair of
boxes Qρ and Qσρ for σ ∈ (0, 1). Tracing the constant dependence on σ gives:
Proposition 12.2 Let u be a solution of the heat equation in Qρ . There exist
constants C and γ, depending only on N , such that for every multi-index α,
for every non-negative integer k, and for all σ ∈ (0, 1)
Z
C |α| |α|!
− |u| dy ds
(1 − σ)N +2+|α| ρ|α| Qρ
Z
C k (2k)!
≤γ
− |u| dy ds.
(1 − σ)N +2+2k ρ2k Qρ
kDα uk∞,Qσρ ≤ γ
∂k
u
∂tk
∞,Qσρ
(12.3)
(12.4)
Remark 12.1 Estimates (12.3)–(12.4) hold for any pair of boxes (xo , to )+Qρ
and (xo , to ) + Qσρ contained in ΩT .
12.2 The Case |α| = 0
We state explicitly the estimate of Proposition 12.2 for the case |α| = 0.
13 The Harnack Inequality
173
Corollary 12.1 Let u ∈ H(ET ) be a local solution of the heat equation in
ET . There exists a constant C depending only on N such that for every box
(xo , to ) + Qρ contained in ET and all σ ∈ (0, 1)
Z
C
sup
|u| ≤
|u| dy ds.
(12.5)
−
(1 − σ)N +2 (xo ,to )+Qρ
(xo ,to )+Qσρ
These estimates have a number of consequences for local or global nonnegative solutions. In the next two sections we present some of them.
13 The Harnack Inequality
Non-negative local solutions of the heat equation in ET satisfy an inequality similar to the Harnack estimate valid for non-negative harmonic functions(Section 5.1 of Chapter 2). This inequality can be stated as follows. For
ρ > 0 consider the box Qρ = Bρ × (−ρ2 , ρ2 ), with its “center” at the origin.
If (xo , to ) ∈ ET , let
(xo , to ) + Qρ = [|x − xo | < ρ] × (to − ρ2 , to + ρ2 )
be the box congruent to Qρ and centered at (xo , to ).
Theorem 13.1. Let u ∈ H(ET ) be a non-negative solution of the heat equation in ET . There exists a constant c depending only upon N such that for
every box (xo , to ) + Q4ρ ⊂ ET
inf
|x−xo |<ρ
u(x, to + ρ2 ) ≥ c u(xo , to ).
(13.1)
(xo , to + ρ2 )
...............................................................................................................................................
•
(xo , to )
Fig. 13.1
Such an estimate can be given different equivalent forms. We illustrate one of
them, assuming for simplicity of notation that (xo , to ) = (0, 0). To distinguish
between the upper part and the lower part of Qρ , let us set
2
Q−
ρ = Bρ × (−ρ , 0),
2
Q+
ρ = Bρ × (0, ρ ).
−
Fix σ ∈ (0, 1), and inside Q+
ρ and Qρ construct the two sub-boxes
2
Q−
Q∗σρ = Bσρ × (1 − σ)ρ2 , ρ2 .
σρ = Bσρ × (−σρ , 0),
174
5 THE HEAT EQUATION
Theorem 13.2. Let u ∈ H(Q4ρ ) be a non-negative solution of the heat equation in Q4ρ . For every σ ∈ (0, 1) there exists a constant c depending only upon
N and σ such that
(13.2)
inf
u ≥ c sup u.
∗
Qσρ
Q−
σρ
.....................................................................................................................................................................
... . . . ∗. . . . . . . . . ..
...
...
... . . . . . . . . . . . ...
..
...
............................ρσ
..
.................................................................................. .... .... .....
....
....
...
..
...
.............................................................................................................................................
...
....
...
...
..
...
..
............................................................................................................ .... .... ....
.
...
..............−
....
.
...
................................................... ....
...
..
.. . . . .ρσ
...
. . . . . . . . . . . . . ..
..............................................................................................................................................................
Q
·
(0, 0)
Q
·
ρ
ρσ
ρ2
(1 − σ)ρ2
−σρ2
−ρ2
Fig. 13.2
In the case of harmonic functions, the main tool in the proof of the Harnack
estimate was the explicit Poisson representation formula of the solution of
the Dirichlet problem for the Laplacean over a ball (formula (3.9) of Chapter 2). The corresponding Dirichlet problem for the heat equation over cylinders whose cross section is a sphere does not have an explicit solution formula.
However, local representations will play a major role via the regularity results
of Proposition 12.1.
The form (13.1) of the Harnack estimate is due independently to Pini
([203]) and Hadamard ([112]). The form (13.2) was introduced by Moser in
a more general context ([187]), which we will extensively deal with in Chapter 12. The proof we present here, based on an idea of Landis [153], is “nonlinear” in nature, and its main ideas can be applied to a large class of parabolic
equations, including degenerate ones ([49], Chapters 6–7, and [55]). Alternative forms of the parabolic Harnack inequality that resemble the mean value
property of harmonic functions are in [56]. We will consider them in Section 10.4 of Chapter 12.
The Harnack inequality is a property also shared by non-negative solutions of parabolic equations in nondivergence form, which are the subject of
Chapter 13. In particular, Section 4 of such a chapter will be devoted to the
Harnack estimate in this context.
13.1 Compactly Supported Sub-Solutions
For given positive numbers M , r, b, and (x, t) ∈ S∞ , consider the function
ψ(x, t) =
M r2b
(4 − |z|2 )2+ ,
(t + r2 )b
where |z|2 =
|x|2
.
t + r2
13 The Harnack Inequality
175
One verifies that ψ ∈ H(S∞ ) ∩ C(S̄∞ ), and it vanishes identically outside the
paraboloid |z| < 2.
Lemma 13.1 The number b > 0 can be chosen so that H(ψ) ≤ 0 in S∞ , for
all M > 0.
Proof. By direct calculation
H(ψ) =
For
4N
N +1
M r2b
|z|4
2
2
−b(4
−
|z|
)
+
4N
−
2
.
(4
−
|z|
)
+
+
(t + r2 )b+1
4 − |z|2
≤ |z|2 < 4, we have H(ψ) ≤ 0. For |z|2 ≤
H(ψ) ≤
4N
N +1
M r2b
4b
2
(1
−
|z|
)
−
+
4N
.
+
(t + r2 )b+1
N +1
To prove the lemma, choose b = N (N + 1).
We will consider a version of ψ “centered” at points (xo , to ) ∈ RN +1 . Precisely
Ψ(xo ,to ) (x, t) = ψ(x − xo , t − to ) =
M r2b
[t − to + r2 ]b
2
|x − xo |2
4−
.
t − to + r 2 +
Corollary 13.1 Let b = N (N + 1). Then
H Ψ(xo ,to ) ≤ 0 in RN × (to , ∞).
13.2 Proof of Theorem 13.1
We may assume that (xo , to ) = (0, 0), that ρ = 1, and that u(xo , to ) = 1. This
is achieved by the change of variables
x→
x − xo
,
ρ
t→
t − to
,
ρ2
u→
u
.
u(0, 0)
Thus we have to show that if u is a solution of the heat equation in the box
Q4 = B4 × (−4, 4) such that u(0, 0) = 1, then u(x, 1) ≥ c, for all x ∈ B1 , for
a positive constant c depending only on N . To prove this we proceed in three
steps.
13.2.1 Locating the Supremum of u in Q1
For s ∈ [0, 1), consider the family of nested and expanding boxes
Qs = [|x| < s] × (−s2 , 0]
and the nondecreasing family of numbers
176
5 THE HEAT EQUATION
Ns = (1 − s)−ξ
Ms = sup u,
Qs
where ξ is a positive constant to be chosen later. One checks that Mo = No =
1, and as s → 1
lim Ms < ∞ and
lim Ns = ∞.
s→1
s→1
Therefore the equation Ms = Ns has roots. Denote by so the largest of such
roots, so that
and Ms < (1 − s)−ξ
sup u = Mso = (1 − so )−ξ
Qso
for s > so .
Since u ∈ C(Q̄4 ), the supremum Mso is achieved at some (xo , to ) ∈ Q̄so , i.e.,
u(xo , to ) = (1 − so )−ξ .
13.2.2 Positivity of u over a Ball
We show next that the “largeness” of u at (xo , to ) spreads over a small ball
centered at xo at the level to .
Lemma 13.2 There exists ε > 0 depending only upon N and independent of
so such that
u(x, to ) ≥
1
(1 − so )−ξ
2
for all |x − xo | < ε
1 − so
.
2
Proof. Costruct the box with “vertex” at (xo , to ) and radius 12 (1 − so )
2 1 − so
1 − so
× to −
, to .
(xo , to ) + Q 21 (1−so ) = |x − xo | <
2
2
By construction, (xo , to ) + Q 12 (1−so ) ⊂ Q 12 (1+so ) , and by the definition of Ms
and Ns
sup
(xo ,to )+Q 1 (1−s
2
o)
u≤
sup
Q 1 (1+s
2
u ≤ N 1+so = 2ξ (1 − so )−ξ .
2
o)
Apply Proposition 12.2 with |α| = 1 over the pair of boxes (xo , to ) + Q 18 (1−so )
and (xo , to ) + Q 21 (1−so ) , to obtain
sup
(xo ,to )+Q 1 (1−s
8
o)
|Du| ≤
C
1 − so
u
sup
(xo ,to )+Q 1 (1−s
2
o)
for a constant C dependent only upon N . Let ε ∈ (0, 14 ) to be chosen later.
Then forall |x − xo | < ε 21 (1 − so )
u(x, to ) ≥ u(xo , to ) − ε
1 − so
2
sup
(xo ,to )+Q 1 (1−s
8
o)
|Du|
≥ (1 − so )−ξ (1 − 2ξ Cε).
To prove the lemma we choose ε small enough that 1 − 2ξ Cε = 12 .
14 Positive Solutions in ST
177
13.2.3 Expansion of the Positivity Set
The point (xo , to ) being fixed, consider the comparison function Ψ(xo ,to ) for
the choice of parameters
M=
1
(1 − so )−ξ ,
2
r=ε
1 − so
.
2
By Lemma 13.2
u(x, to ) ≥
1
(1 − so )−ξ ≥ Ψ(xo ,to ) (x, to ),
2
for |x − xo | < r.
Therefore, by the maximum principle, u ≥ Ψ(xo ,to ) in the box B4 × [to , 4). In
particular, for t = 1 and |x| < 1
u(x, 1) ≥ 21−4b (1 − so )−ξ+b .
The knowledge of so is only qualitative. We render the estimate independent
of so by choosing ξ = b. This gives u(x, 1) ≥ 21−4b = c.
Remark 13.1 A somewhat similar argument will be employed in Section 10.3
of Chapter 12.
14 Positive Solutions in ST
We have shown that uniqueness for the Cauchy problem (1.4) holds within
the class of functions satisfying the growth condition (5.1). However, the representation formula (2.7) is well defined for initial data uo ∈ L1loc (RN ) for
which the integral is convergent. This suggests that we consider the problem
of uniqueness within the class of functions u ∈ H(ST ) such that
Z
|u(y, s)|Γ (x − y; t − s)dy < ∞ for s ∈ (0, t).
(14.1)
RN
It turns out that uniqueness for the Cauchy problem holds for functions in such
a class. More important, every non-negative solution of the heat equation in
ST satisfies (14.1). Therefore, uniqueness for the Cauchy problem (1.4) holds
within the class of non-negative solutions. This was observed by Widder in
one space dimension ([276]). Here we give a different proof, valid in any space
dimension.
Theorem 14.1. Let u ∈ H(ST ) satisfy
H(u) = 0 in ST ,
and
u(·, t) → 0 in L1loc (RN ) as t → 0.
Then, if u satisfies (14.1), it vanishes identically in ST .
178
5 THE HEAT EQUATION
Proof. Let (x, t) ∈ ST be arbitrary but fixed. For ρ > 2|x| consider the balls
B2ρ and let y → ζ(y) ∈ Co2 (B2ρ ) be a non-negative cutoff function in B2ρ
satisfying
1
22
ζ = 1 in Bρ , |Dζ| ≤ , |ζyi yj | ≤ .
ρ
ρ
For δ > 0 let

1 if u > δ

u
if |u| ≤ δ
(14.2)
hδ (u) =

 δ
−1 if u < −δ.
Multiply the PDE by hδ (u)ζ(y)Γ (x − y; t − s), and integrate by parts in dyds
over the cylindrical domain B2ρ × (τ, t − ε) for 0 < τ < t − ε and 0 < ε < t.
This gives
Z u
Z
hδ (ξ)dξ Γ (x − y; ε)ζ(y)dy
B2ρ ×{t−ε}
0
=
1
+
δ
Z
Z
t−ε
τ
Z
B2ρ
Z u
B2ρ ×{τ }
0
Z t−ε Z u
+
τ
0
|Du|2 Γ (x − y; t − s)χ[ |u| < δ]ζ(y) dy ds
hδ (ξ)dξ Γ (x − y; t − τ )ζ(y)dy
hδ (ξ)dξ (Γ ∆ζ + 2DΓ · Dζ) dy ds.
As τ → 0, the first integral on the right-hand side tends to zero, since it is
majorized by
Z
const
|u(y, τ )|dy → 0 as τ → 0.
B2ρ
Discard the second term on the left-hand side since it is non-negative and let
first δ → 0 and then τ → 0 to obtain
Z
|u(y, t − ε)|Γ (x − y; ε)ζ(y)dy
B2ρ
≤
2
ρ2
+
Z
2
ρ
t−ε
0
Z
0
Z
B2ρ
t−ε
Z
|u(y, s)|Γ (x − y; t − s) dy ds
B2ρ −Bρ
|u(y, s)||DΓ (x − y; t − s)| dy ds.
The right-hand side of this inequality tends to zero as ρ → ∞. This is obvious
for the first term in view of (14.1). The second term is majorized by
Z
Z
1 t−ε
|x − y|
dy ds
|u(y, s)|Γ (x − y; t − s)
ρ 0
t−s
ρ<|y|<2ρ
Z
Z
4 t−ε
|u(y, s)|Γ (x − y; t − s) dy ds.
≤
ε 0
ρ<|y|<2ρ
14 Positive Solutions in ST
Letting ρ → ∞ gives
Z
Br
179
|u(y, t − ε)|Γ (x − y; ε)dy = 0
for all r > 2|x| and all ε ∈ (0, t). Finally, we let ε → 0. Arguing as in Section 2,
in the derivation of the representation formula (2.7), gives u(x, t) = 0.
14.1 Non-Negative Solutions
Theorem 14.2. Let u ∈ H(ST ) be a non-negative solution of
H(u) = 0 in ST ,
and
u(·, t) → 0 in L1loc (RN ) as t → 0.
Then u vanishes identically in ST .
It will suffice to prove:
Proposition 14.1 Let u ∈ H(ST ) be a non-negative solution of the heat
equation in ST . Then ∀(xo , to ) ∈ ST
Z
u(y, s)Γ (xo − y; to − s)dy ≤ u(xo , to ) for all 0 < s < to .
(14.3)
RN
Proof. Fix (xo , to ) ∈ ST and s ∈ (0, to ) and introduce the change of variables
τ=
t−s
,
to − s
The function
U (η, τ ) = u xo +
y − xo
η= √
.
to − s
√
to − sη, s + (to − s)τ
satisfies the heat equation in RN × [0, 1]. For such a function, (14.3) becomes
Z
U (η, 0)Γ (η; 1)dη ≤ U (0, 1).
RN
Thus it will be enough to prove that if u ∈ H(S̄1 ) is a non-negative solution
of the heat equation in S1 such that u(·, 0) ∈ C 2 (RN ), then
Z
u(y, 0)Γ (y; 1)dy ≤ u(0, 1).
(14.4)
RN
To prove (14.4) fix ρ > 0 and consider the Cauchy problem
ζ(x)u(x, 0) if |x| < 2ρ
H(v) = 0 in S1 , v(x, 0) =
0
otherwise
(14.5)
where x → ζ(x) ∈ Co∞ (B2ρ ) is non-negative and equals one on the ball Bρ .
Since the initial datum is compactly supported in B2ρ , the unique bounded
solution of (14.5) is given by
Z
v(x, t) =
ζ(y)u(y, 0)Γ (x − y; t)dy.
|y|<2ρ
180
5 THE HEAT EQUATION
Lemma 14.1 u ≥ v in S̄1 .
Assuming this fact for the moment, it follows from the representation of v
and the structure of the cutoff function ζ that
Z
u(y, 0)Γ (y; 1)dy.
u(0, 1) ≥
Bρ
This proves (14.4), since ρ > 0 is arbitrary.
Proof (of Lemma 14.1). The statement would follow from the maximum principle if u satisfed the growth condition (5.1). The positivity of u will replace
such information. Let no be a positive integer larger than 2ρ, and for n ≥ no ,
consider the sequence of homogeneous Dirichlet problems
vn
H(vn ) = 0 in Qn = Bn × (0, 1)
|y|=n
=0
vn (x, 0) =
(14.6)
ζ(x)u(x, 0) if |x| < 2ρ
0
otherwise.
We regard the functions vn as defined in the whole of S1 by defining them
to be zero outside Qn . By the maximum principle applied over the bounded
domains Qn
0 ≤ vn ≤ vn+1 ≤ ku(·, 0)k∞,B2ρ
and
vn ≤ u
(14.7)
for all n ≥ no . By the second of these, the proof of the lemma reduces to
showing that the increasing sequence {vn } converges to the unique solution
of (14.5) uniformly over compact subsets of S1 . Consider compact subsets of
the type K = B̄R × [ε, 1 − ε] for ε ∈ (0, 12 ) and R ≥ 2ρ. By the estimates
of Proposition 12.1 and the uniform upper bound of the first of (14.7), for
every multi-index α and every positive integer k, there exists a constant C
depending only on N , ε, R, |α|, k and independent of n such that
kDα vn k∞,K +
∂k
vn
∂tk
∞,K
≤C
for all n ≥ 2R.
It follows, by a diagonalization process, that {vn } → w, uniformly over compact subsets of S1 , where w ∈ C ∞ (S1 ) and satisfies the heat equation. It
remains to prove that
w(·, t) → ζu(·, 0) in L1loc (RN ) as t → 0.
For this, rewrite (14.6) as
fn = vn − ζu(x, 0), fn,t − ∆fn = ∆ζu(x, 0) in S1
fn |x|=n = 0, fn (x, 0) = 0.
2c Similarity Methods
181
Let hδ (·) be the approximation to the Heaviside function introduced in (14.2).
Multiply the PDE by hδ (fn ) and integrate over Bn × (0, t) for t ∈ (0, 1) to
obtain
Z fn
Z
Z Z
1 t
hδ (ξ)dξ dy +
|Dfn |2 χ[|fn | < δ] dy ds
δ
0
Bn ×{t}
0
Bn
Z tZ
=
∆ζ(y)u(y, 0)hδ (fn ) dy ds.
0
B2ρ
Discard the second term on the left-hand side, which is non-negative, and let
δ → 0 to get
Z
|vn (y, t) − ζ(y)u(y, 0)|dy ≤ t|B2ρ |k∆ζu(x, 0)k∞,B2ρ .
BR
Letting n → ∞
kw(y, t) − ζ(y)u(y, 0)k1,BR ≤ t|B2ρ |k∆ζu(x, 0)k∞,B2ρ .
By the first of (14.7), w is bounded; therefore by uniqueness of bounded
solutions of the Cauchy problem, w = v.
Remark 14.1 The Tychonov function defined in (5.2) is of variable sign.
Problems and Complements
2c Similarity Methods
2.1c The Heat Kernel Has Unit Mass
To verify (2.2), disregard momentarily
the factor π −N/2 , and introduce the
p
change of variables y − x = 2 (t − s)η, whose Jacobian is t[4(t − s)]N/2 . This
transforms the integral into
Z
Z
2
2
2
eη1 +···+ηN dη1 · · · dηN
e−|η| dη =
RN
RN
=
N
Q
j=1
=
=
e
−ηj2
Z
dηj =
R
Z
Z
e
R
−η12
R
2π
dη1
Z
e
R
Z
e
−s2
0
∞
2
re−r dr
−η22
dη2
N/2
ds
N
N/2
=
= π N/2 .
Z
R2
e
−|η|2
dη
N/2
182
5 THE HEAT EQUATION
2.2c The Porous Medium Equation
Find similarity solutions for the nonlinear evolution equation
ut − ∆um = 0
u ≥ 0,
m ≥ 1.
This equation arises in the filtration of a fluid in a porous medium [226].
Similarity solutions were derived independently by Barenblatt [14], and Pattle
2
[199]. Attempt solutions of the form u(x, t) = h(t)f (ξ), where ξ = |x|
tσ and σ
is a positive number to be found.
Solution: By straightforward computation, we have
σ
ut = h′ f − ξ hf ′
t
2x
Dum = mhm f m−1 f ′ σ
t
i 4ξ
nh
2N o
∆um = mhm (m − 1)f m−2 f ′2 + f m−1 f ′′ σ + f m−1 f ′ σ .
t
t
Enforcing the equation gives
h′ f − mhm f m−1 f ′
Stipulate to take
io
2N
ξ n ′
4m m−1 h
m−2 ′2
m−1 ′′
=
h
σf
+
h
(m
−
1)f
f
+
f
f
.
tσ
t
tσ−1
h(t) = tθ
where
θ(m − 1) = σ − 1.
This choice makes both sides homogeneous with respect to t. Next, to compute
f , set to zero the term {· · · }, i.e.,
σf ′ + 4m (m − 1)f m−2 f ′2 + f m−1 f ′′ = 0,
which is equivalent to
σf ′ + 4m f m−1 f ′
′
= 0.
Choose the integration constant to be zero to obtain
σ + 4mf m−2 f ′ = 0
=⇒
1
σ(m − 1) m−1
ξ
f (ξ) = C −
4m
+
for a constant C to be chosen. Setting to zero the left-hand side of the equation
gives
h′ f −
2N m
h mf m−1 f ′ = 0
tσ
=⇒
h′ f −
Using the previously obtained form of f , compute
2N m m ′
h (f ) = 0.
tσ
2c Similarity Methods
183
σ
(f m )′ = − f.
4
Therefore,
σ 2N m
h f =0
4 tσ
h′ f +
h′ +
=⇒
Nσ m
h = 0.
2tσ
This is integrated as
h−m h′ = −
N σ −σ
t
2
=⇒
h1−m =
N (m − 1)σ 1−σ
t
,
1−σ
and it is compatible with the choice h(t) = tθ if
1=
N (m − 1)σ
−N [1 + θ(m − 1)]
=
.
1−σ
2θ
This implies
θ=−
N
κ
and
σ=
2
,
κ
where
κ = N (m − 1) + 2.
Hence, we arrive at
|x|2
t2/κ
1
m−1
, t>0
1 − cγm
tN/κ
+
m−1
=
, κ = N (m − 1) + 2,
2κ
Γm (x, t) =
γm
1
where c > 0 is an arbitrary constant.
2.1. Show that as m → 1, Γm (x, t) tends to the fundamental solution of the
heat equation.
2.2. Find the constant c such that the total mass of Γm is 1, i.e.,
Z
Γm (x − y; t − τ )dy = 1
(c = 4π).
RN
2.3. Show that if m > 1, possible solutions of the Cauchy problem
ut − ∆um = 0 in ST ,
N
u≥0
u(·, 0) = uo ∈ C(R ) ∩ L∞ (RN )
cannot be represented as the convolution of Γm with the initial datum uo .
2.4. Attempt to find similarity solutions when 0 < m < 1.
184
5 THE HEAT EQUATION
2.3c The p-Laplacean Equation
Carry on the same analysis for the nonlinear evolution equation
ut − div |Du|p−2 Du = 0,
p > 2.
A version of this equation arises in modelling certain non-Newtonian fluids
([148]). Then for p = 2 this reduces to the heat equation. The similarity
solutions are
1
"
1 − cγp
tN/λ
1
p−1
1
p−2
γp =
,
λ
p
Γp (x, t) =
|x|
t1/λ
p−1
p #
p−1
p−2
,
t>0
+
λ = N (p − 2) + p.
Prove that Γp → Γ as p → 2. Find the constant c so that Γp has mass 1.
Attempt to find similarity solutions when 1 < p < 2.
2.4c The Error Function
Prove that the unique solution of the Cauchy problem
1 if x ≥ 0
+
ut − uxx = 0 in R × R , u(x, 0) =
0 if x < 0
is given by
u(x, t) =
x
1
1+E √
,
2
4t
where
2
E(s) = √
π
The function s → E(s) is the error function.
2.5c The Appell Transformation ([10])
Let u be a solution of the heat equation in R × R+ . Then
x 1
w(x, t) = Γ (x, t)u
,−
t
t
is also a solution of the heat equation in R × R+ .
Z
0
s
2
e−r dr.
2c Similarity Methods
185
2.6c The Heat Kernel by Fourier Transform
For f ∈ L1 (RN ), let fˆ denote its Fourier transform
Z
1
def
fˆ(x) =
f (y)e−ihx,yidy.
(2π)N/2 RN
Here i is the imaginary unit and hx, yi = xj yj . In general, assuming that
f ∈ L1 (RN ) or even that f is compactly supported in RN , does not guarantee
that fˆ ∈ L1 (RN ), as shown by the following examples.
2.5 Compute the Fourier transform of the characteristic function of the unit
interval in R1 . Show that x → (χ[0,1] )∧ (x) ∈
/ L1 (R).
2.6. Let N = 1, and let m be a positive integer larger than 2. Compute the
Fourier transform of
0
for x < 1
f (x) =
x−m
for x ≥ 1
and show that fˆ ∈
/ L1 (R).
2.7c Rapidly Decreasing Functions
These examples show that L1 (RN ) is not closed under the operation of Fourier
transform, and raise the question of finding a class of functions that is closed
under such an operation. The class of smooth and rapidly decreasing functions
in RN , or the Schwartz class, is defined by ([228])


∞
N
m
α


f
∈
C
(R
)
sup
|x|
|D
f
(x)|
<
∞
def
x∈RN
SN =
.
 for all m ∈ N and all multi-indices α of size |α| ≥ 0 
Proposition 2.1c f ∈ SN =⇒ fˆ ∈ SN .
Proof. For f ∈ SN and multi-indices α and β, compute
Z
xβ
xβ Dα fˆ(x) =
f (y)Dxα e−ihx,yi dy
(2π)N/2 RN
Z
(−i)|α|
xβ y α f (y)e−ihx,yi dy
=
(2π)N/2 RN
Z
(−i)|α|−|β|
=
y α f (y)Dyβ e−ihx,yi dy
(2π)N/2 RN
Z
(−i)|α+β|
Dβ [y β Dα f (y)]e−ihx,yi dy.
=
(2π)N/2 RN
186
5 THE HEAT EQUATION
2.8c The Fourier Transform of the Heat Kernel
2
1
Proposition 2.2c Let ϕ(x) = e− 2 |x| . Then ϕ̂ = ϕ.
Proof. Assume first that N = 1. One verifies that ϕ and ϕ̂ satisfy the same
ODE
ϕ′ + xϕ = 0,
ϕ̂′ + xϕ̂ = 0,
x ∈ R.
Therefore ϕ̂ = Cϕ for a constant C. From (2.2) with t − s =
Z
1 2
1
√
e− 2 y dy = ϕ̂(0) = 1.
2π R
1
2
and N = 1
Since also ϕ(0) = 1, we conclude that C = 1, and the proposition follows in
the case of one dimension. If N ≥ 2, by Fubini’s theorem
Z
2
1
1
e− 2 |y| e−ihx,yi dy
ϕ̂(x) =
(2π)N/2 RN
Z
N
Q
1 2
1
√
e− 2 yj e−ixj yj dyj
=
2π R
j=1
=
N
Q
N
Q
ϕ̂(xj ) =
j=1
ϕ(xj ) = ϕ(x).
j=1
2.7. Prove the rescaling formula ψ̂(εx) = ε−N ψ̂(x/ε), valid for all ψ ∈ SN
and all ε > 0.
2.8. Verify the formula
e−|x−y|
2
(t−τ )
∧
=
1
[2(t − τ )]
N/2
e−|x−y|
2
/4(t−τ )
for all t − τ > 0 fixed.
2.9c The Inversion Formula
Theorem 2.1c. Let f ∈ SN . Then
f (x) =
1
(2π)N/2
Z
fˆ(y)eihx,yi dy.
RN
Proof. The formula follows by computing the limit
Z
Z
2
1
1
ihx,yi
ˆ
f (y)e
dy = lim
fˆ(y)e−|y| (t−τ ) eihx,yi dy.
N/2
N/2
τ
→t
(2π)
(2π)
RN
RN
The integral on the right-hand side is computed by repeated application of
Fubini’s theorem:
3c The Maximum Principle in Bounded Domains
1
(2π)N/2
Z
187
2
fˆ(y)e−|y| (t−τ ) eihx,yi dy
Z
2
1
f (η)e−ihy,ηi e−|y| (t−τ ) eihx,yi dydη
(2π)N RN
Z
Z
1
1
−|y|2 (t−τ ) −ihη−x,yi
f
(η)
e
e
dy
dη
(2π)N/2 RN
(2π)N/2 RN
Z
∧
1
−|y|2 (t−τ )
f
(η)
e
(η − x)dη
(2π)N/2 RN
Z
2
1
f (η)e−|x−η| /4(t−τ ) dη.
N/2
[4π(t − τ )]
RN
RN
=
=
=
=
Therefore
Z
Z
1
ˆ(y)eihx,yi dy = lim
f
Γ (x − η; t − τ )f (η)dη = f (x)
τ →t RN
(2π)N/2 RN
where the last limit is computed by the same technique leading to the representation formula (2.7).
3c The Maximum Principle in Bounded Domains
Let E be a bounded domain in RN with smooth boundary ∂E.
3.1. Let u be a solution of the Dirichlet problem (1.2) with g = 0. Prove
that
1
kuo k1,E .
ku(·, t)k∞,E ≤
(4πt)N/2
3.2. State and prove a maximum principle for u ∈ H(ET ) ∩ C(ĒT ) satisfying
H(u) = v · Du + c in ET .
where v ∈ RN and c ∈ R are given.
3.3. Discuss a possible maximum principle for H(u) = λu for λ ∈ R.
3.4. Let f ∈ C(R+ ) and consider the boundary value problem
u ∈ H(E∞ ) ∩ C(Ē∞ )
|x|2
ut − ∆u = f (t) u −
− 1 in B1 × R+
2N
1
u(·, t) ∂ B =
∗ 1
2N
Prove that this problem has at most one solution, the solution is nonnegative and satisfies
Z t
1
|x|2
exp
.
0 ≤ u(x, t) ≤
f (s)ds +
2N
2N
0
In particular, if f ≤ 0 then u(x, t) ≤ 1/N .
188
5 THE HEAT EQUATION
3.5. In the previous problem assume that
f (t) ≤ −
C
1+t
for all t ≥ t∗
for some C > 0 and some t∗ ≥ 0. Prove that
lim u(x, t) =
t→∞
|x|2
.
2N
Moreover, if u(·, 0) = |x|2 /2N , then u(·, t) = u(·, 0), for all f .
3.6. Let f ∈ C(ĒT ) and α ∈ (0, 1). Prove that a non-negative solution of
H(u) = uα in ET satisfies
kuk∞,ET ≤
1
h
i 1−α
1
kuk∞,∂∗ET + (ediam(E) − 1)kf k∞,ET
.
1−α
3.1c The Blow-Up Phenomenon for Super-Linear Equations
Consider non-negative classical solutions of
ut − ∆u = uα
in E × R+ ,
for some α ≥ 1
that are bounded on the parabolic boundary of ET , say
sup u ≤ M
for some M > 0.
∂∗ E∞
Prove that if α = 1, then u ≤ M et . Therefore if α ∈ [0, 1) the solution remains
bounded for all t ≥ 0, and if α = 1, it remains bounded for all t ≥ 0 with
bound increasing with t. If α > 1, an upper bound is possible only for finite
times.
Lemma 3.1c Let α > 1. Then
u(x, t) ≤
M
.
[1 − (α − 1)M α−1 t]1/(α−1)
Proof (Hint). Divide the PDE by uα and introduce the function
w = u1−α + (α − 1)t.
Using that α > 1, prove that H(w) ≥ 0 in E∞ . Therefore, by the maximum
principle
1
1
+ (α − 1)t ≥ α−1 .
α−1
u
M
Remark 3.1c This estimate is stable as α → 1 in the sense that as α → 1,
the right-hand side converges to the corresponding exponential upper bound
valid for α = 1.
3c The Maximum Principle in Bounded Domains
189
3.1.1c An Example for α = 2
Even though the boundary data are uniformly bounded, the solution might
indeed blow up at interior points of E in finite time, as shown by the following
example ([81]).
ut − uxx = u2 in (0, 1) × (0, ∞), u(·, 0) = uo
u(0, t) = ho (t), u(1, t) = h1 (t),
for all t ≥ 0.
(3.1c)
Assume that
c1
c2
for positive constants c1 and c2 to be chosen. These constants can be chosen such that (3.1c) has no solution that remains bounded for finite times.
Introduce the comparison function
c1
v=
.
c2 − x(1 − x)t
u o , ho , h1 ≥ c =
By direct calculation
2c1 t
2c1 t2 (1 − 2x)2
c1 x(1 − x)
+
−
2
2
[c2 − x(1 − x)t]
[c2 − x(1 − x)t]
[c2 − x(1 − x)t]3
v2 1
+ 2t .
≤
c1 4
vt − vxx =
Taking t ∈ (0, 4c2 ) and choosing c1 sufficiently large, this last term is majorized by v 2 . Therefore
vt − vxx ≤ v 2
in (0, 1) × (0, 4c2 ).
Fix any time T ∈ (0, 4c2 ) and consider the domain ET = (0, 1) × (0, T ). If u
is a solution of (3.1c), the function w = (v − u)e−λt for λ > 0 satisfies
wt − wxx ≤ −(λ − (v + u))w in ET ,
w
∂∗ ET
≤ 0.
Therefore, by choosing λ sufficiently large, the maximum principle implies
that w ≤ 0 in ET .
3.2c The Maximum Principle for General Parabolic Equations
Let Lo (·) be the differential operator introduced in (4.1c) of the Complements
of Chapter 2. By using a technique similar to that of Theorem 4.1c, prove
Theorem 3.1c. Let u ∈ H(ET ) ∩ C(ĒT ) and let c ≤ 0. then
ut − Lo (u) ≤ 0 in ET
=⇒
u(x, t) ≤ sup u in ET .
∂∗ E
3.6. The maximum principle gives one-sided estimates for merely sub(super)solutions of the heat equation. An important class of sub(super)-solutions is
determined as follows. Let u ∈ H(ET ) be a solution of the heat equation in ET .
Prove that for every convex(concave) function ϕ(·) ∈ C 2 (R), the composition
ϕ(u) is a sub(super)-solution of the heat equation in ET .
190
5 THE HEAT EQUATION
4c The Maximum Principle in RN
4.1. Show that u = 0 is the only solution of the Cauchy problem
u ∈ H(ST ) ∩ C(S̄T ) ∩ L2 (ST ),
ut − ∆u = 0 in ST ,
u(·, 0) = 0. (4.1c)
Hint: Let x → ζ(x) ∈ Co2 (B2ρ ) be a non-negative cutoff function in B2ρ
satisfying

0
if |x| < ρ
ζ = 1 in Bρ , |Dζ| ≤ 2
 if ρ < |x| < 2ρ,
ρ

0
if |x| < ρ
4
|ζxi xj | ≤
 2 if ρ < |x| < 2ρ.
ρ
Multiply the PDE by uζ 2 and integrate over B2ρ × (0, t) to derive
Z
u2 (t)ζ 2 dx + 2
B2ρ
Z tZ
0
B2ρ
|Du|2 ζ 2 dx ds = 4
Z tZ
0
ζuDuDζdx ds.
B2ρ
By the Cauchy-Schwarz inequality, the last integral is majorized by
Z tZ
Z tZ
2
|Du|2 ζ 2 dx ds + 2
u2 |Dζ|2 dx ds
0
and
2
Z tZ
0
B2ρ
o
8
u |Dζ| dx ds ≤ 2
ρ
B2ρ
2
2
B2ρ
Z tZ
0
ρ<|x|<2ρ
u2 |Dζ|2 dx ds.
Combine these estimates and let ρ → ∞.
4.2. Prove that the same conclusion holds if in (4.1c) one replaces L2 (ST )
with L1 (ST ).
Hint: Let hδ (·) be the approximation to the Heaviside function introduced
in (14.2). Multiply the PDE by hδ (u)ζ and integrate over Bρ × (0, t) to
obtain
Z u
Z Z
Z
1 t
|Du|2 χ(|u| < δ)ζ dx ds
hδ (ξ)dξ ζdx +
δ
0
B2ρ
B2ρ ×{t}
0
Z u
Z tZ
D
=−
hδ (ξ)dξ Dζ dx ds.
0
B2ρ
0
The last term is transformed and majorized by
Z u
Z Z
Z tZ
const t
|u| dx ds.
hδ (ξ)dξ ∆ζ dx ds ≤
ρ2
0
ρ<|x|<2ρ
0
B2ρ
0
7c Remarks on the Cauchy Problem
191
Combining these estimates and letting δ → 0 we arrive at
Z
Z Z
const t
|u|dxds ≤
|u| dx ds.
ρ2
Bρ ×{t}
0
ρ<|x|<2ρ
To conclude, let ρ → ∞.
4.3. Prove that the same conclusion holds if u satisfies either one of the
weaker conditions
u
∈ L2 (ST ),
(1 + |x|)
u
∈ L1 (ST ).
(1 + |x|2 )
4.1c Counterexamples of the Tychonov Type
4.1. Prove that the function
Z ∞
u(x, t) =
[exy cos(xy + 2ty 2 )
0
+ e−xy cos(xy − 2ty 2 )]ye−y
4/3
√
cos( 3 y 4/3 )dy
is another nontrivial solution of the Cauchy problem in R × R+ with
vanishing initial data ([220]). The interesting feature is that u is of class
C ∞ over the whole plane.
4.2. Prove that the function
Z a+i∞
√
2/3
est+x s−s ds,
a > 0,
u(x, t) =
a−i∞
where the integration is along the line Re s = a of the complex s-plane, is a
third nontrivial solution of the Cauchy problem in R × R+ with vanishing
initial data. Here u has the additional feature that it vanishes identically
for negative t, which does not occur for the previous case. This example
was proposed in [220] as an application of the theory developed in [126].
7c Remarks on the Cauchy Problem
7.1. Write down the explicit solution of
ut − ∆u = u + b · ∇u + et sin(x1 − b1 t) in ST
u(x, 0) = |x|
for a given b ∈ RN .
Hint: The function v(x, t) = u(x − bt, t) satisfies the PDE with b = 0.
192
5 THE HEAT EQUATION
7.2. Using the reflection technique, solve the homogeneous mixed boundary
value problems
ut − uxx = 0 in R+ × R+
ux (0, t) = 0 for t > 0
1
ut − uxx = 0 in R+ × R+
u(0, t) = 0 for t > 0
u(x, 0) = uo ∈ C(R̄+ )
+
u(x, 0) = uo ∈ C (R̄ )
uo,x (0) = 0
uo (0) = 0.
where uo is bounded in R+ .
7.3. Solve the inhomogeneous mixed boundary value problems
ut − uxx = in R+ × R+
ut − uxx = 0 in R+ × R+
u(x, 0) = uo ∈ C 1 (R̄+ )
uo,x (0) = h(0)
u(x, 0) = uo ∈ C(R̄+ )
uo (0) = h(0).
ux (0, t) = h(t) ∈ C 1 (R+ )
u(0, t) = h(t) ∈ C 1 (R+ )
12c On the Local Behavior of Solutions
Proposition 12.1c Let u ∈ H(ET ) be a local solution of the heat equation
in ET . For every p > 0 there exists a constant C, depending only on N and p
such that, for all (xo , to ) + Qρ ⊂ ET
1/p
Z
p
.
|u| dy ds
sup |u| ≤ C −
(xo ,to )+Qρ
(xo ,to )+Q2ρ
Proof. The case p = 1 is the content of Corollary 12.1. The case p > 1 follows
from this and Hölder’s inequality. To prove the estimate for 0 < p < 1, one
may assume that (xo , to ) = (0, 0). Consider the increasing sequence of radii
{ρn }, the family of nested expanding boxes {Qn }, and the nondecreasing
sequence of numbers {Mn }, defined by
ρn = ρ
n
P
i=0
2−i ,
Qn = Bρn × (−ρn2 , 0),
Mn = sup |u|,
n = 0, 1, . . . .
Qn
Apply Corollary 12.1 to the pair of cylinders Qn and Qn+1 to obtain
Z
|u| dy ds.
Mn ≤ C2(n+1)(N +2) −
Qn+1
Fix p ∈ (0, 1). Then by Young’s inequality, for all δ > 0
Z
1−p
|u|p dy ds
−
Mn ≤ C2(n+1)(N +2) Mn+1
Qn+1
≤ δMn+1 + pδ
1− p1
C2
(n+2)(N +2)
1/p Z
−
Q2ρ
p
|u| dy ds
1/p
.
12c On the Local Behavior of Solutions
Setting
1/p
1
K = pδ 1− p C2(N +2)
,
b=2
193
N +2
p
we arrive at the recursive inequalities
Z
Mn ≤ δMn+1 + b K −
n
Q2ρ
p
|u| dy ds
1/p
.
By iteration
Mo ≤ δ n Mn+1 + bK
n
P
Z
(δb)i −
i=0
Q2ρ
|u|p dy ds
Choose δ small enough that δb = 12 , so that the series
Then let n → ∞.
P∞
1/p
.
i
i=0 (δb)
is convergent.
6
THE WAVE EQUATION
1 The One-Dimensional Wave Equation
Consider the hyperbolic equation in two variables
utt − c2 uxx = 0.
(1.1)
The variable t stands for time, and one-dimensional refers to the number of
space variables. A general solution of (1.1) in a convex domain E ⊂ R2 , is
given by
u(x, t) = F (x − ct) + G(x + ct)
(1.2)
where s → F (s), G(s) are of class C 2 within their domain of definition. Indeed,
the change of variables
ξ = x − ct,
η = x + ct
(1.3)
transforms E into a convex domain Ẽ of the (ξ, η)-plane, and in terms of ξ
and η, equation (1.1) becomes
Uξη = 0
where
U (ξ, η) = u
Therefore Uξ = F ′ (ξ) and
U (ξ, η) =
Z
ξ + η η − ξ ,
.
2
2c
F ′ (ξ)dξ + G(η).
Rotating the axes back of an angle θ = arctan(c−1 ), maps Ẽ into E back in
the (x, t)-plane and
u(x, t) = F (x − ct) + G(x + ct).
The graphs of ξ → F (ξ) and η → G(η) are called undistorted waves propagating to the right and left respectively (right and left here refer to the positive
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_7
195
196
6 THE WAVE EQUATION
orientation of the x- and t-axes). The two lines obtained from (1.3) by making
ξ and η constants are called characteristic lines. Write them in the parametric
form
for t ∈ R
x1 (t) = ct + ξ, x2 (t) = −ct + η,
and regard the abscissas t → xi (t) for i = 1, 2 as points traveling on the
x-axis, with velocities ±c respectively.
1.1 A Property of Solutions
Consider any parallelogram of vertices A, B, C, D with sides parallel to the
characteristics x = ±ct + ξ and contained in some convex domain E ⊂ R2 .
C
..
..
...
.
............................
.
.
.
......... ........
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
...
.........
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.
.
..
...
............
...
.................
..
.. ...........................
...
...................
..
................
...
...
...
..
...
..
...
..
..
..
...
...
...
. ........................
...
....................
.
.
.
.
.
.
.
.
.
.
.
..
.
.
.
.
..
.........
.
.
.
.
.
.
.
.
...
.
.
.
.
.
.
..
.
.....
...
.
.................
... ................................
...
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.. ..
.................
...
..
D
B
A
Fig. 1.1
We call it a characteristic parallelogram. Let
A = (x, t),
B = (x + cs, t + s)
C = (x + cs − cτ, t + s + τ ),
D = (x − cτ, t + τ )
be the coordinates of the vertices of a characteristic parallelogram, where s
and τ are positive parameters. If a function u ∈ C(E) is of the form (1.2), for
two continuous functions F (·) and G(·), then
u(A) = F (x − ct) + G(x + ct)
u(C) = F (x − 2cτ − ct) + G(x + 2cs + ct)
u(B) = F (x − ct) + G(x + 2cs + ct)
u(D) = F (x − 2cτ − ct) + G(x + ct).
Therefore
u(A) + u(C) = u(B) + u(D).
(1.4)
2 The Cauchy Problem
197
Therefore any solution of (1.1) satisfies (1.4). Vice versa if u ∈ C 2 (E) is of the
form (1.2) for F and G of class C 2 and satisfies (1.4) for any characteristic
parallelogram, rewrite (1.4) as
[u(x, t) − u(x + cs, t + s)] = [u(x − cτ, t + τ ) − u(x + cs − cτ, t + s + τ )].
Using the Taylor formula one verifies that u satisfies the PDE (1.1). Since
(1.4) only requires that u be continuous, it might be regarded as some sort of
weak formulation of (1.1).
2 The Cauchy Problem
On the noncharacteristic line t = 0, prescribe the shape and speed of the
undistorted waves, and seek to determine the shape and speed of the solution
of (1.1), for all the later and previous times. Formally, seek to solve the Cauchy
problem
utt − c2 uxx = 0
in R2
u(·, 0) = ϕ
(2.1)
in R
ut (·, 0) = ψ
in R
1
2
for given ϕ ∈ C (R) and ψ ∈ C (R). According to (1.2) one has to determine
the form of F and G from the initial data, i.e.,
F + G = ϕ,
From this
F′ =
F ′ + G′ = ϕ′ ,
1
1 ′
ϕ − ψ,
2
2c
−F ′ + G′ =
G′ =
1
ψ.
c
1 ′
1
ϕ + ψ.
2
2c
This, in turn, implies
Z
1
1 ξ
ϕ(ξ) −
ψ(s)ds + c1
2
2c 0
Z
1
1 η
G(η) = ϕ(η) +
ψ(s)ds + c2
2
2c 0
F (ξ) =
for two constants c1 and c2 . Therefore
u(x, t) =
1
1
[ϕ(x − ct) + ϕ(x + ct)] +
2
2c
Z
x+ct
ψ(s)ds
(2.2)
x−ct
since, in view of the second of (2.1), c1 + c2 = 0. Formula (2.2) is the explicit
d’Alembert representation of the unique solution of the Cauchy problem (2.1).
The right-hand side of (2.2) is well defined whenever ϕ ∈ Cloc (R) and ψ ∈
L1loc (R). However, in such a case, the corresponding function (x, t) → u(x, t)
need not satisfy the PDE in the classical sense. For this reason, (2.2) might be
regarded as some sort of weak solution of the Cauchy problem (2.1) whenever
the data satisfy merely the indicated reduced regularity.
198
6 THE WAVE EQUATION
Remark 2.1 (Domain of Dependence) The value of u at (x, t) is determined by the restriction of the initial ϕ and ψ, data to the interval
[x − ct, x + ct]. If the initial speed ψ vanishes on such an interval, then u(x, t)
depends only n the datum ϕ at the points x ± ct of the x-axis.
Remark 2.2 (Propagation of Disturbances) The value of the initial data
ϕ(ξ), ψ(ξ) at a point ξ of the x-axis is felt by the solution only at points (x, t)
within the sector
[x − ct ≤ ξ] ∩ [x + ct ≥ ξ].
If ψ ≡ 0, it is felt only at points of the characteristic curves x = ±ct + ξ.
Remark 2.3 (Well Posedness) The Cauchy problem (2.1) is well posed in
the sense of Hadamard, i.e., (a) there exists a solution; (b) the solution is
unique; (c) the solution is stable. Statement (c) asserts that small perturbations of the data ϕ and ψ yield small changes in the solution u. This is also
referred to as continuous dependence on the data. Such a statement becomes
precise only when a topology is introduced to specify the meaning of “small”
and “continuous”.
Since the problem is linear, to prove (c) it will suffice to show that “small
data” yield “small solutions”. As a smallness condition on ϕ and ψ, take
kϕk∞,R ,
kψk∞,R < ε for some ε > 0.
Then formula (2.2) gives that the solution u corresponding to such data satisfies
ku(·, t)k∞,R ≤ (1 + t)ε.
This proves the continuous dependence on the data in the topology of L∞ (R).
If in addition, the initial velocity ψ is compactly supported in R, say in the
interval (−L, L), then
L
kuk∞,R2 < 1 +
ε.
c
3 Inhomogeneous Problems
Let f ∈ C 1 (R2 ) and consider the inhomogeneous Cauchy problem
utt − c2 uxx = f
u(·, 0) = ϕ
in R2
in R
ut (·, 0) = ψ
in R.
(3.1)
The solution of (3.1) can be constructed by superposing the unique solution
of (2.1) with a solution of
vtt − c2 vxx = f
v(·, 0) = vt (·, 0) = 0
in R2
in R.
(3.2)
3 Inhomogeneous Problems
199
To solve the latter, introduce the change of variables (1.3), which transforms
(3.2) into
ξ + η
1
ξ − η
Uξη (ξ, η) = − 2 F (ξ, η),
.
where F (ξ, η) = f
,−
4c
2
2c
The initial conditions translate into
U (s, s) = Uξ (s, s) = Uη (s, s) = 0
∀s ∈ R.
Integrate the transformed PDE in the first variable, over the interval (η, ξ).
Taking into account the initial conditions
Z ξ
1
Uη (ξ, η) = − 2
F (s, η)ds.
4c η
Next integrate in the second variable, over (ξ, η). This gives
Z ηZ z
1
F (s, z)ds dz.
U (ξ, η) = 2
4c ξ ξ
(3.3)
In (3.3) perform the change of variables
s−z
s+z
−
= τ,
=σ
2c
2
whose Jacobian is 2c. The domain of integration is transformed into
x − ct = ξ < σ − cτ < σ + cτ < η = x + ct.
Therefore, in terms of x and t, (3.3) gives the unique solution of (3.2) in the
form
Z Z
1 t x+c(t−τ )
f (σ, τ )dσ dτ.
(3.4)
v(x, t) =
2c 0 x−c(t−τ )
Remark 3.1 (Duhamel’s Principle ([61])) Consider the one-parameter
family of initial value problems
vtt − c2 vxx = 0
v(·, τ ) = 0
in R × (τ, ∞)
in R
vt (·, τ ) = f (·, τ )
in R.
By the d’Alembert formula (2.2)
v(x, t; τ ) =
1
2c
Z
x+c(t−τ )
f (σ, τ )dσ.
x−c(t−τ )
Therefore, it follows from (3.4), that the solution of (3.2) is given by “superposing” τ → v(x, t; τ ) for τ ∈ (0, t). This is a particular case of Duhamel’s
principle (see Section 3.1c of the Complements).
Remark 3.2 It follows from the solution formula (3.4) that if x → f (x, t) is
odd about some xo , then x → v(x, t) is also odd about xo for all t ∈ R. In
particular, u(xo , t) = 0 for all t ∈ R.
200
6 THE WAVE EQUATION
4 A Boundary Value Problem (Vibrating String)
A string of length L vibrates with its end-points kept fixed. Let (x, t) → u(x, t)
denote the vertical displacement at time t of the point x ∈ (0, L). Assume
that at time t = 0 the shape of the string and its speed are known, say
ϕ, ψ ∈ C 2 [0, L]. At all times t ∈ R the phenomenon is described by the
boundary value problem
utt = c2 uxx
in (0, L) × R
in R
in (0, L).
u(0, ·) = u(L, ·) = 0
u(·, 0) = ϕ, ut (·, 0) = ψ
(4.1)
The data ϕ and ψ are required to satisfy the compatibility conditions
ϕ(0) = ϕ(L) = ψ(0) = ψ(L) = 0.
...
.....
........
....
..
........
.
...
.
.
.
.
.
.... ....
.. .........
.
.
.
.
.
.
......
.
..
....
.
.
.
.
.
......
...
.
......
......
.
....
.......
......
...
...
...... ..............
...
...........
....
.
.
.
.
.
.
.
.
.
.
...
.
......
....
.
.
.
..
.
.
.
.
....
.
......
.
......
... .............
.
......
.. ........
...
......
............
....
......
.. ......
.
.
.
.
.
...... .
.. .......
...... ..
......
..
.......
......
.
......
........
...
......
....... ...
.
.
.
.
...
.
......
.
..
....
.
.
......
.
..
.
.
.
.
...... .........
..
...
..........
...
....
....... ...........
..
.
.
.
.
.
.
.
.
.
..
......
.
....
.
.
.
.
.
.
.
.
..
.
......
..
..
.
......
.......
...... ....
..
.......
.... .
.. ..........
...............................................................................................................................................................
t
γ
M
N
A
α
O
β
Fig. 4.2
L
x
At each point of [0, L] × R, the solution u(x, t) of (4.1), can be determined
by making use of the solution formula (2.2) for the Cauchy problem, and
formula (1.4). First draw the characteristic x = ct originating at (0, 0), and the
characteristic x = −ct+L originating at (L, 0), and let A be their intersection.
As they intersect the vertical axes x = 0 and x = L, reflect them by following
the characteristic of opposite slope, as in Figure 4.2. The solution u(x, t) is
determined for all (x, t) in the closed triangle OAL by means of (2.2). Every
point P of the triangle OAM is a vertex of a parallelogram with sides parallel
to the characteristics, and such that of the three remaining vertices, two lie
on the characteristic x = ct, where u is known, and the other is on the vertical
line x = 0, where u = 0. Thus u(P ) can be calculated from (1.4). Analogously
u can be computed at every point of the closure of LAN. We may now proceed
in this fashion to determine u progressively at every point of the closure of
the regions α, β, etc.
4 A Boundary Value Problem (Vibrating String)
201
4.1 Separation of Variables
Seek a solution of (4.1) in the form u(x, t) = X(x)T (t). The equation yields
T ′′ = c2 λT
X ′′ = λX
in R
in (0, L)
λ ∈ R.
(4.2)
The first of these implies that only negative values of λ yield bounded solutions. Setting λ = −γ 2 , the second gives the one-parameter family of solutions
X(x) = C1 sin γx + C2 cos γx.
These will satisfy the boundary conditions at x = 0 and x = L if C2 = 0 and
γ = nπ/L for n ∈ N. Therefore, the functions
Xn (x) = sin
nπ
x,
L
n∈N
represent a family of solutions for the second of (4.2). With the indicated
choice of γ, the first of (4.2) gives
nπc nπc Tn (t) = An sin
t + Bn cos
t .
L
L
The solutions un = Xn Tn can be superposed to give the general solution in
the form
nπc i
nπ nπc ∞ h
P
t + Bn cos
t sin
x .
(4.3)
u(x, t) =
An sin
L
L
L
n=1
The numbers An and Bn are called the Fourier coefficients of the series in
(4.3), and are computed from the initial conditions, i.e.,
∞
P
Bn sin
n=1
πn
x = ϕ(x),
L
∞
P
An
n=1
nπc
nπ
sin
x = ψ(x).
L
L
Since the system sin nπx
is orthogonal and complete in L2 (0, L) (10.2 of
L
the Complements of Chapter 4), one computes
An =
2
nπc
Z
0
L
sin
nπx
ψ(x)dx,
L
Bn =
2
L
Z
0
L
sin
nπx
ϕ(x)dx.
L
(4.4)
Remark 4.1 We have assumed ϕ, ψ ∈ C 2 [0, L]. Actually, the method leading
to (4.3) requires only that ϕ and ψ be in L2 (0, L). Therefore, one might define
the solutions obtained by (4.3) as weak solutions of (4.1), whenever merely
ϕ, ψ ∈ L2 (0, L). The PDE, however, need not be satisfied in the classical
sense.
202
6 THE WAVE EQUATION
Remark 4.2 The nth term in (4.3) is called the nth mode of vibration or
the nth harmonic. We rewrite the nth harmonic as
Gn sin
nπ
nπc
x cos
(t − τn )
L
L
where Gn and τn are two new constants called amplitude and phase angle
respectively. The solution u can be thought of as the superposition of independent harmonics, each vibrating with amplitude Gn , phase angle τn , and
frequency νn = nπc/L.
The method of separation of variables and the principle of superposition were
introduced by D. Bernoulli ([16, 18]), even though not in the context of a formal PDE. In the context of the wave equation, the method was suggested, on
a more formal basis by d’Alembert; it was employed by Poisson and developed
by Fourier [78].
4.2 Odd Reflection
We describe another method to solve (4.1) by referring to the Cauchy problem
(2.1). If the initial data ϕ and ψ are odd with respect to x = 0, then u is odd
with respect to x = 0. Analogously, if ϕ and ψ are odd about x = L, the same
holds for u. It follows that the solution of the Cauchy problem (2.1) with ϕ
and ψ odd about both points x = 0 and x = L must be zero at x = 0 and
x = L, for all t ∈ R, i.e., it satisfies the boundary conditions at x = 0 and
x = L prescribed by (4.1). This suggests constructing a solution of (4.1) by
converting it into an initial value problem (a Cauchy problem) with initial
data given by the odd extension of ϕ and ψ about both x = 0 and x = L. For
ϕ, such an extension is given by
ϕ(x − nL)
for x ∈ nL, (n + 1)L n ∈ Z even
ϕ̃(x) =
−ϕ (n + 1)L − x for x ∈ nL, (n + 1)L n ∈ Z odd.
An analogous formula holds for ψ̃. Then the solution of (4.1) is given by the
restriction to (0, L) × R of
ũ(x, t) =
1
1
[ϕ̃(x − ct) + ϕ̃(x + ct)] +
2
2c
Z
x+ct
ψ̃(s)ds
x−ct
constructed by the d’Alembert formula.
Remark 4.3 Even if ϕ and ψ are in C 2 [0, L], their odd extensions might fail
to be of class C 2 across x = nL. However, for (x, t) ∈ (0,
L) × R, the points
x ± ct are in the interior of some interval nL, (n + 1)L for some n ∈ N, so
that u is actually a classical solution of (4.1).
5 The Initial Value Problem in N Dimensions
203
4.3 Energy and Uniqueness
Let u ∈ C 2 ([0, L] × R) be a solution of (4.1). The quantity
E(t) =
Z
L
0
(u2t + c2 u2x )(x, t)dx
(4.5)
is called the energy of the system at the instant t. Multiplying the first of (4.1)
by ut , integrating by parts over (0, L), and using the boundary conditions at
x = 0 and x = L gives
d
dt
Z
0
L
(u2t + c2 u2x )(x, t)dx = E ′ (t) = 0.
Thus E(t) = E(0) for all t ∈ R, and the energy is conserved. Also, if ϕ = ψ = 0,
then u = 0 in (0, L) × R. In view of the linearity of the PDE one concludes
that C 2 solutions of (4.1) are unique.
4.4 Inhomogeneous Problems
Let f ∈ C 1 (0, L) × R , and consider the inhomogeneous boundary value
problem
in (0, L) × R
utt − c2 uxx = f
in R
u(0, ·) = u(L, ·) = 0
(4.6)
in (0, L).
u(·, 0) = ϕ, ut (·, 0) = ψ
The solution u(x, t) represents the position, at point x and at time t, of a
string vibrating under the action of a load f applied at time t at its points
x ∈ (0, L). The solution of (4.6) can be constructed by superposing the unique
solution of (4.1) with the unique solution of
vtt − c2 vxx = f
v(0, ·) = v(L, ·) = 0
v(·, 0) = vt (·, 0) = 0
in (0, L) × R
in R
in (0, L).
This, in turn, can be solved by reducing it to an initial value problem, through
an odd reflection of x → f (x, t), for all t ∈ R, about x = 0 and x = L, as
suggested by Remark 3.1.
5 The Initial Value Problem in N Dimensions
Introduce formally the d’Alembertian
2
def ∂
=
∂t2
− c2 ∆.
204
6 THE WAVE EQUATION
and, given ϕ ∈ C 3 (RN ) and ψ ∈ C 2 (RN ), consider the Cauchy problem
u=0
u(·, 0) = ϕ
ut (·, 0) = ψ
in RN × R
(5.1)
in RN .
If N ≥ 3, the problem (5.1) can be solved by the Poisson method of spherical
means, and if N = 2 by the Hadamard method of descent.
5.1 Spherical Means
Let ωN denote the measure of the unit sphere in RN and let dω denote the
surface measure on the unit sphere of RN , that is, the infinitesimal solid angle
in RN . If v ∈ C(RN ), the spherical mean of v at x of radius ρ is
Z
1
M (v; x, ρ) =
vdσ
meas[∂Bρ (x)] ∂Bρ (x)
Z
Z
1
1
v(y)dσ(y) =
v(x + ρν)dω
=
ωN ρN −1 |x−y|=ρ
ωN |ν|=1
where ν ranges over the unit sphere of RN .
Remark 5.1 The function ρ → M (v; x, ρ) can be defined in all of R by an
even reflection about the origin since
Z
Z
Z
v(x − ρν)dω.
v(x − ρ(−ν))dω =
v(x + ρν)dω =
|ν|=1
|ν|=1
|ν|=1
Remark 5.2 If v ∈ C s (RN ) for some s ∈ N, then x → M (v; x, ρ) ∈ C s (RN ).
Remark 5.3 Knowing (x, ρ) → M (v; x, ρ) permits one to recover x → v(x),
since
lim M (v; x, ρ) = v(x) for all x ∈ RN .
ρ→0
5.2 The Darboux Formula
Assume that v ∈ C 2 (RN ). By the divergence theorem
Z
Z
∇v(y) · νdσ(y)
∆v(y)dy =
|x−y|=ρ
|x−y|<ρ
Z
N −1
=ρ
∇v(x + ρν) · νdω
|ν|=1
Z
d
= ρN −1
v(x + ρν)dω.
dρ |ν|=1
6 The Cauchy Problem in R3
205
Therefore
∂
1
M (v; x, ρ) =
∂ρ
ωN ρN −1
=
1
ωN ρN −1
Z
∆v(y)dy
|x−y|<ρ
Z ρ
N −1
r
0
∆x
Z
v(x + rν)dωdr.
|ν|=1
Multiplying by ρN −1 and taking the derivative with respect to ρ yields
∂
N −1 ∂
ρ
M (v; x, ρ) = ∆x (ρN −1 M (v; x, ρ)).
∂ρ
∂ρ
This, in turn, gives Darboux’s formula
2
∂
N −1 ∂
+
M (v; x, ρ) = ∆x M (v; x, ρ)
∂ρ2
ρ ∂ρ
(5.2)
valid for all v ∈ C 2 (RN ).
5.3 An Equivalent Formulation of the Cauchy Problem
Let u ∈ C 2 (RN × R) be a solution of (5.1). Then for all x ∈ RN and for all
ρ>0
Z
1
∆x u(x + ρν, t)dω
∆x M (u; x, ρ) =
ωN |ν|=1
Z
1 ∂2
∂2
1
u(x + ρν, t) dω = 2 2 M (u; x, ρ).
= 2
2
c ωN |ν|=1 ∂t
c ∂t
Therefore, setting
M (ρ, t) = M (u(x, t); x, ρ)
and recalling Remarks 5.1 and 5.3, one concludes that u ∈ C 2 (RN × R) is a
solution of (5.1) if and only if
2
∂2
∂
N −1 ∂
2
M (ρ, t)
M (ρ, t) = c
+
∂t2
∂ρ2
ρ ∂ρ
(5.3)
M (ρ, 0) = M (ϕ; x, ρ) = Mϕ (x, ρ)
Mt (ρ, 0) = M (ψ; x, ρ) = Mψ (x, ρ).
6 The Cauchy Problem in R3
If N = 3, the initial value problem (5.3) becomes, on multiplication by ρ
206
6 THE WAVE EQUATION
2
∂2
2 ∂
(ρM
(ρ,
t))
=
c
(ρM (ρ, t)) in R × R
∂t2
∂ρ2
ρM (ρ, 0) = ρMϕ (x, ρ)
(6.1)
ρMt (ρ, 0) = ρMψ (x, ρ).
By the d’Alembert formula (2.2)
ρM (ρ, t) =
1
[(ρ − ct)Mϕ (x, ρ − ct) + (ρ + ct)Mϕ (x, ρ + ct)]
2
Z ρ+ct
1
+
sMψ (x, s)ds.
2c ρ−ct
Differentiating with respect to ρ
1
M (ρ, t) + ρMρ (ρ, t) = [Mϕ (x, ρ − ct) + Mϕ (x, ρ + ct)]
2
1
∂
∂
+ (ρ − ct) Mϕ (x, ρ − ct) + (ρ + ct) Mϕ (x, ρ + ct)
2
∂ρ
∂ρ
1
(ρ + ct)Mψ (x, ρ + ct) − (ρ − ct)Mψ (x, ρ − ct) .
+
2c
Letting ρ → 0 gives the solution formula for (5.1)
Z
Z
1
u(x, t) =
ϕ(x + cνt)dω +
ϕ(x − cνt)dω
(6.2)
8π
|ν|=1
|ν|=1
Z
Z
1
∇ϕ(x − cνt) · ν dω
ct
∇ϕ(x + νct) · ν dω − ct
+
8π
|ν|=1
|ν|=1
Z
Z
1
ψ(x + νct)dω + t
ψ(x − cνt)dω .
t
+
8π
|ν|=1
|ν|=1
From this and Remark 5.1
Z
Z
1 ∂
1
u(x, t) =
t
ϕ(x + νct)dω +
t
ψ(x + νct)dω.
4π ∂t
4π |ν|=1
|ν|=1
This can be written in the equivalent form
Z
Z
1
∂ 1
2
ϕ(y)dσ +
ψ(y)dσ.
4πc u(x, t) =
∂t t |x−y|=ct
t |x−y|=ct
By carrying out the differentiation under the integral in (6.3)
Z
1
2
4πc u(x, t) = 2
[tψ(y) + ϕ(y) + ∇ϕ · (x − y)]dσ.
t |x−y|=ct
(6.3)
(6.4)
(6.5)
Theorem 6.1. Let N = 3 and assume that ϕ ∈ C 3 (R3 ) and ψ ∈ C 2 (R3 ).
Then there exists a unique solution to the Cauchy problem (5.1), and it is
given by (6.2)–(6.5).
6 The Cauchy Problem in R3
207
Proof. We have only to prove the uniqueness. If u, v ∈ C 2 (R3 × R) are two
solutions, the spherical mean of their difference
Z
3
M̃ =
(u − v)(x + ρν)dω
4π |ν|=1
satisfies (6.1) with homogeneous data. By the uniqueness of solutions to the
one dimensional Cauchy problem, M̃ = 0 for all ρ > 0. Thus u = v.
Formulas (6.2)–(6.5) are the Kirchoff formulas; they permit one to read the
relevant properties of the solution u.
Remark 6.1 (Domain of Dependence) The solution at a point (x, t) ∈
RN +1 for N = 3 depends on the data ϕ and ψ and the derivatives ϕxi on the
sphere |x − y| = ct. Unlike the 1-dimensional case, the data in the interior of
Bct (x) are not relevant to the value of u at (x, t).
Remark 6.2 (Regularity) In the case N = 1 the solution is as regular as
the data. If N = 3, because of the t-derivative intervening in the representation
(6.4), solutions of (5.1) are less regular than the data ϕ and ψ. In general,
if ϕ ∈ C m+1 (R3 ) and ψ ∈ C m (R3 ) for some m ∈ N, then u ∈ C m (R3 × R).
Thus if ϕ and ψ are merely of class C 2 in R3 , then uxi xi might blow up at
some point (x, t) ∈ R3 × R even though ϕxi xj , and ψxi xj are bounded. This is
known as the focussing effect. In view of Remark 6.1, the set of singularities
might become compressed for t > 0 into a smaller set called the caustic.
Remark 6.3 (Compactly Supported Data) In the remainder of this section we assume that the initial data ϕ and ψ are compactly supported, say
in the ball Br (0), and discuss the stability in L∞ (R3 ) for all t ∈ R. From the
solution formula (6.3), it follows that x → u(x, t) is supported in the spherical
annulus (ct − r)+ ≤ |x| ≤ r + ct. A disturbance concentrated in Br (0) affects
the solution only within such a spherical annulus.
Remark 6.4 (Decay for Large Times) We continue to assume that the
data ϕ and ψ are supported in the ball Br (0). By Remark 6.3, the solution
x → u(x, t) is also compactly supported in R3 . The solution is also compactly
supported in the t variable, in the following sense:
t → u(x, t) = 0 if for fixed |x|, |t| is sufficiently large.
A stronger statement holds, i.e., kuk∞,R (t) → 0 as t → ∞. Indeed, from (6.5),
for large times
ku(t)k∞,R3 ≤
(1 + c)r2
(kϕk∞,R3 + k∇ϕk∞,R3 + kψk∞,R3 )
c2 t
(6.6)
since the sphere |x − y| = ct intersects the support of the data, at most in a
disc of radius r.
208
6 THE WAVE EQUATION
Remark 6.5 (Energy) Let E(t) denote the energy of the system at time t
Z
E(t) =
ut2 + c2 |Du|2 dx
R3
where D denotes the gradient with respect to the space variables only. Multiplying the PDE u = 0 by ut and integrating by parts in R3 yields
d
E(t) = 0.
dt
The compactly supported nature of x → u(x, t) is employed here in justifying the integration by parts. The same result would hold for a solution
u ∈ C 2 (R3 × R) satisfying
|Du|(·, t) ∈ L2 (R3 )
for all t ∈ R.
(6.7)
A consequence is
Lemma 6.1 There exists at most one solution to the Cauchy problem (5.1)
within the class (6.7).
Also, taking into account (6.6) and Theorem 6.1,
Theorem 6.2. Let N = 3 and assume that ϕ and ψ are supported in the ball
Br for some r > 0. Assume further that ϕ ∈ C 3 (R3 ) and ψ ∈ C 2 (R3 ). Then
there exists a unique solution to the Cauchy problem (5.1), and it is given by
(6.2)–(6.4). Moreover, such a solution is stable in L∞ (R3 ).
Therefore, for smooth and compactly supported initial data, (5.1) is well posed
in the sense of Hadamard, in the topology of L∞ (R3 ).
7 The Cauchy Problem in R2
Consider the Cauchy problem for the wave equation in two space dimensions
utt − c2 (ux1 x1 + ux2 x2 ) = 0
u(·, 0) = ϕ
ut (·, 0) = ψ
in R2 × R
in R2
(7.1)
2
in R .
Theorem 7.1. Assume that ϕ ∈ C 3 (R2 ) and ψ ∈ C 2 (R2 ). Then the Cauchy
problem (7.1) has the unique solution
Z
1
ϕ(y1 , y2 )dy1 dy2
∂
p
u(x1 , x2 , t) =
∂t 2πc Dct (x1 ,x2 ) c2 t2 − (y1 − x1 )2 − (y2 − x2 )2
(7.2)
Z
ψ(y1 , y2 )dy1 y2
1
p
+
2πc Dct (x1 ,x2 ) c2 t2 − (y1 − x1 )2 − (y2 − x2 )2
where Dct (x1 , x2 ) is the disc of center (x1 , x2 ) and radius ct.
7 The Cauchy Problem in R2
209
The Hadamard method of descent ([111]), consists in viewing the solution of
(7.1) as an x3 -independent solution of (5.1) for N = 3, for which one has
the explicit representations (6.2)–(6.5). Let S be the sphere in R3 , of center
(x1 , x2 , 0) and radius ct
S = (y1 , y2 , y3 ) ∈ R3 (x1 − y1 )2 + (x2 − y2 )2 + y32 = c2 t2 .
From (6.5)
u(x1 , x2 , t) = u(x1 , x2 , 0, t)
Z
Z
∂
1
1
=
ϕ(y1 , y2 )dσ +
ψ(y1 , y2 ) dσ.
∂t 4πc2 t S
4πc2 t S
If P = (y1 , y2 , y3 ) ∈ S and if ν(P ) is the outward unit normal to S at P , then
for |y3 | > 0
y3
y3
ct
ν·
=
and
dσ =
dy1 dy2
|y3 |
ct
|y3 |
where dy = dy1 dy2 is the Lebesgue measure in R2 and (y1 , y2 ) ranges over the
disc (y1 − x1 )2 + (y2 − x2 )2 < (ct)2 . Also
p
|y3 | = c2 t2 − [(y1 − x1 )2 + (y2 − x2 )2 ].
Carry these remarks in the previous formula and denote by x = (x1 , x2 ) and
y = (y1 , y2 ) points in R2 to obtain
Z
1
∂
ϕ(y)
p
u(x, t) =
dy
∂t 2πc |y−x|<ct c2 t2 − |y − x|2
(7.3)
Z
ψ(y)
1
p
dy
+
2πc |y−x|<ct c2 t2 − |y − x|2
where we have used that as P = (y1 , y2 , y3 ) = (y, y3 ) runs over S, y runs twice
over the disc |y − x| < ct. Formula (7.3) is the Poisson formula for the solution
of (7.1).
Remark 7.1 (Domain of Dependence) The solution u at a point (x, t) ∈
R2 × R depends on the values of the initial data ϕ, ∇ϕ, and ψ on the whole
disc |y − x| < ct. This is in contrast to the three-dimensional case in which
only the values on the sphere of center x and radius ct were relevant.
Remark 7.2 (Disturbances and the Huygens Principle) The values of
the data ϕ, ∇ϕ, and ψ at some xo ∈ R2 (initial disturbances at xo ) will not
affect a point x until time ct(x) = |x − xo |, and will affect u(x, t) at all further
times t > t(x). Therefore a signal starting at xo at time t = 0 is received by
x at t = t(x) and keeps being “received” thereafter. This explains the propagation of circular waves in still water originating from a “nearly-a-point” disturbance. In the three-dimensional case, an initial disturbance ϕ(xo ), ∇ϕ(xo ),
210
6 THE WAVE EQUATION
and ψ(xo ) at xo ∈ R3 reaches x at time ct = |x − xo | and will not affect u(x, t)
for all later times. This is a special case of the Huygens principle, which states
that if N ≥ 3 and N is odd, signals originating at some xo ∈ RN are received
by an observer at x ∈ RN only at a single instant.
8 The Inhomogeneous Cauchy Problem
Consider the inhomogeneous initial value problem
in RN × R, N = 2, 3
u = f ∈ C 2 (RN × R)
in RN
u(·, 0) = ϕ ∈ C 3 (RN )
2
(8.1)
N
N
in R .
ut (·, 0) = ψ ∈ C (R )
The solution is the sum of the unique solution of (5.1) (f = 0), and
v=f
v(x, 0) = vt (x, 0) = 0
in RN × R
in RN .
(8.2)
The Duhamel principle permits one to reduce the solution of (8.2) to the
solution of the family of homogeneous problems (f = 0)
w(x, t; τ ) = 0
w(·, τ ; τ ) = 0
wt (·, τ ; τ ) = f (·, τ )
in RN × (t > τ )
in RN
By Duhamel’s principle, the solution of (8.2) is given by
Z t
w(x, t; τ )dτ.
v(x, t) =
0
Indeed, by direct calculation
vt (x, t) =
Z
t
wt (x, t; τ )dτ
0
since w(x, t; t) = 0. Therefore v(x, 0) = vt (x, 0) = 0. Next
Z t
vtt = wt (x, t; t) +
wtt (x, t; τ )dτ
0
Z t
= f (x, t) + c2
∆w(x, t; τ )dτ = f + c2 ∆v
0
so that (8.2) holds. If N = 3 and t ≥ 0
Z
Z t
1
1
f (y, τ )dσdτ.
v(x, t) =
4πc2 0 (t − τ ) |x−y|=c(t−τ )
(8.3)
9 The Cauchy Problem for Inhomogeneous Surfaces
If N = 2 and t ≥ 0
1
v(x, t) =
2πc
Z tZ
0
|x−y|≤c(t−τ )
f (y, τ )
p
dy dτ.
2
c (t − τ )2 − |x − y|2
211
(8.4)
Remark 8.1 (Domain of Dependence) If N = 3, the value of v at a point
(x, t), for t > 0, depends only on the values of the forcing term f on the surface
of the truncated backward characteristic cone
[|x − y| = c(t − τ )] ∩ [0 ≤ τ ≤ t].
If N = 2, the domain of dependence is the full truncated backward characteristic cone
[|x − y| < c(t − τ )] ∩ [0 ≤ τ ≤ t].
Remark 8.2 (Disturbances) The effect of a source disturbance at a point
(xo , to ) is not felt at x until the time
1
t(x) = to + |x − xo |.
c
Notice that 1c |x − xo | is the time it takes for an initial disturbance at xo to
affect x. Thus f (xo , to ) can be viewed as an initial datum delayed to a time to .
For this reason, the solution formulas (8.3), (8.4) are referred to as retarded
potentials.
9 The Cauchy Problem for Inhomogeneous Surfaces
The methods introduced for the inhomogeneous initial value problem permit
one to solve the following noncharacteristic Cauchy problem
u=f
u(·, Φ) = ϕ
ut (·, Φ) = ψ
in R3 × (t > Φ)
in R3
(9.1)
3
in R .
The data ϕ, and ψ are now given on the surface Σ = [t = Φ]. Such a surface
must be noncharacteristic in the sense that c|∇Φ| 6= 1 in R3 . We require that
Σ is nearly flat, in the sense
ck∇Φk∞,R3 < 1.
(9.2)
To convey the main ideas of the technique, we will assume that ϕ, ψ, and Φ
are as smooth as needed to carry out the calculations below. Finally, without
loss of generality, we may assume that Φ ≥ 0.
212
6 THE WAVE EQUATION
9.1 Reduction to Homogeneous Data on t = Φ
First consider the problem of finding v ∈ C 3 (R3 × R), a solution of
( v − f)
= ( v − f )t
t=Φ
v(·, Φ) = ϕ,
t=Φ
= ( v − f )tt
t=Φ
=0
(9.3)
vt (·, Φ) = ψ.
Lemma 9.1 Let (9.2) hold. Then there exists a solution to problem (9.3).
Proof. Seek v of the form
v(x, t) =
4
P
i=0
ai (x)(t − Φ(x))i
where x → ai (x), for i = 1, . . . , 4, are smooth functions to be calculated. The
last two of (9.3) give ao = ϕ and a1 = ψ. Next, by direct calculation
v=
4
P
i=2
i(i − 1)ai (x)(t − Φ(x))i−2
− c2
− 2c
− c2
− c2
4
P
∆ai (x)(t − Φ(x))i
i=0
4
2 P
i=1
4
P
i=1
4
P
i=2
i∇ai (x)∇(t − Φ(x))(t − Φ(x))i−1
ai (t − Φ(x))i−1 ∆(t − Φ(x))
i(i − 1)ai (x)(t − Φ(x))i−2 |∇(t − Φ(x))|2 .
From this and (9.2)–(9.3)
2(1 − c2 |∇Φ|2 )a2 = c2 [∆ (ϕ − a1 Φ) + Φ∆a1 ] + f
6(1 − c2 |∇Φ|2 )a3 = c2 [∆(ψ − 2a2 Φ) + 2Φ∆a2 ] + ft
24(1 − c2 |∇Φ|2 )a4 = 2c2 [∆(a2 − 3a3 Φ) + 3Φ∆a3 ] + ftt .
9.2 The Problem with Homogeneous Data
Look for a solution of (9.1) of the form w = u − v, and set F = f −
w satisfies
in R3 × (t > Φ)
w=F
w(·, Φ) = wt (·, Φ) = 0
in R3 .
v. Then
(9.4)
By the construction process of the solution of (9.3) F = Ft = Ftt = 0 on
t = Φ, so that the function
10 Solutions in Half Space. The Reflection Technique
Fo (x, t) =
F (x, t)
0
213
for t ≥ Φ(x)
for t ≤ Φ(x)
is of class C 2 in R3 × R. Then solve
w̄ = Fo
w̄(x, 0) = w̄t (x, 0) = 0
in R3 × (t > 0)
in R3
whose solution is given by the representation formula (8.3). The restriction of
w̄ to [t > Φ] is the solution of (9.4). This will follow from (8.3) and the next
lemma.
Lemma 9.2 Let (9.2) hold. Then w̄(x, t) = 0 for t ≤ Φ(x).
Proof. In (8.3), written for w̄ and Fo , fix x and t ≤ Φ(x). For all y on the
lateral surface of the backward truncated characteristic cone
[|x − y| = c(t − τ )] ∩ [0 ≤ τ < t ≤ Φ(x)]
we must have τ < Φ(y). Indeed, if not
|x − y| ≤ c(Φ(x) − Φ(y)) ≤ c|∇Φ(ξ)||x − y|
for some ξ on the line segment τ x + (1 − τ )y for τ ∈ (0, 1). In view of (9.2)
this yields a contradiction. Since Fo vanishes for (y, τ ) such that τ ≤ Φ(y),
the lemma follows.
The solution obtained this way is unique. This is shown as in Theorem 6.1.
Unlike the Cauchy–Kowalewski theorem, the data are not required to be analytic and the solution is global. Analytic data would yield analytic solutions
only near Σ.
10 Solutions in Half Space. The Reflection Technique
Consider the initial boundary value problem
u=f
u(·, 0) = ϕ
ut (·, 0) = ψ
u(x1 , x2 , 0, t) = h(x1 , x2 , t)
in (R2 × R+ ) × R
for x3 ≥ 0
for x3 ≥ 0
(10.1)
for x3 = 0, t ≥ 0.
If the data are sufficiently smooth, and there is a solution of class C 3 in the
closed half-space R2 × [x3 ≥ 0]× [t ≥ 0], the following compatibility conditions
must be satisfied
214
6 THE WAVE EQUATION
h(x1 , x2 , 0) = ϕ(x1 , x2 )
ht (x1 , x2 , 0) = ψ(x1 , x2 )
(10.2)
2
c ∆ϕ + f (x1 , x2 , 0, t) = htt (x1 , x2 , 0)
c2 ∆ψ + ft (x1 , x2 , 0, t) = httt (x1 , x2 , 0)
Assume henceforth that (10.2) are satisfied and reduce the problem to one
with homogeneous data on the hyperplane x3 = 0.
10.1 An Auxiliary Problem
First find a solution v ∈ C 3 (R3 × R) of the problem
v
( v − f)
x3 =0
= h, vx3 x3 = 0 = vx3 x3
= ( v − f )x3
x3 =0
x3 =0
x3 =0
=0
= ( v − f )x3 x3
(10.3)
x3 =0
= 0.
Lemma 10.1 There exists a smooth solution to (10.3).
Proof. Look for solutions of the form
v(x, t) = h(x1 , x2 , t) +
4
P
i=2
and calculate
ai−1 (x1 , x2 , t)xi3
v − f = (htt − c2 ∆h)(x1 , x2 , t) +
− c2
4
P
i=2
3
P
[ ai−1 (x1 , x2 , t)]xi3
i=2
i(i − 1)ai−1 xi−2
− f (x, t).
3
Therefore the conditions (10.3) yield
2c2 a1 =
h−f
x3 =0
, 6c2 a2 = −fx3
x3 =0
, 24c2 a3 = −fx3 x3
x3 =0
.
10.2 Homogeneous Data on the Hyperplane x3 = 0
Set w = u − v and F = f −
v. Then
w=F
def
w(·, 0) = ϕo = ϕ − v(·, 0)
def
wt (·, 0) = ψo = ψ − vt (·, 0)
w
x3 =0
=0
in (R2 × [x3 > 0]) × [t > 0]
in R2 × [x3 ≥ 0]
in R2 × [x3 ≥ 0]
for x3 = 0, t ≥ 0.
Let F̃ , ϕ̃o , and ψ̃o be the odd extensions of F , ϕo , and ψo about x3 = 0, and
consider the problem
11 A Boundary Value Problem
215
in R3 × R
w̃ = F̃
w(·,
e 0) = ϕ̃o (x)
in R3
in R3 .
w
et (·, 0) = ψ̃o (x)
If this problem has a smooth solution w̃, it must be odd about x3 = 0, that is,
w̃(x1 , x2 , 0, t) = 0, so that the restriction of w̃ to x3 ≥ 0 is the unique solution
of the indicated problem with homogeneous data on x3 = 0. To establish the
existence of w̃ we have only to check that ϕ̃o ∈ C 3 (R3 ), ψ̃o ∈ C 2 (R3 ) and
F̃ ∈ C 2 (R3 × R). For this it will suffice to check that
F = Fx3 = Fx3 x3 = 0
ϕ̃o = ϕ̃0,x3 = ϕ̃0,x3 x3 = 0
for x3 = 0.
ψ̃o = ψ̃0,x3 = ψ̃0,x3 x3 = 0
These conditions follow from the definition of odd reflection about x3 = 0, the
compatibility conditions (10.2), and the construction (10.3) of the auxiliary
function v.
11 A Boundary Value Problem
Let E be a bounded open set in RN with smooth boundary ∂E and consider
the initial boundary value problem
u=0
u(·, t)
∂E
=0
u(·, 0) = ϕ
ut (·, 0) = ψ
in E × R+
in R+
in E
(11.1)
in E.
Here u(x, t) represents the displacement, at the point x at time t, of a vibrating
ideal body, kept at rest at the boundary at ∂E. By the energy method, (11.1)
has at most one solution. To find such a solution we use an N -dimensional
version of the method of separation of variables of Section 4.1. Solutions of
the type T (t)X(x) yield
−∆Xn = λn X in E
Xn = 0
on ∂E
n∈N
(11.2)
and
Tn′′ (t) = −c2 λn Tn (t)
for t > 0,
n ∈ N.
(11.3)
The next proposition is a consequence of Theorem 11.1 of Section 11 of Chapter 4.
Proposition 11.1 There exists an increasing sequence {λn } of positive numbers and a sequence of corresponding functions {vn } ⊂ C 2 (E) satisfying
(11.2). Moreover {vn } form a complete orthonormal system in L2 (E).
216
6 THE WAVE EQUATION
Using this fact, write the solution u as
P
u(x, t) = Tn (t)vn (x)
(11.4)
and deduce that the initial conditions to be associated to (11.3) are derived
from (11.4) and the initial data in (11.1), i.e.,
Z
Z
vn ϕ dx,
Tn′ (0) =
vn ψ dx.
Tn (0) =
E
E
Thus
√
p
sin(c λn t)
√
ψ
Tn (t) =
+ ϕ cos(c λn t) vn dx.
c λn
E
Even though the method is elegant and simple, the eigenvalues and eigenfunctions for the Laplace operator in E can be calculated explicitly only for
domains with a simple geometry (see Section 8c of the Complements of Chapter 3). The approximate solutions
Z
un (x, t) =
n
P
Ti (t)vi (x)
i=1
satisfy, for all i ∈ N, the approximating problems
un = 0
un (·, t)
∂E
in E × R
=0
n
def P
un (x, 0) = ϕn (x) =
def
un,t (x, 0) = ψn (x) =
in R
hϕ, vi ivi (x)
in E
in E
i=1
n
P
hψ, vi ivi (x)
(11.5)
i=1
The function u(·, t) defined by (11.4) is meant as the limit of un (·, t) in L2 (E),
uniformly in t ∈ R. The PDE in (11.1) and the initial data are verified in the
following weak sense. Let f be any function in C 2 (Ē × R), and vanishing on
∂E. Multiply the PDE in (11.5) by any such f and integrate by parts over
E × (0, t), where t ∈ R is arbitrary but fixed. This gives
Z tZ
Z
un (x, t)f (x, t) dx +
un (x, t)(x, τ ) f dx dτ
E
Z
Z0 E
ϕn ft (x, 0) dx.
=
ψn f (x, 0) dx −
E
E
Letting n → ∞ gives the weak form of (11.1)
Z tZ
Z
u(x, t)f (x, t) dx +
u(x, t)(x, τ ) f dx dt
E
Z
Z0 E
ϕft (x, 0) dx
ψf (x, 0) dx −
=
E
2
for all f ∈ C (Ē × R) vanishing on ∂E.
E
13 The Characteristic Goursat Problem
217
12 Hyperbolic Equations in Two Variables
The most general linear hyperbolic equation in two variables x = (x1 , x2 )
takes the form
∂2u
L(u) =
+ b · ∇u + cu = f
(12.1)
∂x1 ∂x2
where b = (b1 , b2 ) and c, f are given continuous functions in R2 . For this, the
characteristics are the lines xi = (const)i for i = 1, 2. If b = c = f = 0, then,
up to a change of variables, (12.1) can be rewritten in the form of the wave
equation
vtt − vxx = 0 in R2
(12.2)
where
x1 = x − t
x2 = x + t
and
v(x, t) = u(x − t, x + t).
Therefore if v is prescribed on the characteristics x ± t = const, the method
of the characteristic parallelograms of Section 1.1 permits one to solve (12.2)
in the whole of R2 .
13 The Characteristic Goursat Problem
The characteristic Goursat problem consists in finding u ∈ C 2 (R2 ) satisfying1
L(u) = f in R2 ,
u
xi =0
= ϕi ∈ C 2 (R),
i = 1, 2.
(13.1)
Theorem 13.1. There exists a unique solution to the characteristic Goursat
problem (13.1).
13.1 Proof of Theorem 13.1: Existence
Setting ∇u = (w1 , w2 ) = w, by virtue of (12.1)
∂
∂
w1 =
w2 = f − b · w − cu.
∂x2
∂x1
Integrate the first of these equations over (0, x2 ) and the second over (0, x1 ).
Taking into account the data ϕi on the characteristics xi = 0, i = 1, 2, recast
(13.1) into the equivalent form
1
The problem is also referred to as the Darboux–Goursat problem. For L(·)
linear, the problem was posed and solved by Darboux, [42](Tome II, pages 91-94).
The nonlinear case of ux1 x2 = F (x1 , x2 , u, xx1 , ux2 ) was solved by E. Goursat, [108,
Vol. 3 part I]. See also J. Hadamard, [110](pages 107-108).
218
6 THE WAVE EQUATION
w1 (x) =
ϕ′2 (x1 )
+
w2 (x) = ϕ′1 (x2 ) +
u(x) = ϕ2 (x1 ) +
Z
x2
Z0 x1
Z0 x2
(f − b · w − cu)(x1 , s)ds
(f − b · w − cu)(s, x2 )ds
(13.2)
w2 (x1 , s)ds.
0
The last equation could be equivalently replaced by
Z x1
w1 (s, x2 )ds.
u(x) = ϕ1 (x2 ) +
0
To solve (13.2), define
uo = ϕ2 ,
w1,o = ϕ′2 ,
w2,o = ϕ1′
and recursively, for n = 0, 1, . . .
Z x2
[f − b · (w1,n , w2,n ) − cun ](x1 , s)ds
w1,n+1 (x) = ϕ′2 (x1 ) +
0
Z x1
[f − b · (w1,n , w2,n ) − cun ](s, x2 )ds
w2,n+1 (x) = ϕ′1 (x2 ) +
0
Z x2
w2,n (x1 , s)ds.
un+1 = ϕ2 (x1 ) +
(13.2)n
0
A solution of (13.2) can be found by letting n → ∞ in (13.2)n , provided the
sequences {un } and {wi,n } for i = 1, 2 are uniformly convergent over compact
subsets of R2 . For this it suffices to prove that the telescopic series
P
P
uo + (un − un−1 ) and wi,o + (wi,n − wi,n−1 )
(13.3)
are absolutely and uniformly convergent on compact subsets K ⊂ R2 . Having
fixed one such K, one may assume that it is a square about the origin with
sides parallel to the coordinate axes, and such that meas(K) ≤ 1. Set
Vn = (un , w1,n , w2,n ), |x| = |x1 | + |x2 |
kVn − Vn−1 k = |un − un−1 | + |w1,n − w1,n−1 | + |w2,n − w2,n−1 |
CK = 1 + kbk∞,K + kck∞,K + kf k∞,K ,
AK = 1 + kVo k∞,K .
Lemma 13.1 For all x ∈ K and all n ∈ N
kVn − Vn−1 k(x) ≤ AK (2CK )n
Proof. From (13.2)n=0
|x|n
.
n!
(13.4)
13 The Characteristic Goursat Problem
w1,1 − w1,o =
w2,1 − w2,o =
u1 − uo =
From this
Z
219
x2
Z0 x1
Z0 x2
[f − b · (w1,o , w2,o ) − cuo ](x1 , s)ds
[f − b · (w1,o , w2,o ) − cuo ](s, x2 )ds
w2,o (x1 , s)ds.
0
kV1 − Vo k(x) ≤ kf k∞,K + (kbk∞,K + kck∞,K )kVo k∞,K
Z
x2
ds
0
+ kf k∞,K + (kbk∞,K + kck∞,K )kVo k∞,K
Z x2
+ kVo k∞,K
ds
Z
x1
ds
0
0
≤ AK CK |x|.
Therefore (13.4) holds for n = 1. We show by induction that if it does hold
for n it continues to hold for n + 1. From (13.2), for all x ∈ K
kVn+1 − Vn k(x)
Z
≤ CK
Z x1
kVn − Vn−1 k(s, x2 )ds
kVn − Vn−1 k(x1 , s)ds +
0
0
Z x2
Z x1
n n+1
2 CK
≤ AK
|(s, x2 )|n ds
|(x1 , s)|n ds +
(n − 1)!
0
0
n+1
n+1 |x|
≤ AK (2CK )
.
(n + 1)!
x2
Returning to the absolute convergence of the series in (13.3), it follows from
the lemma that for all x ∈ K
P
P
|x|n
kVo k(x) + kVn − Vn−1 k(x) ≤ AK 1 + (2CK )n
= AK e2CK |x| .
n!
13.2 Proof of Theorem 13.1: Uniqueness
Let us assume that there exist two locally bounded solutions of the system
(i)
(i)
(13.2), say (u(i) , w1 , w2 ) = V (i) for i = 1, 2, and set
(1)
(2)
(1)
(2)
kV (1) − V (2) k = |u(1) − u(2) | + |w1 − w1 | + |w2 − w2 |.
Write the system (13.2) for V (1) and V (2) , and subtract the resulting equations, to obtain for all x ∈ K
kV (1) − V (2) k(x) ≤ kV (1) − V (2) k∞,K BK |x|
where BK = kbk∞,K + kck∞,K . Since K is an arbitrary compact subset of
R2 , this implies V (1) = V (2) identically.
220
6 THE WAVE EQUATION
13.3 Goursat Problems in Rectangles
Let α1 < β1 and α2 < β2 , and let R be the rectangle [α1 , β1 ] × [α2 , β2 ].
Prescribe data ϕ1 ∈ C 2 [α1 , β1 ] and ϕ2 ∈ C 2 [α2 , β2 ] on the segments [α1 , β1 ]
and [α2 , β2 ], and consider the problem of finding u ∈ C 2 (R) satisfying
L(u) = f
u(x1 , α2 ) = ϕ2 (x1 )
in R
for x1 ∈ [α1 , β1 ]
(13.5)
for x2 ∈ [α2 , β2 ].
u(α1 , x2 ) = ϕ1 (x2 )
The same proof applies, and one may conclude that (13.5) has a unique solution. Analogously, there exists a unique solution to the characteristic problem
L(u) = f
u(x1 , β2 ) = ϕ2 (x1 )
in R
for x1 ∈ [α1 , β1 ]
(13.6)
for x2 ∈ [α2 , β2 ].
u(β1 , x2 ) = ϕ1 (x2 )
14 The Noncharacteristic Cauchy Problem and the
Riemann Function
Let Γ be a regular curve in R2 whose tangent is nowhere parallel to either of
the coordinate axes. For example

s∈R
 x1 = s
Γ = x2 = h(s) ∈ C 1 (R)
 ′
h (s) < 0
for all s ∈ R.
Consider the problem of finding u ∈ C 2 (R2 ) satisfying
L(u) = f in R2 ,
u
Γ
= ux2
Γ
=0
(14.1)
where L(·) is defined in (12.1). As an example, take the case b = c = 0, and
Γ is the line x2 = −x1 . Then (14.1) reduces to the Cauchy problem for the
wave equation
in R2
vtt − vxx = f˜
t = x1 + x2
v(·, 0) = 0
x = −x1 + x2 ,
vt (·, 0) = 0
where
v(x, t) = u
t − x t + x
,
,
2
2
t − x t + x
f˜(x, t) = f
.
,
2
2
This problem has a unique solution is given by the representation formula
(3.4). We will prove that (14.1) has a unique solution and will exhibit a representation formula for it.
14 The Noncharacteristic Cauchy Problem and the Riemann Function
221
Through a point x ∈ R2 − Γ , draw two lines parallel to the coordinate
axes and let Ex be the region enclosed by these lines and Γ , as in Figure 14.3
Ex = {(σ, s) h(σ) < s < x2 ; α < σ < x1 }.
( ; x2 )
..
..
...
...
...
...
........................................................................................................................................................................................................
.
........
..
. ......
..
....
.........
.
.
.............
.
..
.
...............
.
................
.
...
.
...............
...
..
..............
.
...............
..
.
....
.............
.
.............
..
.
.
..............
.
...
............
..
........
.
..
..... ..
.
..... .
.
....
..
..........................................................................................................................................................................................................
...
...
...
..
..
.
(x1 ; x2 ) = x
(x1 ; )
Fig. 14.3
Let L∗ (·) denote the adjoint operator to L(·)
L∗ (v) =
∂2v
− div(bv) + cv.
∂x1 ∂x2
This is well defined if bi ∈ C 1 (R2 ), which we assume henceforth. Let u, v be
a pair of functions in C 2 (R2 ), and compute the quantity
ZZ
[vL(u) − uL∗ (v)]dy.
Ex
√
The outward unit normal to Ex on Γ is n = (−h′ , 1)/ 1 + h′2 . Therefore by
Green’s theorem
ZZ
[vL(u) − uL∗ (v)]dy = (uv)(x) − (uv)(α, x2 )
E
Z xx2
Z x1
−
u[vx2 − vb1 ](x1 , s)ds −
u[vx1 − vb2 ](s, x2 )ds
(14.2)
β
α
Z x1
Z x1
−
vh′ (s)[ux2 + ub1 ](s, h(s))ds −
u[vb2 − vx1 ](s, h(s))ds.
α
α
If u is a solution to (14.1), then (14.2) reduces to
ZZ
Z x2
[vf − uL∗ (v)]dy = (uv)(x) −
u[vx2 − b1 v](x1 , s)ds
Ex
−
Z
β
x1
α
(14.3)
u[vx1 − b2 v](s, x2 )ds.
Next, in (14.3), we make a particular choice of the function v. For each fixed
x ∈ R2 , let y → R(y; x) ∈ C 2 (R2 ) satisfy
222
6 THE WAVE EQUATION
in R2
Z y2
R(x1 , y2 ; x) = exp
b1 (x1 , s)ds
x2
Z y1
b2 (s, x2 )ds .
R(y1 , x2 ; x) = exp
Ly∗ [R(y; x)] = 0
(14.4)
x1
Such a function exists, and it can be constructed by the method of successive
approximations of the previous section. The last two of (14.4) imply that
R(x; x) = 1. Therefore, writing (14.3) for y → v(y) = R(y; x) yields the
representation formula
ZZ
(14.5)
R(y; x)f (y)dy.
u(x) =
Ex
This formula, derived under the assumption that a solution of (14.1) exists,
indeed does give the unique solution of such a noncharacteristic problem, as
can be verified by direct calculation. The function y → R(y; x) is called the
Riemann function ([218]), with pole at x, for the operator L(·) in R2 .2
Remark 14.1 The integral formula (14.3) and the Riemann function R(·; ·)
permit us to give a representation formula for the unique solution of the
noncharacteristic problem (12.1) with inhomogeneous data on Γ
L(u) = f in R2 ,
u
Γ
= ϕ, ux2
Γ
=ψ
(14.1)′
for given smooth functions in R.
15 Symmetry of the Riemann Function
The Riemann function y → R∗ (y; x), with pole at x, for L∗ (·) satisfies
L[R∗ (y; x)] = 0 in R2
Z y1
∗
b2 (s, x2 )ds
R (y1 , x2 ; x) = exp −
x
Z y1 2
∗
b1 (x1 , s)ds .
R (x1 , y2 ; x) = exp −
(15.1)
x2
It follows from this that R∗ (x; x) = 1.
Lemma 15.1 R(y; x) = R∗ (x; y).
Proof. Let x = (x1 , x2 ) and y = (y1 , y2 ) be fixed in R2 and be such that the
line through them is not parallel to either coordinate axis. Without loss of
generality may assume that y1 < x1 and y2 < x2 , and construct the rectangle
2
For an N -dimensional version of the Riemann function, see Hadamard [110].
3c Inhomogeneous Problems
223
Qx,y = [y1 < s < x1 ] × [y2 < τ < x2 ].
By Green’s theorem, for every pair of functions u, v ∈ C 2 (R2 )
ZZ
[vL(u) − uL∗ (v)]dsdτ = (uv)(x) − (uv)(y)
Qx,y
−
−
Z
x2
y
Z 2x2
y2
v[ux2 + b1 u](y1 , τ )dτ −
u[vx2 − b1 v](x1 , τ )dτ −
Z
x1
v[ux1 + b2 u](s, y2 )ds
y
Z 1x1
y1
(15.2)
u[vx1 − b2 v](s, x2 )dτ.
Write this identity for v = R(·; x) and u = R∗ (·; y).
Remark 15.1 (The Characteristic Goursat Problem) The integral formula (15.2) and the Riemann function permit one to give a representation
formula in terms of R(·; ·) of the characteristic Goursat problems (13.5) and
(13.6).
Problems and Complements
2c The d’Alembert Formula
2.1. Solve the Cauchy problems
utt − uxx = f in R × R
u(·, 0) = ut (·, 0) = 0
for
f (x, t) = ex−t
f (x, t) = x2 .
3c Inhomogeneous Problems
3.1c The Duhamel Principle ([61])
A linear differential operator with constant coefficients and of order n ∈ N in
the space variables x = (x1 , . . . , xN ) is defined by
P
Aα Dα w, Aα ∈ R, w ∈ C n (RN ).
L(w) =
|α|≤n
Let f ∈ C(RN +1 ), and for a positive integer m ≥ 2 let
224
6 THE WAVE EQUATION
(x, t; τ ) → v(x, t; τ ),
x ∈ RN , t ∈ (τ, ∞), τ ∈ R
be a family of solutions to the homogeneous Cauchy problems
∂m
v = L(u)
∂tm
in RN × (τ, ∞), m ≥ 2
∂j
v(·, τ ; τ ) = 0
∂tj
for j = 0, 1, . . . , m − 2
(3.1c)
∂ m−1
v(·, τ ; τ ) = f (·, τ )
∂tm−1
parametrized with τ ∈ R. Then, the inhomogeneous Cauchy problem
∂m
u = L(u) + f (x, t)
∂tm
in RN × R
∂j
u(·, 0) = 0
∂tj
for j = 0, 1, . . . , m − 1
(3.2c)
has a solution given by
u(x, t) =
Z
t
v(x, t; τ )dτ
0
(x, t) ∈ RN × R.
Formulate a general Duhamel’s principle ([61]) if m = 1.
4c Solutions for the Vibrating String
4.1. Solve the boundary value problems
utt − uxx = f in (0, L) × R
u(0, ·) = u(L, ·) = 0
for
u(·, 0) = ut (·, 0) = 0
f (x, t) = ex
f (x, t) = sin πx
f (x, t) = x2 .
4.2. Solve the boundary value problem
utt − uxx = x in (0, 1) × R
u(·, 0) = x2 (1 − x), ut (·, 0) = 0
ux (0, ·) = 0, u(1, ·) = 0.
4.3. Let β ∈ R be a given constant. Solve
utt − uxx = β(2ut − βu) in (0, 1) × R
u(·, 0) = ϕ ∈ C 2 (0, 1), ut (·, 0) = 0
u(0, ·) = u(1, ·) = 0.
(3.3c)
4c Solutions for the Vibrating String
225
4.4. Solve the previous problem for ϕ ∈ C(0, 1) but not necessarily of class
C 2 (0, 1). Take, for example
2hx
for x ∈ (0, 21 )
ϕ(x) =
2h(1 − x) for x ∈ ( 12 , 1)
where h is a given positive constant.
4.5. Let a ∈ R, and consider the boundary value problem
utt + aut − uxx = 0
u(0, t) = u(1, t) = 0
u(·, 0) = ϕ, ut (·, 0) = ψ
in (0, 1) × (t > 0)
for t > 0
in (0, 1).
Find an expression for the energy E(t), introduced in (4.5) in terms of ut ,
and estimate E(t) in terms of the initial data only.
Hint: Setting
Z tZ 1
f (t) =
u2t (x, s) dx ds
0
0
derive a differential inequality f ′ ≤ A − Bf , for suitable constants A, B.
4.6. In the previous problem take
a = 1,
ϕ(x) = sin πx + 2 sin 5πx,
ψ = 0.
Write down the explicit solution. Find constants c1 and c2 such that |u| +
|ut | ≤ c1 ec2 t .
4.7. Solve by the separation of variables
utt − uxx = cos 2t
u(0, t) = u(1, t) = 0
u(x, 0) = 0
∞
P
ut (x, 0) =
sin 2nπx
in (0, 1) × R+
for t > 0
in (0, 1)
in (0, 1).
n=1
4.8. Solve by the separation of variables
utt − uxx = 0
u(0, t) = u(1, t) = 0
3
3
u(x, 0) = x (1 − x)
ut (x, 0) = 0
in (0, 1) × R+
for t > 0
in (0, 1)
in (0, 1).
Discuss the regularity of u.
4.9. Solve by the separation of variables
utt − uxx = 0
u(0, t) = u(π, t) = 0
2
u(x, 0) = 3 sin x
ut (x, 0) = 0
in (0, π) × R+
for t > 0
in (0, π)
in (0, π).
226
6 THE WAVE EQUATION
Discuss the regularity of u.
4.10. Solve by the separation of variables
utt − uxx = 0
in (0, π) × R+
ux (0, t) = ux (π, t) = 0
π
π
u(x, 0) = − |x − |
2
2
ut (x, 0) = 0
for t > 0
in (0, π)
in (0, π).
Hint: It might be useful to draw a graph of f (x) =
4.11. Solve by the separation of variables
utt − uxx + u = 0
u(0, t) = u(π, t) = 0
u(x, 0) = x
ut (x, 0) = χ[0, π2 ] (x)
π
2
− |x − π2 |.
in (0, π) × R+
for t > 0
in (0, π)
in (0, π).
4.12. Relying on (1.4) solve the problem
utt − uxx = 0
u(0, t) = u(1, t) = 0
u(x, 0) = 0
ut (x, 0) = 1
in (0, 1) × R+
for t > 0
in (0, 1)
in (0, 1).
Hint: It might be useful to solve first in [0, 1]×[0, 1] and then in [0, 1]×R+ .
4.13. Relying on (1.4) in D = [0, 1] × [0, 1] solve the problem
in (0, 1) × R+
for t > 0
utt − uxx = 0
u(0, t) = 1
u(1, t) = 0
for t > 0
2
in (0, 1)
in (0, 1).
u(x, 0) = 1 − x
ut (x, 0) = 0
4.14. Let u be the solution of (4.1) defined in (4.3)–(4.4). Discuss questions
of convergence of the formal approximating solutions
un =
n
P
(Aj sin jπt + Bj cos jπt) sin jπx.
j=1
Take L = c = 1 and verify that for all p, q, j ∈ N
∂p ∂q
un
∂xp ∂tq
∞,(0,1)×R
≤
n
P
j=1
(jπ)p+q (|Aj | + |Bj |).
m
4.15. Let m be a positive even integer, and let Codd
(0, 1) be defined as
in 10.3 of the Complements of Chapter 4. Assume that ϕ and ψ are in
m
Codd
(0, 1), and prove that un → u in C m [(0, 1) × R].
6c Cauchy Problems in R3
227
6c Cauchy Problems in R3
6.1. Solve the Cauchy problem
u = 0 in R3 × R,
u(x, 0) = |x|2 ,
6.2. Find a space-independent solution of
find the solution of
u = e−t in R3 × R,
ut (x, 0) = x3 in R3 .
u = e−t in R3 × R, and use it to
u(·, 0) = x1 ,
ut (·, 0) = x2 x3 .
6.1c Asymptotic Behavior
6.3. Let u be the solution of
u = 0 in R3 × R,
u(·, 0) = 0,
ut (·, 0) = |x|k
for some k > 0. Compute the limit of u(0, t) as t → ∞. Prove that as
|x| → ∞ the solution has the form
u(x, t) = a(x, t)(1 + |x|k )
for a smooth function a(x, t) uniformly bounded on compact subsets of
the t-axis.
6.4. Let uε be the unique solution of uε = 0 in R3 × R, with initial data
(
−ε2
∂
ε2 −|x|2
e
for |x| < ε
uε (x, 0) =
uε (·, 0) = 0,
∂t
0
for |x| ≥ ε.
Study the limit of uε , as ε → 0, in some appropriate topology.
6.2c Radial Solutions
6.5. Let B be the unit ball about the origin in R3 and consider the problem
(internal vibrations of a contracted sphere)
utt − ∆u = 0
u(·, t)
∂B
in B × R
=0
u(x, 0) = 0
ut (x, 0) = cos
π
|x|
2
for t ∈ R
in B
(6.1c)
in B.
Find a radial solution of (6.1c) by the following steps:
(i) Set |x| = ρ and recast the problem as
2
utt − uρρ − uρ = 0
ρ
u(1, t) = uρ (0, t) = 0
π
u(ρ, 0) = 0, ut (ρ, 0) = cos ρ
2
in (0, 1) × R
for t ∈ R
for ρ ∈ (0, 1).
(6.2c)
228
6 THE WAVE EQUATION
(ii) Let v be the symmetric extension of u(·, t) about the origin
0<ρ<1
for
u(ρ, t)
v(ρ, t) =
u(−ρ, t) for − 1 < ρ < 0.
Verify that vρ (0, t) = 0, and that v solves
2
vtt − vρρ − vρ = 0
ρ
v(−1, t) = v(1, t) = 0
π
v(ρ, 0) = 0, vt (ρ, 0) = cos ρ
2
in (−1, 1) × R
for t ∈ R
for ρ ∈ (−1, 1).
(6.3c)
(iii) Solve (6.3c) and verify that
v(0, t) = t cos
π
t
2
for |t| < 1.
6.6. Now consider (6.2c) in the whole of R3 , i.e.,
utt − ∆u = 0
u(x, 0) = 0
in R3 × R
π
ut (x, 0) = cos |x|
2
in R3
in R3 .
Write down the explicit solution and check that u = v.
6.7. Prove that all radial solutions of the wave equation in R3 × R are of the
form
F (|x| − ct) + G(|x| + ct)
u(x, t) =
|x|
for functions F (·) and G(·) of class C 2 (R).
6.8. Write down the explicit solution of
u = 0 in R3 × R,
u(·.0) = 0, ut (·, 0) = ψ,
where ψ is radial.
6.9. In the case c = 1 and
ψ(|x|) =
prove that the unique solution

t


 1 − (|x| − t)2
u(x, t) =

4|x|


0
1 for |x| < 1
0 for |x| ≥ 1
of (6.4c) is
if 0 < |x| < 1 − t,
t ∈ (0, 1]
if |1 − t| < |x| < t + 1,
t≥0
if 0 ≤ |x| ≤ t + 1.
In particular, the solution is discontinuous at (0, 1).
(6.4c)
6c Cauchy Problems in R3
229
6.3c Solving the Cauchy Problem by Fourier Transform
We will use here notions and techniques introduced in Sections 2.6c–2.9c of
the Complements of Chapter 5. Consider the Cauchy problem
vtt − ∆v = 0 in RN × R+ ,
v(·, 0) = 0,
vt (·, 0) = ψ.
(6.5c)
We assume that ψ is in the class of the rapidly decreasing functions or the
Schwartz class SN , and seek a solution v(·, t) in the same class with respect to
the space variables. Taking the Fourier transform of the PDE in (6.5c) with
respect to the space variables gives
v̂tt + |y|2 v̂ = 0,
v̂(y, 0) = 0,
v̂t (y, 0) = ψ̂.
This can be solved explicitly to give
v̂(y, t) =
sin |y|t
ψ̂(y).
|y|
Prove that
sup Dα
y∈RN
sin |y|t
< ∞,
|y|
for every multi-index α.
Deduce that v̂(·, t) ∈ SN . By the inversion formula obtain the solution of
(6.5c) in the form
Z
sin |y|t
1
ψ̂(y)eihx,yi dy.
(6.6c)
v(x, t) =
(2π)N RN |y|
Now consider the general Cauchy problem
utt − ∆u = 0 in RN × R+ ,
u(·, 0) = ϕ,
ut (·, 0) = ψ
(6.7c)
with both ϕ and ψ in SN . Verify that if w solves (6.5c) with the initial condition wt (·, 0) = ϕ, then the solution of (6.7c) is given by
u(x, t) = v(x, t) + wt (x, t).
It follows from (6.6c) that the solution of (6.7c) can be represented by the
formula
Z 1
sin |y|t
u(x, t) =
ψ̂(y)
+
cos
|y|t
ϕ̂(y)
eihx,yi dy.
(2π)N RN
|y|
6.3.1c The 1-Dimensional Case
If N = 1, by the inversion formula
230
6 THE WAVE EQUATION
Z iy(x+t)
1
e
+ eiy(x−t)
u(x, t) = √
ϕ̂(y)dy
2
2π R
Z iy(x+t)
1
e
− eiy(x−t)
+√
ψ̂(y)dy
2iy
2π R
Z
Z
1 x+t
1
1
iηy
√
e ψ̂(y)dy dη
= [ϕ(x + t) + ϕ(x − t)] +
2
2 x−t
2π R
Z x+t
1
1
= [ϕ(x + t) + ϕ(x − t)] +
ψ(y)dy.
2
2 x−t
6.3.2c The Case N = 3
We refer to the representation formula (6.6c). Let N = 3 and prove the formula
Z
Z
sin |y|t
1
t
ihη,yi
=
e
dσ =
eithη,yi dσ.
|y|
4πt |η|=t
4π |η|=1
Hint: If T is a rotation matrix in R3 , then
Z
Z
eihη,T yi dσ(y).
eihη,yi dσ(η) =
|η|=t
|η|=t
Next, choose T such that T y = |y|(0, 0, 1) and compute the integral by introducing polar coordinates. By the Fubini theorem and the inversion formula,
one computes from (6.6c)
Z
Z 1
1
ihy,ηi
e
dσ ψ̂(y)eihx,yi dy
v(x, t) =
4πt R3 (2π)3/2 |η|=t
Z
Z
1
1
ψ(y)dσ(y).
=
ψ(x + η)dσ =
4πt |η|=t
4πt
|x−y|=t
The solution of (6.7c) is given by
Z
Z
∂
1
1
u(x, t) =
ϕ(y)dσ(y) +
ψ(y)dσ(y).
∂t 4πt |x−y|=t
4πt |x−y|=t
7c Cauchy Problems in R2 and the Method of Descent
7.1. Solve the Cauchy problem
utt − ∆u = 0 in R2 × R,
u(x, 0) = |x|2 ,
ut (x, 0) = 1 in R2 .
7.2. Write down the explicit solution of
utt − ∆u = 0 in R2 × R,
u(·, 0) = p(·),
ut (·, 0) = 0
where p is a homogeneous polynomial of degree 10 in x1 and x2 . Write
down u(0, 0, t) in the case p(x1 , x2 ) = (x21 + x22 )5 .
231
8c Inhomogeneous Cauchy Problems
7.3. Solve the problem
utt − ∆u = 0 in R2 × R,
u(x, 0) = 0,
ut (x, 0) = sin (x1 + x2 ).
Find the solution in the form u(x, t) = a(t) sin (x1 + x2 ).
7.4. Recover the d’Alembert formula (2.2) from the Poisson formula (7.3)
and the method of descent.
7.1c The Cauchy Problem for N = 4, 5
In the Darboux formula (5.2) take N = 5 and let
w(x, ρ, t) = ρ2
∂
M (u; x, ρ, t) + 3ρM (u; x, ρ, t).
∂ρ
Verify that wtt = c2 wρρ and solve (5.1) for N = 5. Prove that the solution of
(5.1) is given by
Z
1 ∂
1
u(x, t) = t2 + t
ψ(y)dσ
3 ∂t
ω5 (ct)4 |x−y|=ct
Z
1
∂ 1 2∂
t
+t
ϕ(y)dσ
.
+
∂t 3 ∂t
ω5 (ct)4 |x−y|=ct
Use the previous result and the method of descent to solve (5.1) for N = 4.
8c Inhomogeneous Cauchy Problems
8.1c The Wave Equation for the N and (N + 1)-Laplacean
Denote by ∆N the Laplacean with respect to the N variables x = (x1 , . . . , xN )
and by ∆N +1 the Laplacean with respect to the (N + 1) variables (x, xN +1 ).
Let k ∈ R be a given constant and let u ∈ C 2 (RN × R) be a solution of
utt = c2 ∆N u − k 2 u
in RN × R.
Then, for any two given constants A and B, the function
solves
k
i
h
k
xN +1 + B sin
xN +1 u(x, t)
v(x, xN +1 , t) = A cos
c
c
vtt = c2 ∆N +1 v
in RN +1 × R.
Similarly, if u solves
utt = c2 ∆N u + k 2 u
in RN × R
(8.1c)
232
6 THE WAVE EQUATION
then
h
k
k
i
v(x, xN +1 , t) = A cosh
xN +1 + B sinh
xN +1 u(x, t)
c
c
solves (8.1c). Use these remarks and the method of descent to solve the Cauchy
problems
utt = c2 ∆2 u ± λ2 u
in R2 × R
u(·, 0) = ϕ ∈ C 3 (R2 )
in R2
ut (·, 0) = ψ ∈ C 2 (R2 )
in R2 ,
where λ ∈ R is a given constant.
8.1.1c The Telegraph Equation
Solve the Cauchy problems
3
utt = uxx ± λ2 u
u(·, 0) = ϕ ∈ C (R)
in R × R
in R
in R.
2
ut (·, 0) = ψ ∈ C (R)
(T)±
The equation (T)− is called the telegraph equation. Set
2
B (s) =
π
+
Z
π/2
cos(s sin θ)dθ,
0
2
B (s) =
π
−
Z
π/2
cosh(s sin θ)dθ
0
where s is a real parameter. Prove that the solutions u± of (T)± are
1
u± (x, t) =
2
Z
x+t
x−t
p
ψ(s)B ± λ t2 − (x − s)2 ds
Z
p
∂ 1 x+t
+
ϕ(s)B ± λ t2 − (x − s)2 ds .
∂t 2 x−t
The functions B ± (·) are the Bessel functions of order zero [19].
8.2c Miscellaneous Problems
8.1. Let a, b, c be given constants. Solve
utt = uxx + aux + but + cu in R × R
u(·, 0) = ϕ ∈ C 3 (R), ut (·, 0) = ψ ∈ C 2 (R).
Hint: Reduce the problem to the previous one by exponential shifts in x
and t. The solution is
8c Inhomogeneous Cauchy Problems
233
p
b
ψ(s) − ϕ(s) A λ t2 − (x − s)2 ds
2
x−t
Z x+t
p
bt−ax ∂
as
+e 2
e 2 ψ(s)A λ t2 − (x − s)2 ds ,
∂t
x−t
2u(x, t) = e
where
bt−ax
2
2
2
2
2
Z
x+t
e
if a ≥ b + 4c,
if a ≤ b + 4c,
as
2
λ=
λ=
r
r
a2 − b 2
− c,
4
A(s) = B − (s)
a2 − b 2
,
4
A(s) = B + (s).
c−
8.2. Solve
utt = uxx + aux + but + cu + f in R × R
u(·, 0) = ϕ ∈ C 3 (R),
ut (·, 0) = ψ ∈ C 2 (R),
f ∈ C 2 (R2 ).
8.3. Let β be a given constant. Solve the boundary value problem
utt − uxx = −βut in (0, 1) × R+ ,
u(0, ·) = u(1, ·) = 0
u(·, 0) = ϕ ∈ C 2 (0, 1) ∩ C[0, 1],
ut (·, 0) = 0.
8.4. Solve the problem
2
ux = 0
x
u(−1, ·) = u(1, ·) = 0
u(x, 0) = 0
π
ut (x, 0) = cos x
2
utt − uxx −
in (−1, 1) × R
in R
for x ∈ (−1, 1)
for x ∈ (−1, 1).
Verify that the solution is symmetric about the origin, that ux (0, t) = 0
for all t, and moreover
π
u(0, t) = t cos t for |t| < 1.
2
Hint:
(xu) = 0.
8.5. Using the Fourier Transform, solve the problem
utt − uxx = −a2 u
lim u(x, t) = 0
in R × R+
for t > 0
|x|→+∞
u(x, 0) = e−b|x| ,
ut (x, 0) = 0
0<b<a
for x ∈ R
for x ∈ R.
Hint: After determining the expression of û(y, t), use the Inversion Theorem for the Fourier Transform and leave u(x, t) written as an integral.
234
6 THE WAVE EQUATION
8.6. Solve explicitly
utt − (uxx + uyy ) = 0
π
u ± , y, t = 0
2
u(x, y, 0) = cos x cos y
ut (x, y, 0) = 0
π π
− ,
× R × R+
2 2
for t > 0 and y ∈ R
π π
×R
in − ,
2 2
π
π
in − ,
×R
2 2
in
Solution: Direct inspection from the data suggests that the solution might
be
√
u(x, y, t) = cos x cos y cos 2 t.
The point is to arrive at such a solution in a constructive way. Periodic
extension of cos x reduces the problem to a Cauchy problem set in R2 ×R+
with the same initial data. The solution of such a Cauchy problem is
Z
∂
cos ξ cos η
p
2πu(x, y, t) =
dξdη
2
∂t Dt (x,y) t − (x − ξ)2 − (y − η)2
Z
∂
cos(x − u) cos(y − v)
√
dudv.
=
∂t Dt (0,0)
t2 − u 2 − v 2
Therefore, the expression of u(x, y, t) hinges upon computing the integral
Z
cos(x − u) cos(y − v)
√
I=
dudv.
t2 − u 2 − v 2
Dt (0,0)
The numerator of the integrand separates into
cos(x − u) cos(y − v) = (cos x cos u + sin x sin u)(cos y cos v + sin y sin v)
= cos x cos y cos u cos v + cos x sin y cos u sin v
+ sin x cos y sin u cos v + sin x sin y sin u sin v.
Therefore, I is the sum of the four integrals
Z
cos u cos v
p
dudv
I1 = cos x cos y
2 − (u2 + v 2 )
t
Dt (0,0)
Z
cos u sin v
p
I2 = cos x sin y
dudv
2
t − (u2 + v 2 )
Dt (0,0)
Z
sin u cos v
p
I3 = sin x cos y
dudv
2
t − (u2 + v 2 )
Dt (0,0)
Z
sin u sin v
p
I4 = sin x sin y
dudv.
2
t − (u2 + v 2 )
Dt (0,0)
We claim that I2 = I3 = I4 = 0. To see this, disregard the (x, y)-dependent
coefficients and call by Ij′ the resulting integrals for j = 2, 3, 4. In all of
these effect a polar change of variables u = r cos θ and v = r sin θ, to get
8c Inhomogeneous Cauchy Problems
I2′ =
I3′ =
I4′ =
Z
Z
Z
t
0
t
0
t
0
235
Z 2π
r
√
cos(r cos θ) sin(r sin θ)dθ dr
t2 − r 2
0
Z 2π
r
√
sin(r cos θ) cos(r sin θ)dθ dr
t2 − r 2
0
Z 2π
r
√
sin(r cos θ) sin(r sin θ)dθ dr.
t2 − r 2
0
Now, I2′ = I4′ = 0 since the integrand is odd. The integrand of I3′ is even
and hence
Z
Z π2
1 2π
sin(r cos θ) cos(r sin θ)dθ =
sin(r cos θ) cos(r sin θ)dθ
2 0
0
Z π
sin(r cos θ) cos(r sin θ)dθ
+
π
2
=
′
I3,1
′
+ I3,2
.
In I3,2 effect the change of variables θ → 12 π + α to find
′
′
I3,2
= −I3,1
=⇒
I3′ = 0.
Thus, the only nonzero integral is I1′ . To compute it recall that I4′ = 0
and write
I1′ = I1′ + I4′
Z
cos u cos v + sin u sin v
p
dudv
=
t2 − (u2 + v 2 )
Dt (0,0)
Z
cos(u − v)
p
=
dudv.
t2 − (u2 + v 2 )
Dt (0,0)
In this last integral effect the change of variables
√
√
u + v = 2ξ,
u − v = 2η.
This is a rotation of (u, v) of 14 π about the origin. Therefore,
{u2 + v 2 < t2 } −→ {ξ 2 + η 2 < t2 }
and
dudv = dξdη
since a rotation does not deform the areas. Thus,
√
Z
cos 2η
p
I1′ =
dξdη.
t2 − η 2 − ξ 2
Dt (0,0)
Since the integrand is even in ξ and η
236
6 THE WAVE EQUATION
I1′ = 4
=4
Z
t
0
Z
√ Z
cos 2η
t −η 2
0
t
√
cos 2η
Z
Z
π
2
0
0
= 2π
√2
t
0
dξ
p
dη
(t2 − η 2 ) − ξ 2
dα dη
√
√
2π
cos 2ηdη = √ sin 2t.
2
Combining these calculations, the solution has the form
√
d sin 2t
√
u(x, y, t) = cos x cos y
.
dt
2
8.7. Solve the problem
uxy = xy in y > x,
u(s, s) = 0, u(s, −s) = s3
√
2s(1 − s) if s ∈ (0, 1)
∇u(s, s) · (−1, 1) =
0
otherwise.
10c The Reflection Technique
10.1. Find the solution of the boundary value problem
utt − uxx = 0
u(0, ·) = h
u(·, 0) = ϕ
ut (·, 0) = ψ
in R+ × R+
for t > 0
for x > 0
for x > 0
where the data h, ϕ, ψ are smooth and satisfy the compatibility conditions
h′ (0) = ψ(0),
h(0) = ϕ(0),
h′′ (0) = c2 ϕ′′ (0).
10.2. Transform the problem
utt − uxx = 0,
in [x > 0] × [t > 0]
u(0, t) = 0 for t > 0
0
in [0 ≤ x ≤ 1] ∪ [x ≥ 2]
u(x, 0) =
(2 − x)(x − 1) in [1 ≤ x ≤ 2].
ut (x, 0) = 0 in x ≥ 0
into another one in the whole of R. Find the times t such that u(3, t) 6= 0.
Find the extrema of x → u(x, 10).
11c Problems in Bounded Domains
237
11c Problems in Bounded Domains
11.1c Uniqueness
Let E ⊂ RN be bounded, open, and with boundary ∂E of class C 1 . Prove
that there exists at most one solution of the boundary value problem
utt − ∆u + k(x, t)u = 0, in E × R
∂
u + q(x, t)u = 0, on ∂E × R
∂n
u(·, 0) = ϕ, in E
ut (·, 0) = ψ, in E
where ϕ and ψ are smooth and k(·, ·) and q(·, ·) are bounded and non-negative
in their domain of definition.
Hint: Multiply the PDE by ut and integrate by parts over E × (0, t) to get
Z tZ
Z tZ
quut dσdτ = Eo ,
(11.1c)
kuut dxdτ + 2
E(t) + 2
0
where
E(t) =
Z
E
0
E
[u2t + |∇u|2 ](·, t)dx
∂E
Eo =
and
Z
E
(ψ 2 + |∇ϕ|2 )dx.
(11.2c)
Next, multiply the PDE by u and integrate by parts over E × (0, t) to get
Z
Z tZ
uut (·, t)dx +
|∇u|2 ] − u2t dxdτ
E
0
E
(11.3c)
Z tZ
Z tZ
Z
2
2
+
ku dxdτ +
qu dσdτ =
ϕψdx.
0
E
0
∂E
E
Integrate (11.1c) in dτ over (0, t), multiply by 2 and add it to (11.3c) to get
Z
Z t
Z tZ
Z tZ
1 d
2
2
u (·, t) dx +
E(τ )dτ +
ku dxdτ +
qu2 dσdτ
2 dt E
0
0
E
0
∂E
Z tZ τ Z
Z tZ τ Z
= 2tEo + E1 − 4
kuuτ dxdsdτ − 4
quuτ dσdsdτ.
0
0
E
0
0
∂E
(11.4c)
By the Cauchy–Schwarz inequality and the assumptions on k
Z tZ τ Z
Z tZ
Z t
4
kuut dxdsdτ ≤ 2γt
ku2 dxdτ + 2t
E(τ ) dτ,
0
0
E
0
E
0
where γ = sup k. Stipulate taking t so small that
1 − 2t max{1; γ} ≥ 12 .
(11.5c)
238
6 THE WAVE EQUATION
Then, inserting this estimation in (11.4c) and integrating in dτ over (0, t) gives
Z tZ τ
Z tZ τ Z
Z
2
ku2 dxdsdτ
u (·, t) dx +
E(s) dsdτ +
0
E
E
0
0
0
Z tZ τ Z
(11.6c)
+2
qu2 dσdsdτ
= 4t2 Eo + 2tE1 +
Z
E
0
0
∂E
ϕ2 dx − 8
Z tZ
τ
0
0
Z
s
Z
quuτ dσdℓdsdτ
∂E
0
To establish uniqueness assume ϕ = ψ = 0 and infer from (11.6c)
Z tZ τ Z
Z
2
u (·, t)dx + 2
qu2 dσdsdτ
E
0
∂E
0
Z tZ τ Z sZ
quuτ dσdℓdsdτ
≤ −8
0
0
0
(11.7c)
∂E
Formulate assumptions for uniqueness to hold. Possible assumptions are:
qt ≤ Cq m
for given non-negative constants C and m.
11.2c Separation of Variables
11.2. Let R = [0, 1] × [0, π], and consider the problem
utt − ∆u = 0,
u(·, t)
∂R
=0
in R × R+
u(x, 0) = 0
ut (x, 0) = f (x1 )g(x2 ),
where f (0) = f (1) = g(0) = g(π) = 0. Solve by the separation of variables.
In particular, write down the explicit solution for the data
∞
P
if 0 ≤ x ≤ 34
x
f (x) =
g(y) =
sin ny.
−3(x − 1) if 34 < x ≤ 1,
n=1
11.3. Solve (11.1) for E = [0, 1]3 in terms of the eigenvalues and eigenfunctions (λ2i , vi ) of the Laplacean in E
∆vi = λ2i vi .
(11.8c)
11.3-(i). Solve (11.8c) by the separation of variables. Denote by x, y, z the
coordinates in R3 and seek a solution of the form vi = Xi (x)Wi (y, z).
Then,
Xi′′ = −ξi2 Xi and ∆(y,z) Wi = −νi2 Wi
where ξi and νi are positive numbers linked by ξi2 + νi2 = λ2i .
14c Goursat Problems
239
11.3-(ii). Find Wi of the form Wi (y, z) = Yi (y)Zi (z). Then,
Yi′′ = −ηi2 Yi
Zi′′ = −ζi2 Zi ,
and
where ηi and ζi are positive numbers linked by ηi2 + ζi2 = νi2 .
11.3-(iii). Verify that for all triples (m, n, ℓ) of positive integers, the pairs
λ2i = π 2 m2 + n2 + ℓ2 , vi = sin πm sin πn sin πℓ
are eigenvalues and eigenfunctions of (11.8c). Prove that these are all the
eigenvalues and eigenfunctions of (11.8c).
12c Hyperbolic Equations in Two Variables
12.1c The General Telegraph Equation
Let u(s, t) be the intensity of electric current in a conductor, considered as a
function of t and the distance s from a fixed point of the conductor. Let α
denote the capacity and β the induction coefficients. Then,
utt − c2 uss + (α + β)ut + αβu = 0 in R × R.
Setting
e1/2(α+β)t u(s, t) = v(x, y),
x = s + ct, y = s − ct
transforms the equation into
vxy + λv = 0,
λ=
α − β 2
4c
.
14c Goursat Problems
14.1. Prove that (14.5) is the unique solution of (14.1)
14.2. Prove that (14.1)′ has a unique solution and give a representation
formula in terms of R(·; ·).
14.3. Give a representation formula for the characteristic Goursat problems
(13.5) and (13.6) in terms of R(·; ·).
14.4. Use the method of successive approximations of Section 13, to find the
Riemann function for the operator
L(v) =
∂2
v + b · ∇v + cv,
∂x1 ∂x2
where b1 , b2 , and c are constants.
240
6 THE WAVE EQUATION
14.1c The Riemann Function and the Fundamental Solution of the
Heat Equation
The fundamental solution of the heat equation uy = uxx can be recovered
as the limit, as ε → 0, of the Riemann function for the hyperbolic equation
([110], 145–147).
uxx + εuxy − uy = 0.
The change of variables ξ = y and η = x − 1ε y transforms this equation into
1
εuξη + uη − uξ = 0.
ε
Using the previous problem, show that the Riemann function, with pole at
the origin, for such an equation is given by
r !
ξ
η
ξη
−
R (ξ, η); (0, 0) = e ε2 ε Jo 2
ε3
where Jo (·) is the Bessel function of order zero. Returning to the original
coordinates
2y
x
2p
y(εx
−
y)
.
R (x, y); (0, 0) = e ε2 − ε Jo
ε2
Let ε → 0 to recover the fundamental solution of the heat equation in one
space dimension with pole at the origin. For the asymptotic behavior of Jo (s)
as s → ∞, see Bowman [19].
7
QUASI-LINEAR EQUATIONS OF FIRST
ORDER
1 Quasi-Linear Equations
A first-order quasi-linear PDE is an expression of the form
ai (x, u(x))uxi = ao (x, u(x))
(1.1)
where x ranges over a region E ⊂ RN , the function u is in C 1 (E), and (x, z) →
ai (x, z) are given smooth functions of their arguments. Introduce the vector
a = (a1 , . . . , aN ), and rewrite (1.1) as
(a, ao ) · (∇u, −1) = 0.
(1.1)′
Thus if u is a solution of (1.1), the vector (a, ao ) is tangent to the graph of
u at each of its points. For this reason, the graph of u is called an integral
surface for (1.1). More generally, an N -dimensional surface Σ of class C 1 is
an integral surface
for (1.1) if for every point P = (x, z) ∈ Σ, the vector
a(P ), ao (P ) is tangent to Σ at P . The curves
ẋi (t) = ai (x(t), z(t)), i = 1, . . . , N
(−δ, δ) ∋ t →
ż(t) = ao (x(t), z(t))
(1.2)
(x(0), z(0)) = (xo , zo ) ∈ E × R
defined for some δ > 0, are the characteristics associated to (1.1), originating
at (xo , zo ). The solution of (1.2) is local in t, and the number δ that defines
the interval of existence might depend upon (xo , zo ). For simplicity we assume
that there exists some δ > 0 such that the range of the parameter t is (−δ, δ),
for all (xo , zo ) ∈ E × R.
Proposition 1.1 An N -dimensional hypersurface Σ is an integral surface
for (1.1) if and only if it is the union of characteristics.
Proof. Up to possibly relabeling the coordinate variables and the components
ai , represent Σ, locally as z = u(x) for some u of class C 1 . For (xo , zo ) ∈ Σ,
let t → (x(t), z(t)) be the characteristic trough (xo , zo ), set
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_8
241
242
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
w(t) = z(t) − u(x(t))
and compute
ẇ = ż − uxi ẋi = ao (x, z) − ai (x, z)uxi (x)
= ao (x, u(x) + w) − ai (x, u(x) + w)uxi (x).
Since (xo , zo ) ∈ Σ, w(0) = 0. Therefore w satisfies the initial value problem
ẇ = ao (x, u(x) + w) − ai (x, u(x) + w)uxi (x)
w(0) = 0.
(1.3)
This problem has a unique solution. If Σ, represented as z = u(x), is an integral surface, then w = 0 is a solution of (1.3) and therefore is its only solution.
Thus z(t) = u(x(t), t), and Σ is the union of characteristics. Conversely, if Σ
is the union of characteristics, then w = 0, and (1.3) implies that Σ is an
integral surface.
2 The Cauchy Problem
Let s = (s1 , . . . , sN −1 ) be an (N − 1)-dimensional parameter ranging over the
cube Qδ = (−δ, δ)N −1 . The Cauchy problem associated with (1.1) consists
in assigning an (N − 1)-dimensional hypersurface Γ ⊂ RN +1 of parametric
equations
x = ξ(s) = ξ1 (s), . . . ,ξN (s)
Qδ ∋ s →
(2.1)
z = ζ(s),
ξ(s), ζ(s) ∈ E × R
and seeking a function u ∈ C 1 (E) such that ζ(s) = u(ξ(s)) for s ∈ Qδ and
the graph z = u(x) is an integral surface of (1.1).
2.1 The Case of Two Independent Variables
If N = 2, then s is a scalar parameter and Γ is a curve in R3 , say for example
(−δ, δ) ∋ s → r(s) = (ξ1 , ξ2 , ζ)(s).
Any such a curve is noncharacteristic if the two vectors (a1 , a2 , ζ)(r(s)) and
(ξ1′ , ξ2′ , ζ ′ )(s) are not parallel for all s ∈ (−δ, δ). The projection of s → r(s)
into the plane [z = 0] is the planar curve
(−δ, δ) ∋ s → ro (s) = (ξ1 , ξ2 )(s)
of tangent vector (ξ1′ , ξ2′ )(s). The projections of the characteristics through
r(s) into the plane [z = 0] are called characteristic projections, and have
tangent vector (a1 , a2 )(r(s)). We impose on s → r(s) that its projection into
the plane [z = 0] be nowhere parallel to the characteristic projections, that
is, the two vectors (a1 , a2 )(r(s)) and (ξ1′ , ξ2′ )(s) are required to be linearly
independent for all s ∈ (−δ, δ).
3 Solving the Cauchy Problem
243
2.2 The Case of N Independent Variables
Returning to Γ as given in (2.1), one may freeze all the components of s but
the ith , and consider the map
si → (ξ1 , . . . , ξN , ζ)(s1 , . . . , si , . . . , sN −1 ).
This is a curve traced on Γ with the tangent vector
∂ξ ∂ζ
∂ξN ∂ζ
∂ξ1
=
.
,
,...,
,
∂si ∂si
∂si
∂si ∂si
Introduce the (N − 1) × (N + 1) matrix
∂ξ(s) ∂ζ(s)
∂ξj (s) ∂ζ(s)
=
.
∂s
∂s
∂si
∂si
The (N − 1)-dimensional surface Γ is noncharacteristic if the vectors
∂ξ ∂ζ
,
(s)
(a, ao )(ξ(s), ζ(s)),
∂si ∂si
are linearly independent for all s ∈ Qδ , equivalently, if the N × (N + 1) matrix
!
a(ξ(s), ζ(s)) ao (ξ(s), ζ(s))
(2.2)
∂ξ(s)
∂ζ(s)
∂s
∂s
has rank N . We impose that the characteristic projections be nowhere parallel
to s → ξ(s), that is
!
a(ξ(s), ζ(s))
6= 0 for all s ∈ Qδ .
(2.3)
det
∂ξ(s)
∂s
Thus we require that the first N × N minor of the matrix (2.2) be nontrivial.
3 Solving the Cauchy Problem
In view of Proposition 1.1, the integral surface Σ is constructed as the union
of the characteristics drawn from points (ξ, ζ)(s) ∈ Γ , that is, Σ is the surface
(−δ, δ) × Qδ ∋ (t, s) → x(t, s), z(t, s)
given by
d
x(t, s) = a(x(t, s), z(t, s)),
dt
x(0, s) = ξ(s)
d
z(t, s) = ao (x(t, s), z(t, s)),
dt
z(0, s) = ζ(s).
(3.1)
244
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
..
.......
......
......
............
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
........... .. ........
................. ........
...
.............
.
...
....
...........
..
......... ...........................................
..
.......................... .......
....
.
.
...
.
.
.
....
..................
.... .............
...
.
.
.
.
.
.
.
.
.
.
....
...
...
... ............
...
.
.
.
.
.
.
..
..
..... ........................................................... ....
.
.
.
.
..
.
.
.
...
............... .... ....
..
.......................
.
.
.
.
.
.
.
.
.
.
.... ... ...
..
..
................... .....................................
.
.
..
.
.
.
...
... .. ..
.
... ..................
.
.
.
.
..
.
.
.
.
.
.
.
.
.
.
.
.. ... ..
... .............
...
..
....................
.
.
.
.
.
.
.
.
.
.
.
..
.. ... ...
..
... .......................................... .... ....
.
.
.
.
.
.
.
.
.
.
.
.
..
.. .. .
..
...
............. ..................................... .... ....
.
.
.
.
.
..
.... .. ..
......
.. .
.
....
.
.
.
.
............
.
.
.
.
.
.
.
.
.... .. ..
.
....................
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
... .. .
....
.... ...........
.
.
.
.
. .
.
...
.
Fig. 3.1
The solutions of (3.1) are local in t. That is, for each s ∈ Qδ , (3.1) is solvable
for t ranging in some interval (−t(s), t(s)). By taking δ smaller if necessary,
we may assume that t(s) = δ for all s ∈ Qδ . If the map
M : (−δ, δ) × Qδ ∋ (t, s) → x(t, s)
is invertible, then there exist functions S : E → Qδ and T : E → (−δ, δ) such
that s = S(x) and t = T (x) and the unique solution of the Cauchy problem
(1.1), (2.1) is given by
def
u(x) = z(t, s) = z(T (x), S(x)).
The invertibility of M must be realized in particular at Γ , so that the determinant of the Jacobian matrix
t
d x(0, s) ∂ξ(s)
J=
dt
∂s
must not vanish, that is, Γ cannot contain characteristics. In view of (3.1) for
t = 0, this is precisely condition (2.3).
The actual computation of the solution involves solving (3.1), calculating
the expressions of s and t in terms of x, and substituting them into the
expression of z(t, s). The method is best illustrated by some specific examples.
3.1 Constant Coefficients
In (1.1), assume that the coefficients ai for i = 0, . . . , N are constant. The
characteristics are lines of parametric equations
x(t) = xo + at,
z(t) = zo + ao , t
t ∈ R.
The first N of these are the characteristic projections. It follows from (1.1)′
that the function f (x, z) = u(x) − z is constant along such lines. If Γ is given
as in (2.1), the integral surface is
x(t, s) = ξ(s) + at
z(t, s) = ζ(s) + ao t
s = (s1 , . . . , sN −1 ).
(3.2)
3 Solving the Cauchy Problem
245
The solution z = u(x) is obtained from the last of these upon substitution of
s and t calculated from the first N . As an example, let N = 2 and let Γ be a
curve in the plane x2 = 0, say
Γ = {ξ1 (s) = s; ξ2 (s) = 0; ζ(s) ∈ C 1 (R)}.
The characteristics are the lines of symmetric equations
x1 − x1,o
x2 − x2,o
z − zo
=
=
,
a1
a2
ao
(x1,o , x2,o , zo ) ∈ R3
with the obvious modifications if some of the ai are zero. The characteristic
projections are the lines
a2 x1 = a1 x2 + const.
These are not parallel to the projection of Γ on the plane z = 0, provided
a2 6= 0, which we assume. Then (3.2) implies
x2 = a2 t,
ξ1 (s) = s = x1 −
a1
x2
a2
and the solution is given by
a1 ao
u(x1 , x2 ) = ζ x1 − x2 + x2 .
a2
a2
3.2 Solutions in Implicit Form
Consider the quasi-linear equation
a(u) · ∇u = 0,
a = (a1 , a2 , . . . , aN )
(3.3)
where ai ∈ C(R), and aN 6= 0. The characteristics through points (xo , zo ) ∈
RN +1 are the lines
x(t) = xo + a(zo )t
lying on the hyperplane z = zo .
A solution u of (3.3) is constant along these lines. Consider the Cauchy problem with data on the hyperplane xN = 0, i.e.,
u(x1 , . . . , xN −1 , 0) = ζ(x1 , . . . , xN −1 ) ∈ C 1 (RN −1 ).
In such a case the hypersurface Γ is given by

 xi = si , i = 1, . . . , N − 1
RN −1 ∋ s → xN = 0

z(s) = ζ(s).
(3.4)
Setting x̄ = (x1 , . . . , xN −1 ), and ā = (a1 , . . . , aN −1 ), the integral surface associated with (3.3) and Γ as in (3.4), is
246
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
x̄(t, s) = s + ā(ζ(s))t
(3.5)
xN (t, s) = aN (ζ(s))t
z(t, s) = ζ(s).
From the first two compute
s = x̄ −
ā(ζ(s))
xN .
aN (ζ(s))
Since a solution u of (3.3) must be constant along z(s, t) = ζ(s), we have
ζ(s) = u(x). Substitute this in the expression of s, and substitute the resulting
s into the third of (3.5). This gives the solution of the Cauchy problem (3.3)–
(3.4) in the implicit form
ā(u(x))
u(x) = ζ x̄ −
xN
aN (u(x))
(3.6)
as long as this defines a function u of class C 1 . By the implicit function
theorem this is the case in a neighborhood of xN = 0. In general, however,
(3.6) fails to give a solution global in xN .
4 Equations in Divergence Form and Weak Solutions
Let (x, u) → F(x, u) be a measurable vector-valued function in RN × R and
consider formally, equations of the type
div F(x, u) = 0
in RN .
(4.1)
Ru
The equation (3.3) can be written in this form for F(u) =
a(σ)dσ. A
measurable function u is a weak solution of (4.1) if F(·, u) ∈ [L1loc (RN )]N , and
Z
F(x, u) · ∇ϕ dx = 0 for all ϕ ∈ Co∞ (RN ).
(4.2)
RN
This is formally obtained from (4.1) by multiplying by ϕ and integrating by
parts. Every classical solution is a weak solution. Every weak solution such
that F(·, u) is of class C 1 in some open set E ⊂ RN is a classical solution of
(4.1) in E. Indeed, writing (4.2) for all ϕ ∈ Co∞ (E) implies that (4.1) holds in
the classical sense within E. Weak solutions could be classical in sub-domains
of RN . In general, however, weak solutions fail to be classical in the whole of
RN as shown by the following example. Denote by (x, y) the coordinates in
R2 and consider the Burgers equation ([24, 25]
1 ∂ 2
∂
u+
u = 0.
∂y
2 ∂x
One verifies that the function
(4.3)
4 Equations in Divergence Form and Weak Solutions
u(x, y) =
− 32
0
p
y + 3x + y 2
solves the PDE in the weak form
Z
uϕy + 21 u2 ϕx dxdy = 0
R2
247
for 4x + y 2 > 0
for 4x + y 2 < 0
for all ϕ ∈ Co∞ (R2 ).
The solution is discontinuous across the parabola 3x + y 2 = 0.
4.1 Surfaces of Discontinuity
Let RN be divided into two parts, E1 and E2 , by a smooth surface Γ of unit
normal ν oriented, say, toward E2 . Let u ∈ C 1 (Ēi ) for i = 1, 2 be a weak
solution of (4.1), discontinuous across Γ . Assume also that F(·, u) ∈ C 1 (Ēi ),
so that
div F(x, u) = 0 in Ei for i = 1, 2
in the classical sense. Let [F(·, u)] denote the jump of F(·, u) across Γ , i.e.,
[F(x, u)] =
lim
E1 ∋x→Γ
Rewrite (4.2) as
Z
Z
F(x, u) · ∇ϕdx +
E2
E1
F(x, u) −
lim
E2 ∋x→Γ
F(x, u) · ∇ϕdx = 0
F(x, u).
for all ϕ ∈ Co∞ (RN ).
Integrating by parts with the aid of Green’s theorem gives
Z
ϕ[F(x, u)] · νdσ = 0 for all ϕ ∈ Co∞ (RN ).
Γ
Thus if a weak solution suffers a discontinuity across a smooth surface Γ , then
[F(x, u)] · ν = 0
on Γ.
(4.4)
Even though this equation has been derived globally, it has a local thrust, and
it can be used to find possible local discontinuities of weak solutions.
4.2 The Shock Line
Consider the PDE in two independent variables
uy + a(u)ux = 0
and rewrite it as
for some a ∈ C(R)
∂
∂
R(u) +
S(u) = 0
∂y
∂x
in R2
(4.5)
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
248
where
R(u) = u
and
S(u) =
Z
u
a(s)ds.
More generally, R(·) and S(·) could be any two functions satisfying
S ′ (u) = a(u)R′ (u).
Let u be a weak solution of (4.5) in R2 , discontinuous across a smooth curve
of parametric equations Γ = {x = x(t), y = y(t)}. Then, according to (4.4),
Γ must satisfy the shock condition1
[R(u)]x′ − [S(u)]y ′ = 0.
(4.6)
In particular, if Γ is the graph of a function y = y(x), then y(·) satisfies the
differential equation
[R(u)]
y′ =
.
[S(u)]
As an example, consider the case of the Burgers equation (4.3). Let u be a
weak solution of (4.3), discontinuous across a smooth curve Γ parametrized
locally as Γ = {x = x(t), y = t}. Then (4.6) gives the differential equation of
the shock line2
x′ (t) =
[u+ (x(t), t) + u− (x(t), t)]
,
2
u± =
lim
x→x(t)±
u(x, t).
(4.7)
5 The Initial Value Problem
Denote by (x, t) points in RN × R+ , and consider the quasi-linear equation in
N + 1 variables
ut + ai (x, t, u)uxi = ao (x, t, u)
(5.1)
with data prescribed on the N -dimensional surface [t = 0], say
u(x, 0) = uo (x) ∈ C 1 (RN ).
(5.2)
Using the (N + 1)st variable t as a parameter, the characteristic projections
are
x′i (t) = ai (x(t), t, z(t)) for i = 1, . . . , N
xN +1 = t
for t ∈ (−δ, δ) for some δ > 0
x(0) = xo ∈ RN .
Therefore t = 0 is noncharacteristic and the Cauchy problem (5.1)–(5.2) is
solvable. If the coefficients ai are constant, the integral surface is given by
1
2
This is a special case of the Rankine–Hugoniot shock condition ([211, 129]).
The notion of shock will be made more precise in Section 13.3.
6 Conservation Laws in One Space Dimension
x(t, s) = s + at,
249
z(t, s) = uo (s) + ao t
and the solution is
u(x, t) = uo (x − at) + ao t.
In the case N = 1 and ao = 0, this is a traveling wave in the sense that the
graph of uo travels with velocity a1 in the positive direction of the x-axis,
keeping the same shape.
5.1 Conservation Laws
Let (x, t, u) → F(x, t, u) be a measurable vector-valued function in RN × R+ ×
R and consider formally homogeneous, initial value problems of the type
ut + div F(x, t, u) = 0 in RN × R+
u(·, 0) = uo ∈ L1 (RN ).
(5.3)
These are called conservation laws. The variable t represents the time, and u
is prescribed at some initial time t = 0.
Remark 5.1 The method of integral surfaces outlined in Section 2, gives
solutions near the noncharacteristic surface t = 0. Because of the physics underlying these problems we are interested in solutions defined only for positive
times, that is defined only on one side of the surface carrying the data.
A function u is a weak solution of the initial value problem (5.3) if
(a) u(·, t) ∈ L1loc (RN ) for all t ≥ 0, and for all i = 1, . . . , N we have
1
Fi (·, ·, u) ∈ Lloc
(RN × R+ ),
(b) the PDE is satisfied in the sense
Z ∞Z
(5.4)
uϕt + F(x, t, u) · Dϕ dxdt = 0
0
RN
for all ϕ ∈ Co∞ (RN × R+ ), and where D denotes the gradient with respect
to the space variables only,
(c) the initial datum is taken in the sense of L1loc (RN ), that is
lim
R+ ∋t→0
ku(·, t) − uo k1,K = 0
(5.5)
for all compact sets K ⊂ RN .
6 Conservation Laws in One Space Dimension
Let a(·) be a continuous function in R and consider the initial value problem
ut + a(u)ux = 0 in R × R+ ,
u(·, 0) = uo ∈ C(R).
(6.1)
250
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
The characteristic through (xo , 0, zo ), using t as a parameter, is
z(t) = zo ,
x(t) = xo + a(zo )t
and the integral surface is
x(s, t) = s + a(uo (s))t,
z(s, t) = uo (s).
Therefore the solution, whenever it is well defined, can be written implicitly
as
(6.2)
u(x, t) = uo (x − a(u)t).
The characteristic projections through points (s, 0) of the x-axis are the lines
x = s + a(uo (s))t
and u remains constant along such lines. Two of these characteristic projections, say
x = si + a(uo (si ))t
i = 1, 2, such that a(uo (s1 )) 6= a(uo (s2 ))
(γi )
intersect at (ξ, η) given by
a(uo (s1 ))s2 − a(uo (s2 ))s1
a(uo (s1 )) − a(uo (s2 ))
s1 − s2
.
η=−
a(uo (s1 )) − a(uo (s2 ))
ξ=
(6.3)
.
........
....
..
... .....
......
........
.
.
.
... ...
.
.
.. ...
o
.
.
.. ...
...
...
....
.
...
.................. ....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
........
..... ......... ..
.
.
.
.
.........
.... ........ .
.
.
.
... .......
.
...
...... .....
... .......
............................. ........
....
...
.
....
..
...
.
.
....
.
.
...
.
...
....
.
.
.
...
...
...
.
.
.
.
.
.
.
...
...
.
..................................
.
.
.
.
.
.
...
.
.
.
.
.....
.
.... ..... .
.
...
.
.
.
.
.
.
.
.....
..
...
...
.
.. ..
.
.
.
.
.
.
.
.
.
.
.
.
.
....
...
.
. ...
.....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.......................
.....
.
..................................................................................................................................
.
.
.
.
.
....
.
.
..
...
.......... .....
...
...........
....
..
....
... ...... ..................
.......................................................................................
....
. .
..
.
........ ..........
....
........
...................
...
.....
........
.....
..
.
.
........
.
..
.......
.
.
.
.
.
.
.
.
.
.
.
.
......
....
.
.
.
.
.
.
.
...
.
.
.
....
........
....
........
.... ........ ...............
. ...... ...........
. .
..........................
....
....
.......
.......
u (x)
t
x
Fig. 6.1
6 Conservation Laws in One Space Dimension
251
Since u is constant along each of the γi , it must be discontinuous at (ξ, η),
unless uo (s) = const. Therefore the solution exists only in a neighborhood of
the x-axis. It follows from (6.3) and Remark 5.1 that the solution exists for
all t > 0 if the function s → a(uo (s)) is increasing. Indeed, in such a case, the
intersection point of the characteristic lines γ1 and γ2 occurs in the half-plane
t < 0. If a(·) and uo (·) are differentiable, compute from (6.2)
u′o (x − a(u)t)a(u)
1 + u′o (x − a(u)t)a′ (u)t
u′o (x − a(u)t)
ux =
.
1 + u′o (x − a(u)t)a′ (u)t
ut = −
These are implicitly well defined if a(·) and uo (·) are increasing functions, and
when substituted into (6.1) satisfy the PDE for all t > 0. Rewrite the initial
value problem (6.1) as
ut + F (u)x = 0 in R × R+
u(·, 0) = uo
where F (u) =
Z
u
a(s)ds.
(6.4)
0
Proposition 6.1 Let F (·) be convex and of class C 2 , and assume that the
initial datum uo (·) is nondecreasing and of class C 1 . Then the initial value
problem (6.4) has a unique classical solution in R × R+ .
6.1 Weak Solutions and Shocks
If the initial datum uo is decreasing, then a solution global in time is necessarily a weak solution. The shock condition (4.6) might be used to construct
weak solutions, as shown by the following example. The initial value problem
ut + 12 (u2 )x = 0 in R × R+

for x < 0
1
u(x, 0) = 1 − x for 0 ≤ x ≤ 1

0
for x ≥ 1
has a unique weak solution for 0 < t < 1, given by

1
for x < t

x−1
u=
for t < x < 1

 t−1
0
for x ≥ 1.
(6.5)
(6.6)
For t > 1 the geometric construction of (6.2) fails for the sector 1 < x < t.
252
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
.
.. ... ...
. ....
.... .... .... .... .... ....... ......... .
.
..... ......... ......... ......... ......... .........
.... .
.
.
.
.
.
.
.
.
.
.
.
.
.
.. .. .. . .
.. .. .. ..
..... ..... ..... ..... ..... ..... ...... . . ..
.... .... .... .... .... .... ......... ... ... ..
. . .. ..
.... ......... ........ ........ ......... ......... ...................
.
.
.
.
.
... .... .... .... ... .... ... .. . . .. . .
..... .... .... .... ..... .... ..... .. .. . . ... ....
..... ..... ..... ..... ..... ..... ..... .... .. .. .... ..
..
.... .... ..... ..... ..... .... .... ... ... .. .
..... ......... ........ ........ ........ ........ ........
. ..
.
.
.. .. .. . .
.
.
.
. .
... .. ... ... .
.... .... .... ..... ..... .... ....
....................................................................................................................
......................................
......................................................................
0
1
x
Fig. 6.2
The jump discontinuity across the lines x = 1 and x = t is 1. Therefore,
starting at (1, 1) we draw a curve satisfying (4.7). This gives the shock line
2x = t + 1, and we define the weak solution u for t > 1 as
1 for 2x < t + 1
u=
(6.7)
0 for 2x > t + 1.
Remark 6.1 For t > 1 fixed, the solution x → u(x, t) drops from 1 to 0 as
the increasing variable x crosses the shock line.
6.2 Lack of Uniqueness
If uo is nondecreasing and somewhere discontinuous, then (6.4) has, in general, more than one weak solution. This is shown by the following Riemann
problem:
1
ut + (u2 )x = 0 in R × R+
2
(6.8)
0 for x ≤ 0
u(x, 0) =
1 for x > 0.
No points of the sector 0 < x < t can be reached by characteristics originating
from the x-axis and carrying the data (Figure 6.3). The solution is zero for
x < 0, and it is 1 for x > t. Enforcing the shock condition (4.7) gives
0 for 2x < t
u(x, t) =
(6.9)
1 for 2x > t.
However, the continuous function


 0x for x < 0
for 0 ≤ x ≤ t
u(x, t) =

t
1 for x > t
is also a weak solution of (6.8).
(6.10)
7 Hopf Solution of The Burgers Equation
253
.
........
..
..
...
...
.
.
.
... ... ... ... ... ....
.... ... ... ... ...
... .... .... .... ....
.. .. .. .. .. ..
.... ... ... .... ...
... ... ... ... ... ...
... ...... ...... ..... ......
.
.
.. .. .. .. .. ..
.... .... .... .... ....
.. .. .. .. .. ...
... ... ... ... ... ..
.... .... ... .... ....
... ... ... ... ...
.. .. .. .. .. ...
.... ...... ...... ...... ......
.
.. .. .. .. .. ..
.
... ... ... ... ... ... ...... ...... ...... ...... .....
.. .. .. .. .. .. .... .... .... .... .....
.. .. .. .. .. ... .... .... ..... .... ....
.. .. .. .. .. .. .... .... .... .... ....
................................................................................................................................................................................................
t
0
x
Fig. 6.3
7 Hopf Solution of The Burgers Equation
Insight into the solvability of the initial value problem (6.4) is gained by
considering first the special case of the Burgers equation, for which F (u) =
1 2
2 u . Hopf’s method [122] consists in solving first the regularized parabolic
problems
1
un,t − un,xx = −un un,x in R × R+
n
(7.1)
un (·, 0) = uo
and then letting n → ∞ in a suitable topology. Setting
Z x
un (y, t)dy
U (x, t) =
xo
for some arbitrary but fixed xo ∈ R transforms the Cauchy problem (7.1) into
1
1
Uxx = − (Ux )2 in R × R+
n
Z 2x
U (x, 0) =
uo (s)ds.
Ut −
xo
n
Next, one introduces the new unknown function w = e− 2 U and verifies that
w is a positive solution of the Cauchy problem
1
wxx = 0 in R × R+
n
Rx
n
w(x, 0) = e− 2 xo uo (s)ds .
wt −
(7.2)
Such a positive solution is uniquely determined by the representation formula
Z
Ry
|x−y|2
n
1
e− 2 xo uo (s)ds e−n 4t dy
w(x, t) = √
4πt R
provided uo satisfies the growth condition3
3
See (2.7) and Theorem 2.1 of Chapter V, and Section 14 of the same Chapter.
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
254
|uo (s)| ≤ Co |s|1−εo
for all |s| ≥ ro
for some given positive constants Co , ro , and εo . The unique solutions un of
(7.1) are then given explicitly by
Z
(x − y)
un (x, t) =
dλn (y)
t
R
where dλn (y) are the probability measures
dλn (y) = R
e
−n
2
Re
Ry
uo (s)ds+ |x−y|
2t
R
y
uo (s)ds+ |x−y|
2t
−n
2
xo
xo
2
2
dy.
dy
The a priori estimates needed to pass to the limit can be derived either from
the parabolic problems (7.1)–(7.2) or from the explicit representation of un
and the corresponding probability measures λn (y). In either case they depend
on the fact that F (·) is convex and F ′ = a(·) is strictly increasing.4
Because of the parabolic regularization (7.1), it is reasonable to expect
that those solutions of (6.4) constructed in this way satisfy some form of the
maximum principle.5 It turns out that Hopf’s approach, and in particular the
explicit representation of the approximating solutions un and the corresponding probability measures λn (y), continues to hold for the more general initial
value problem (6.4). It has been observed that these problems fail, in general,
to have a unique solution. It turns out that those solutions of (6.4) that satisfy the maximum principle, form a special subclass of solutions within which
uniqueness holds. These are called entropy solutions.
8 Weak Solutions to (6.4) When a(·) is Strictly
Increasing
We let a(·) be continuous and strictly increasing in R, that is, there exists a
positive constant L such that
1
a.e. s ∈ R.
(8.1)
a′ (s) ≥
L
Assume that the initial datum uo satisfies
uo ∈ L∞ (R) ∩ L1 (−∞, x) for all x ∈ R
lim sup |uo (x)| = 0
x→−∞
inf
x∈R
4
Z
(8.2)
x
−∞
uo (s)ds ≥ −C for some C > 0.
Some cases of nonconvex F are in [133].
By 3.2. of the Complements of Chapter 5, the presence of the term un un,x in
(7.1) is immaterial for a maximum principle to hold.
5
8 Weak Solutions to (6.4) When a(·) is Strictly Increasing
255
For example, the datum of the Riemann problem (6.8) satisfies such a condition. The initial datum is not required to be increasing, nor in L1 (R). Since
F (·) is convex ([50], Chapter IV, Section 13)
F (u) − F (v) ≥ a(v)(u − v)
u, v ∈ R,
a(v) = F ′ (v)
(8.3)
and since F ′ is strictly increasing, equality holds only if u = v. This inequality
permits one to solve (6.4) in a weak sense and to identify a class of solutions,
called entropy solutions, within which uniqueness holds ([157, 158]).
8.1 Lax Variational Solution
To illustrate the method assume first that F (·) is of class C 2 and that uo is
regular, increasing and satisfies
uo (x) = 0 for all x < b for some b < 0.
The geometric construction of (6.2) guarantees that a solution must vanish
for x < b for all t > 0. Therefore the function
Z x
u(s, t)ds
U (x, t) =
−∞
is well defined in R × R+ . Integrating (6.4) in dx over (−∞, x) shows that U
satisfies the initial value problem
Z x
+
uo (s)ds.
Ut + F (Ux ) = 0 in R × R , U (x, 0) =
−∞
It follows from (8.3), with u = Ux and all v ∈ R, that
Ut + a(v)Ux ≤ a(v)v − F (v)
(8.4)
and equality holds only if v = u(x, t). For (x, t) ∈ R × R+ fixed, consider the
line of slope 1/a(v) through (x, t). Denoting by (ξ, τ ) the variables, such a
line has equation x − ξ = a(v)(t − τ ), and it intersects the axis τ = 0 at the
abscissa
η = x − a(v)t.
(8.5)
The left-hand side of (8.4) is the derivative of U along such a line. Therefore
d
U x − a(v)(t − τ ), τ = Ut + a(v)Ux ≤ a(v)v − F (v).
dτ
Integrating this over τ ∈ (0, t) gives
Z η
uo (s)ds + t a(v)v − F (v)
U (x, t) ≤
−∞
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
256
valid for all v ∈ R, and equality holds only for v = u(x, t). From (8.5) compute
−1 x − η
v=a
(8.6)
t
and rewrite the previous inequality for U (x, t) in terms of η only, that is
U (x, t) ≤ Ψ (x, t; η)
for all η ∈ R
(8.7)
where
Ψ (x, t; η) =
Z
η
uo (s)ds
x − η −1 x − η
−1 x − η
a
−F a
.
+t
t
t
t
−∞
(8.8)
Therefore, having fixed (x, t), for that value of η = η(x, t) for which v in
(8.6) equals u(x, t), equality must hold in (8.7). Returning now to F (·) convex and uo satisfying (8.1)–(8.2), the arguments leading to (8.7) suggest the
construction of the weak solution of (6.4) in the following two steps:
Step 1: For (x, t) fixed, minimize the function Ψ (x, t; η), i.e., find η = η(x, t)
such that
Ψ (x, t; η(x, t)) ≤ Ψ (x, t; s) for all s ∈ R.
(8.9)
Step 2: Compute u(x, t) from (8.6), that is
x − η(x, t)
u(x, t) = a−1
.
t
(8.10)
9 Constructing Variational Solutions I
Proposition 9.1 For fixed t > 0 and a.e. x ∈ R there exists a unique η =
η(x, t) that minimizes Ψ (x, t; ·). The function x → u(x, t) defined by (8.10) is
a.e. differentiable in R and satisfies
u(x2 , t) − u(x1 , t)
L
≤
x2 − x1
t
for a.e. x1 < x2 ∈ R.
Moreover, for a.e. (x, t) ∈ R × R+
r Z x−a(o)t
1/2
Z y
2L
uo (s)ds − inf
uo (s)ds
|u(x, t)| ≤
.
y∈R −∞
t
−∞
(9.1)
(9.2)
Proof. The function η → Ψ (x, t; η) is bounded below. Indeed, by the expressions (8.6) and (8.8) and the assumptions (8.1)–(8.2)
9 Constructing Variational Solutions I
Ψ (x, t; η) ≥ inf
y∈R
Z
y
−∞
uo (s)ds + t[va(v) − F (v)] ≥ −C +
t 2
v
2L
257
(9.3)
for η = x − a(v)t. A minimizer can be found by a minimizing sequences {ηn },
that is one for which
Ψ (x, t; ηn ) > Ψ (x, t; ηn+1 )
and
lim Ψ (x, t; ηn ) = inf Ψ (x, t; η) .
η
By (9.3), the sequence {ηn } is bounded. Therefore, a subsequence can be
selected and relabeled with n such that {ηn } → η(x, t). Since Ψ (x, t; ·) is
continuous in R
lim Ψ (x, t; ηn ) = Ψ (x, t; η(x, t)) ≤ Ψ (x, t; η) ,
n→∞
for all η ∈ R.
This process guarantees the existence of at least one minimizer for every fixed
x ∈ R. Next we prove that such a minimizer is unique, for a.e. x ∈ R.
Let H(x) denote the set of all the minimizers of Ψ (x, t; ·), and define a function
x → η(x, t) as an arbitrary selection out of H(x).
Lemma 9.1 If x1 < x2 , then η(x1 , t) < η(x2 , t).
Proof (of Proposition 9.1 assuming Lemma 9.1). Since x → η(x, t) is increasing, it is continuous in R except possibly for countably many points. Therefore,
η(x, t) is uniquely defined for a.e. x ∈ R. From (8.10) it follows that for a.e.
x1 < x2 and some ξ ∈ R
′
a−1 (ξ)
{(x2 − x1 ) − [η(x2 , t) − η(x1 , t)]}
t
′
a−1 (ξ)
L
≤
(x2 − x1 ) ≤ (x2 − x1 ).
t
t
u(x2 , t) − u(x1 , t) ≤
This proves (9.1). To prove (9.2), write (9.3) for η = η(x, t), the unique minimizer of Ψ (x, t; ·). For such a choice, by (8.10), v = u. Therefore
Z y
t 2
u (x, t) ≤ Ψ x, t; η(x, t) − inf
uo (s)ds
y∈R −∞
2L
Z y
≤ Ψ (x, t; η) − inf
uo (s)ds
y∈R
−∞
for all η ∈ R, since η(x, t) is a minimizer. Taking η = x − a(0)t and recalling
the definitions (8.8) of Ψ (x, t; ·) proves (9.2).
9.1 Proof of Lemma 9.1
Let ηi = η(xi , t) for i = 1, 2. It will suffice to prove that
258
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
Ψ (x2 , t; η1 ) < Ψ (x2 , t; η)
for all η < η1 .
(9.4)
By minimality, Ψ (x1 , t; η1 ) ≤ Ψ (x1 , t; η) for all η < η1 . From this
Ψ (x2 , t; η1 )+[Ψ (x1 , t; η1 ) − Ψ (x2 , t; η1 )]
≤ Ψ (x2 , t; η) + [Ψ (x1 , t; η) − Ψ (x2 , t; η)].
Therefore inequality (9.4) will follow if the function
η → L(η) = Ψ (x1 , t; η) − Ψ (x2 , t; η)
is increasing. Rewrite L(η) in terms of vi = vi (η) given by (8.6) with x = xi
for i = 1, 2. This gives
Z v1
sa′ (s)ds.
L(η) = t[v1 a(v1 ) − F (v1 )] − t[v2 a(v2 ) − F (v2 )] = t
v2
From this one computes
∂v1
∂v2
L′ (η) = t v1 a′ (v1 )
− v2 a′ (v2 )
∂η
∂η
x1 − η
x2 − η
− a−1
> 0.
= a−1
t
t
10 Constructing Variational Solutions II
For fixed t > 0, the minimizer η(x, t) of Ψ (x, t; ·) exists and is unique for a.e.
x ∈ R. We will establish that for all such (x, t)
Z
x − η(x, t)
x−η
a−1
= lim
a−1
dλn (η)
(10.1)
n→∞ R
t
t
where dλn (η) are the probability measures on R
e−nΨ (x,t;η)
dη
−nΨ (x,t;η) dη
Re
dλn (η) = R
for n ∈ N.
(10.2)
Therefore the expected solution
u(x, t) = a−1
x − η(x, t)
t
can be constructed by the limiting process (10.1). More generally, we will
establish the following
10 Constructing Variational Solutions II
259
Lemma 10.1 Let f be a continuous function in R satisfying the growth condition
Rv ′
(10.3)
|f (v)| ≤ Co |v|eco o sa (s)ds for all |v| ≥ γo
for given positive constants Co , co and γo . Then for fixed t > 0 and a.e. x ∈ R
Z −1 x − η
f [u(x, t)] = lim
f a
dλn (η).
n→∞ R
t
Proof. Introduce the change of variables
x−η
x − η(x, t)
v = a−1
,
vo = a−1
t
t
(10.4)
and rewrite Ψ (x, t; η) as
Ψ (v) =
Z
x−a(v)t
−∞
uo (s)ds + t[va(v) − F (v)].
(10.5)
The probability measures dλn (η) are transformed into the probability measures
e−n[Ψ (v)−Ψ (vo )] a′ (v)
e−nΨ (v) a′ (v)
R
dv
=
dv
−nΨ (v) a′ (v)dv
−n[Ψ (v)−Ψ (vo )] a′ (v)dv
Re
Re
dµn (v) = R
(10.6)
and the statement of the lemma is equivalent to
Z
f (vo ) = lim
f (v)dµn (v)
n→∞
R
where vo is the unique minimizer of v → Ψ (v). For this it suffices to show that
Z
def
|f (v) − f (vo )|dµn (v) → 0 as n → ∞.
In =
R
By (9.3) the function Ψ (·) grows to infinity as |v| → ∞. Since vo is the only
minimizer, for each ε > 0 there exists δ = δ(ε) > 0 such that
Ψ (v) > Ψ (vo ) + δ
for all |v − vo | > ε.
(10.7)
Moreover, the numbers ε and δ being fixed, there exists some positive number
σ such that
Ψ (v) ≤ Ψ (vo ) + 12 δ for all |v − vo | < σ.
From this we estimate from below
Z
Z
2σ − 1 nδ
1
−n[Ψ (v)−Ψ (vo )] ′
e 2 .
e−n[Ψ (v)−Ψ (vo)] dv ≥
e
a (v)dv ≥
L
L
|v−vo |<σ
R
Therefore
260
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
L 1 nδ −n[Ψ (v)−Ψ (vo )] ′
a (v)dv.
e2 e
2σ
Next estimate In by using these remarks as
Z
In ≤
|f (v) − f (vo )|dµn (v)
|v−vo |<ε
Z
L 1
+ sup |f (v)| e 2 nδ
e−n[Ψ (v)−Ψ (vo )] a′ (v)dv
2σ
|v−vo |<2γ
ε<|v−vo |<2γ
Z
L 1 nδ+nΨ (vo )
+
(|f (v)| + |f (vo )|)e−nΨ (v) a′ (v)dv
e2
2σ
|v−vo |>2γ
dµn (v) ≤
= In(1) + In(2) + In(3)
where γ is a positive number to be chosen. Denoting by ω(·) the modulus of
(1)
continuity of f , estimate In ≤ ω(ε), since dµn (v) is a probability measure.
(2)
The second term In is estimated by means of (10.7) as
Z
L 1
In(2) ≤ sup |f (v)|a′ (v) e 2 nδ
e−n[Ψ (v)−Ψ (vo)] dv
2σ
|v−vo |<2γ
ε<|v−vo |<2γ
1
≤ C(γ, σ, L)e− 2 nδ
(2)
for a constant C depending only on the indicated quantities. Thus In
(3)
as n → ∞. The last term In is estimated using the lower bound
Z v
Ψ (v) ≥ −C + t[va(v) − F (v)] ≥ −C + t
sa′ (s)ds.
→0
0
Also choose γ ≥ γo , where γo is the constant in the growth condition (10.3).
By choosing γ even larger if necessary, we may ensure that [|v − vo | > 2γ] ⊂
[|v| > γ]. For this choice
Z
Rv ′
Co L n[δ+C+Ψ (vo )]
(3)
In ≤
e
e−(nt−co ) o sa (s)ds |v|a′ (v)dv.
(10.8)
σ
|v|>γ
If n is so large that nt − co > 0, the integral on the right-hand side of (10.8)
can be computed explicitly, and estimated as follows
Z
Z ∞
Z −γ
Rv ′
e−(nt−co ) o sa (s)ds |v|a′ (v)dv =
· · · dv +
· · · dv
|v|>γ
γ
−∞
Rγ ′
γ2
2
2
e−(nt−co ) o sa (s)ds ≤
e−(nt−co ) 2L .
=
nt − co
nt − co
This in (10.8) gives
co γ 2
In(3)
2
−δ−C−Ψ (vo )+ tγ
2L
2Co Le 2L −n
≤
e
σ(nt − co )
.
11 The Theorems of Existence and Stability
261
The number t > 0 being fixed, choose γ large enough that
−δ − C − Ψ (vo ) +
tγ 2
> 0.
2L
Then let n → ∞ to conclude that limn→∞ In ≤ ω(ε) for all ε > 0.
11 The Theorems of Existence and Stability
11.1 Existence of Variational Solutions
Theorem 11.1 (Existence). Let the assumptions (8.1)–(8.2) hold, and let
u(·, t) denote the function constructed in Sections 8–10. Then
ku(·, t)k∞,R ≤ kuo k∞,R
for all t > 0.
(11.1)
The function u solves the initial value problem (6.4) in the weak sense
Z tZ
Z
Z
u(x, t)ϕ(x, t)dx − uo (x)ϕ(x, 0)dx (11.2)
[uϕt + F (u)ϕx ]dxdτ =
0
R
R
R
for all ϕ ∈ C 1 [R+ ; Co∞ (R)] and a.e. t > 0. Moreover, u takes the initial datum
uo in the sense of L1loc (R), that is, for every compact subset K ⊂ R
lim ku(·, t) − uo k1,K = 0.
t→0
(11.3)
Finally, if uo (·) is continuous, then for all t > 0
u(x, t) = uo (x − a[u(x, t)]t)
for a.e. x ∈ R.
(11.4)
11.2 Stability of Variational Solutions
Assuming the existence theorem for the moment, we establish that the solutions constructed by the method of Sections 8–10 are stable in L1loc (R). Let
{uo,m } be a sequence of functions satisfying (8.2) and in addition
kuo,mk∞,R ≤ γkuok∞,R for all m, for some γ > 0
(11.5)
uo,m → uo weakly in L1 (−∞, x) for all x ∈ R.
Denote by um the functions constructed by the methods of Sections 8–10, corresponding to the initial datum uo,m . Specifically, first consider the functions
Ψm (x, t; ·) defined as in (8.8), with uo replaced by uo,m . For fixed t > 0, let
ηm (x, t) be a minimizer of Ψm (x, t; ·). Such a minimizer is unique for almost
all x ∈ R. Then set
x − ηm (x, t)
−1
um (x, t) = a
.
t
262
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
1
(R)). For fixed t > 0 and all compact subTheorem 11.2 (Stability in Lloc
sets K ⊂ R
kum (·, t) − u(·, t)k1,K → 0 as m → ∞.
Proof. Denote by Eo and Em theSsubsets of R where u(·, t) and um (·, t) are not
uniquely defined. The set E = Em has measure zero and {um (·, t), u(·, t)}
are all uniquely well defined in R − E. We claim that
lim um (x, t) = u(x, t)
m→∞
and
lim ηm (x, t) = η(x, t)
m→∞
for all x ∈ R − E, where η(x, t) is the unique minimizer of Ψ (x, t; ·). By (11.1)
and the first of (11.5), {um (x, t)} is bounded. Therefore also {ηm (x, t)} is
bounded, and a subsequence {ηm′ (x, t)} contains in turn a convergent subsequence, say for example {ηm′′ (x, t)} → ηo (x, t). By minimality
Ψm′′ (x, t; ηm′′ (x, t)) ≤ Ψm′′ (x, t; η(x, t)).
Letting m′′ → ∞
Ψ (x, t; ηo (x, t)) ≤ Ψ (x, t; η(x, t)).
Therefore ηo (x, t) = η(x, t), since the minimizer of Ψ (x, t; ·) is unique. Therefore any subsequence out of {ηm (x, t)} contains in turn a subsequence convergent to the same limit η(x, t). Thus the entire sequence converges to η(x, t).
Such a convergence holds for all x ∈ R − E, i.e., {um (·, t)} → u(·, t) a.e. in R.
Since {um (·, t)} is uniformly bounded in R, the stability theorem in L1loc (R)
follows from the Lebesgue dominated convergence theorem.
12 Proof of Theorem 11.1
12.1 The Representation Formula (11.4)
Let dλn (η) and dµn (v) be the probability measures introduced in (10.2) and
(10.6), and set
Z
Z
−1 x − η
vdµn (v)
dλn (η) =
a
un (x, t) =
t
R
R
(12.1)
Z
Z
x−η
dλn (v) =
F (v)dµn (v)
Fn (x, t) =
F a−1
t
R
R
Z
Z
Hn (x, t) = ln e−nΨ (x,t;η) dη = ln e−nΨ (v) a′ (v)dv
R
R
where the integrals on the right are computed from those on the left by the
change of variables (10.4)–(10.5). From the definitions (8.8) and (10.5) of
Ψ (x, t; η) and Ψ (v), and recalling that F ′ = a(·)
12 Proof of Theorem 11.1
x−η
=v
t
−1 x − η
= −F (v).
Ψt (x, t; η) = −F a
t
Ψx (x, t; η) = a−1
263
(12.2)
Then compute
∂
Hn (x, t) = −n
∂x
= −n
∂
Hn (x, t) = −n
∂t
Z
R
Z
ZR
Ψx (x, t; η)dλn (η) = −n
uo (x − a(v)t)dµn (v)
Ψt (x, t; η)dλn (η) = n
R
Z
Z
vdµn (v)
R
F (v)dµn (v).
R
Therefore
un (x, t) = −
These imply
and
1 ∂
Hn (x, t),
n ∂x
Fn (x, t) =
1 ∂
Hn (x, t).
n ∂t
∂
∂
un +
Fn = 0 in R × R+
∂t
∂x
Z
un (x, t) =
uo (x − a(v)t)dµn (v).
(12.3)
(12.4)
R
Since uo ∈ L∞ (R) and dµn (v) is a probability measure
kun (·, t)k∞,R ≤ kuo k∞,R
for all t > 0.
Therefore by Lemma 10.1 and Lebesgue’s dominated convergence theorem,
{un (·, t)} and {Fn (·, t)} converge to u(·, t) and F [u(·, t)] respectively in
L1loc (R), for all t > 0. Moreover {un } and {Fn } converge to u and F (u)
respectively, in L1loc (R × R+ ). This also proves (11.1).
If uo is continuous, the representation formula (11.4) follows from (12.4)
and Lemma 10.1, upon letting n → ∞.
12.2 Initial Datum in the Sense of L1loc (R)
Assume first uo ∈ C(R). Then by the representation formula (11.4)
Z
lim ku(·, t) − uo k1,K = lim
|uo (x − a[u(x, t)]t) − uo (x)|dx = 0
t→0
t→0
K
since u is uniformly bounded in K for all t > 0. If uo merely satisfies (8.2),
construct a sequence of smooth functions uo,m satisfying (11.5), and in addition {uo,m } → uo in L1loc (R). Such a construction may be realized through
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
264
a mollification kernel J1/m , by setting uo,m = J1/m ∗ uo . By the stability
Theorem 11.2
ku(·, t) − um (·, t)k1,K →
for all t > 0.
Moreover, since uo,m are continuous
kum (·, t) − uo,m k1,K →
as t → 0.
This last limit is actually uniform in m. Indeed
Z
Z
|um (x, t) − uo,m (x)|dx =
|uo,m (x − a[um (x, t)]t) − uo,m (x)|dx
K
ZK
=
|J1/m ∗ [uo (x − a[um (x, t)]t) − uo (x)]|dx
ZK
≤
|uo (x − a[um (x, t)]t) − uo (x)|dx.
K
Since a[um (x, t)] is uniformly bounded in K for all t > 0, the right-hand side
tends to zero as t → 0, uniformly in m.
Fix a compact subset K ⊂ R and ε > 0. Then choose t > 0 such that
kum (·, t) − uo,m k1,K ≤ ε.
Such a time t can be chosen independent of m, in view of the indicated uniform
convergence. Then we write
ku(t) − uo k1,K ≤ ku(t) − um (t)k1,K + kum (t) − uo,m k1,K + kuo,m − uo k1,K .
Letting m → ∞ gives ku(·, t) − uo k1,K ≤ ε.
12.3 Weak Forms of the PDE
Multiply (12.3) by ϕ ∈ C 1 [R+ ; Co∞ (R)] and integrate over (ε, t) × R for some
fixed ε > 0. Integrating by parts and letting n → ∞ gives
Z tZ
Z
Z
u(x, t)ϕ(x, t)dx −
u(x, ε)ϕ(x, ε)dx.
[uϕt + F (u)ϕx ]dxdτ =
ε
R
R
R
L1loc (R)
Now (11.2) follows, since u(·, t) → uo in
as t → 0.
The following proposition provides another weak form of the PDE.
Proposition 12.1 For all t > 0 and a.e. x ∈ R
Z x
u(s, t)ds = Ψ [x, t; u(x, t)]
−∞
=
Z
x−a[u(x,t)]t
−∞
uo (s)ds + t[ua(u) − F (u)](x, t).
(12.5)
13 The Entropy Condition
265
Proof. Integrate (12.3) in dτ over (ε, t) and then in ds over (k, x), where k
is a negative integer, and in the resulting expression let n → ∞. Taking into
account the expression (12.1) of Fn and the second of (12.2), compute
Z x
Z x
Z tZ
[Ψτ (x, τ ; η) − Ψτ (k, τ ; η)] dλn (η)dτ
u(s, ε)ds = lim
u(s, t)ds −
k
n→∞
k
ε
R
= Ψ [x, t; u(x, t)] − Ψ [k, ε; u(k, ε)]
by virtue of Lemma 10.1. To prove the proposition first let ε → 0 and then
k → −∞.
13 The Entropy Condition
A consequence of (9.1) is that the variational solution claimed by Theorem 11.1 satisfies the entropy condition
lim sup [u(x + h, t) − u(x, t)] ≤ 0
(13.1)
0<h→0
for all fixed t > 0 and a.e. x ∈ R. The notion of a weak solution introduced
in Section 5.1 does not require that (13.1) be satisfied. However, as shown by
the examples in Section 6.2, weak solutions need not be unique. We will prove
that weak solutions of the initial value problem (6.4) that in addition satisfy
the entropy condition (13.1), are unique. The method, due to Kruzhkov [141],
is N -dimensional and uses a notion of entropy condition more general than
(13.1).
13.1 Entropy Solutions
Consider the initial value problem
ut + div F(u) = 0 in ST = RN × (0, T ]
u(·, 0) = uo ∈ L1loc (RN )
(13.2)
where F ∈ [C 1 (R)]N . A weak solution of (13.2), in the sense of (5.4)–(5.5), is
an entropy solution if
ZZ
sign(u − k){(u − k)ϕt + [F(u) − F(k)] · Dϕ}dxdt ≥ 0
(13.3)
ST
for all non-negative ϕ ∈ Co1 (ST ) and all k ∈ R, where D denotes the gradient
with respect to the space variables only.
The first notion of entropy solution is due to Lax [157, 158], and it amounts
to (13.1). A more general notion, that would cover some cases of nonconvex
F (·), and would ensure stability, still in one space dimension, was introduced
by Oleinik [197, 198]. A formal derivation and a motivation of Kruzhkov notion
of entropy solution (13.3) is in Section 13c of the Complements. When N = 1
the Kruzhkov and Lax notions are equivalent, as we show next.
266
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
13.2 Variational Solutions of (6.4) Are Entropy Solutions
Proposition 13.1 Let u be the weak variational solution claimed by Theorem 11.1. Then for every convex function Φ ∈ C 2 (R) and all non-negative
ϕ ∈ Co∞ (R × R+ )
Z u
ZZ
Φ(u)ϕt +
F ′ (s)Φ′ (s)ds ϕx dxdt ≥ 0 for all k ∈ R.
R×R+
k
Corollary 13.1 The variational solutions claimed by Theorem 11.1 are entropy solutions.
Proof. Apply the proposition with Φ(s) = |s − k|, modulo an approximation
procedure. Then Φ′ (s) = sign(s − k) for s 6= k.
The proof of Proposition 13.1 uses the notion of Steklov averages of a function
f ∈ L1loc (R × R+ ). These are defined as
Z x+h
fh (x, t) = −
f (s, t)ds,
Z t+ℓ
fℓ (x, t) = −
f (x, τ )dτ
x
t
Z t+ℓ Z x+h
fhℓ (x, t) = −
−
f (s, τ )dsdτ
t
x
for all h ∈ R and all ℓ ∈ R such that t + ℓ > 0. One verifies that as h, ℓ → 0
fh (·, t) → f (·, t) in L1loc (R)
a.e. t ∈ R+
1
+
fℓ (x, ·) → f (x, ·) in Lloc (R )
a.e. x ∈ R
fhℓ
→f
in L1loc (R × R+ ).
Lemma 13.1 The variational solutions of Theorem 11.1 satisfy the weak formulation
∂
∂
uhℓ +
Fhℓ (u) = 0 in R × R+
∂t
∂x
Z ℓ
(13.4)
uhℓ (·, 0) = − uh (·, τ )dτ.
0
Moreover, uhℓ (·, 0) → uo in
L1loc (R)
as h, ℓ → 0.
Proof. Fix (x, t) ∈ R × R+ and h ∈ R and ℓ > 0. Integrate (12.3) in dτ over
(t, t + ℓ) and in ds over (x, x + h), and divide by hℓ. Letting n → ∞ proves
the lemma.
Proof (of Proposition 13.1). Let Φ ∈ C 2 (R) be convex and let ϕ ∈ Co∞ (R ×
R+ ) be non-negative. Multiplying the first of (13.4) by Φ′ (uhℓ )ϕ and integrating over R × R+ , gives
13 The Entropy Condition
−
ZZ
R×R+
267
[Φ(uhℓ )ϕt − F ′ (uhℓ )Φ′ (uhℓ )uhℓ x ϕ] dxdt
ZZ
[Fh,ℓ (u) − F (uhℓ )]Φ′′ (uhℓ )uhℓ x ϕdxdt
=
R×R+
ZZ
+
[Fh,ℓ (u) − F (uhℓ )]Φ′ (uhℓ )ϕx dxdt.
R×R+
The second term on the left-hand side is transformed by an integration by
parts and equals
Z uhℓ
ZZ
F ′ (s)Φ′ (s)ds ϕx dxdt
R×R+
k
where k is an arbitrary constant. Then let ℓ → 0 and h → 0 in the indicated
order to obtain
Z u
ZZ
Φ(u)ϕt +
F ′ (s)Φ′ (s)ds ϕx dxdt
R×R+
ZZ k
= − lim
[Fh (u) − F (uh )]Φ′′ (uh )(uh )x ϕdxdt.
h→0
R×R+
It remains to show that the right-hand side is non-negative. Since F (·) is
convex, by Jensen’s inequality Fh (u) ≥ F (uh ). By (9.1), for a.e. (x, t) ∈ R×R+
Z
∂ x+h
u(x + h, t) − u(x, t)
L
(uh )x =
−
u(s, t)ds =
≤ .
∂x x
h
t
− lim
h→0
ZZ
[Fh (u) − F (uh )]Φ′′ (uh )(uh )x ϕdxdt
ZZ
L
≥ lim
Φ′′ (uh )[Fh (u) − F (uh )]ϕ dxdt = 0.
h→0
τ
+
R×R
R×R+
13.3 Remarks on the Shock and the Entropy Conditions
Let u be an entropy solution of (13.2), discontinuous across a smooth hypersurface Γ . The notion (13.3) contains information on the nature of the
discontinuities of u across Γ . In particular, it does include the shock condition (4.4) and a weak form of the entropy condition (13.1).
If P ∈ Γ , the ball Bρ (P ) centered at P with radius ρ is divided by
Γ , at least for small ρ, into Bρ+ and Bρ− as in Figure 13.1. Let ν =
(νt ; νx1 , . . . , νxN ) = (νt ; ν x ) denote the unit normal oriented toward Bρ+ .
We assume that u ∈ C 1 (B̄ρ± ) and that it satisfies the equation in (13.2) in
the classical sense in Bρ± . In (13.3) take a non-negative test function ϕ ∈
268
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
Γ
..
..
..
.
................................................
.
.
.
.
.
.
.
.
.
.
........
......
.
.
.
.......
.
.
....
.
......
....
.
.
.
.
..
.
......
.
..
.....
.....
..
.....
.....
.
.
.
.
.
.
....
.
....
.
.
...
.
.
.
..
...
.
.
.
.
...
..
...
−
.
.
...
.
..
.
...
.
.
ρ
...
..
..
.
.
...
.
.
+
.
.
.
.
...
.
..
...
.
ρ
..
..
..
....
.
.
..
.
..
.
..
.. .
...
.
..
...........
...
.
.
.
..
.. ...........................
...
..
.
.
..
..
...
..
.
.
.
.
...
.
..
.
.
.
.
...
...
...
...
...
...
...
...
...
..
...
...
...
....
.
.
.
.
...
.
.
....
...
...
....
.....
...
.....
....
...
..... ........
.....
.
.
.
..... ......
.
..
...
.....
.........
.....
..... ............
......
.........
.....
.......
..............
.........
.....
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
....................
....
.....
.....
B
B
ν
Fig. 13.1
Co∞ (Bρ (P )) and integrate by parts by means of Green’s theorem. This gives,
for all k ∈ R
Z
sign(u+ − k){(u+ − k)νt + [F(u+ ) − F(k)] · ν x }ϕdσ
Γ
Z
sign(u− − k){(u− − k)νt + [F(u− ) − F(k)] · ν x }ϕdσ
≤
Γ
where dσ is the surface measure on Γ and u± are the limits of u(x, t) as
(x, t) tends to Γ from Bρ± . Since ϕ ≥ 0 is arbitrary, this gives the pointwise
inequality
sign(u+ − k){(u+ − k)νt + [F(u+ ) − F(k)] · ν x }
≤ sign(u− − k){(u− − k)νt + [F(u− ) − F(k)] · ν x }
(13.5)
on Γ . If k > max{u+ , u− }, (13.5) implies
([u+ − u− ], [F(u+ ) − F(u− )]) · ν ≥ 0
and if k < min{u+ , u− }
([u+ − u− ], [F(u+ ) − F(u− )]) · ν ≤ 0.
Therefore, the surface of discontinuity Γ must satisfy the shock condition
(4.4). Next, in (13.5), take k = 12 [u+ + u− ], to obtain
sign[u+ − u− ] F(u+ ) + F(u− ) − 2F(k) ν x ≤ 0.
(13.6)
This is an N -dimensional generalized version of the entropy condition (13.1).
Lemma 13.2 If N = 1 and F (·) is convex, then (13.6) implies (13.1).
14 The Kruzhkov Uniqueness Theorem
269
Proof. If N = 1, Γ is a curve in R2 , and we may orient it, locally, so that
ν = (νt , νx ), and νx ≥ 0. Since F (·) is convex, (8.3) implies that
F (u± ) − F (k) ≥ F ′ (k)(u± − k).
Adding these two inequalities gives
[F (u+ ) + F (u− ) − 2F (k)]νx ≥ 0.
This in (13.6) implies sign[u+ − u− ] ≤ 0.
14 The Kruzhkov Uniqueness Theorem
Theorem 14.1. Let u and v be two entropy solutions of (13.2) satisfying in
addition
F(u) − F(v)
≤ M for some M > 0.
(14.1)
u−v
∞,ST
Then u = v.
Remark 14.1 The assumption (14.1) is satisfied if F ∈ C 1 (R) and the solutions are bounded. In particular
Corollary 14.1 There exists at most one bounded entropy solution to the
initial value problem (6.4).
14.1 Proof of the Uniqueness Theorem I
Lemma 14.1 Let u and v be any two entropy solutions of (13.2). Then for
every non-negative ϕ ∈ Co∞ (ST )
ZZ
sign(u − v){(u − v)ϕt + [F(u) − F(v)] · Dϕ}dxdt ≥ 0.
(14.2)
ST
Proof. For ε > 0, let Jε be the Friedrichs mollifying kernels, and set
t−τ
|x − y|
x−y t−τ
= Jε
Jε
.
,
δε
2
2
2
2
Let ϕ ∈ Co∞ (ST ) be non-negative and assume that its support is contained in
the cylinder BR × (s1 , s2 ) for some R > 0 and ε < s1 < s2 < T − ε. Set
x−y t−τ
x+y t+τ
δε
.
(14.3)
,
,
λ(x, t; y, τ ) = ϕ
2
2
2
2
The function λ is compactly supported in ST × ST , with support contained in
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
270
|x + y|
|t + τ |
|t − τ |
|x − y|
<R ∩
< ε ; s1 <
< s2 ∩
<ε .
2
2
2
2
The variables of integration in (13.3) are x and t. We take k = v(y, τ ) for a.e.
(y, τ ) ∈ ST and integrate in dydτ over ST . This gives
ZZ ZZ
sign[u(x, t) − v(y, τ )]{[u(x, t) − v(y, τ )]λt
ST
ST
+ [F(u(x, t)) − F(v(y, τ ))] · ∇x λ}dxdtdydτ ≥ 0.
Analogously, one may write (13.3) for v in the variables of integration y, τ ,
and take k = u(x, t). Integrating in dxdt over ST gives an analogous inequality
with λt and Dx λ replaced by λτ and Dy λ. Adding these two inequalities gives
ZZ ZZ
{|u(x, t) − v(y, τ )|(λt + λτ ) + sign[u(x, t) − v(y, τ )]
(14.4)
ST
ST
+ [F(u(x, t) − F(v(y, τ )] · (Dx λ + Dy λ)}dxdtdydτ ≥ 0.
To transform this integral, compute from (14.3)
x−y t−τ
x+y t+τ
δε
,
,
λt + λτ = ϕt
2
2
2
2
x+y t+τ
x−y t−τ
Dx λ + Dy λ = Dϕ
δε
.
,
,
2
2
2
2
Then, in the resulting integral, make the change of variables
x+y
= ξ,
2
t+τ
= s;
2
x−y
= η,
2
t−τ
= σ.
2
The domain of integration is mapped into
{[|ξ| < R] × [s1 , s2 ]} × { [|η| < ε] × [|σ| < ε]}
and (14.4) is transformed into
ZZ
n ZZ
o
ϕt (ξ, s)
|u(ξ + η, s + σ) − v(ξ − η, s − σ)|δε (η, σ)dηdσ dξds
ST
ST
ZZ
n ZZ
Dϕ(ξ, s) ·
+
sign[u(ξ + η, s + σ) − v(ξ − η, s − σ)]
ST
ST
o
× [F(u(ξ + η, s + σ) − F(v(ξ − η, s − σ)]δε (η, σ)dηdσ dξds ≥ 0.
By the properties of mollifiers the integrals in {· · · } converge respectively to
|u(ξ, s) − v(ξ, s)|
and
sign[u(ξ, s) − v(ξ, s)][F(u(ξ, s) − F(v(ξ, s))]
for a.e. (ξ, s) ∈ ST . Moreover, they are uniformly bounded in ε, for a.e. (ξ, s) ∈
supp(ϕ). Therefore (14.2) follows by letting ε → 0 in the previous expression
and passing to the limit under the integrals.
14 The Kruzhkov Uniqueness Theorem
271
(xo , T )
·
.....................................................................................................................................................................................................
. .
.... ........
.....
.....
....
.....
.....
.....
....
.....
.
.
.
.
.....
.....
.
...
.
.
..........................................................................
.
.
.
.....
o
.....
.
.....
.
.
.....
.....
.
.
.
.....
...
.....
.....
.....
.....
.
.
.
.
.
.
.....
.....
.....
....
.....
.....
.....
.....
.....
.....
.....
.....
.....
.
.
.
.....
.....
.
.
.....
.
.
....
.
.
.
.
............................................................... ....................................................................................................................................................................................................................................................................
|x − x | < M (T − t)
·
xo
IRN
MT
Fig. 14.1
14.2 Proof of the Uniqueness Theorem II
Fix xo ∈ RN and R > 0 and construct the backward characteristic cone of
“slope” M
[|x − xo | < M (T − t)] × [0 < t < T ].
The cross section of this cone with the hyperplane t = const, for 0 < t < T ,
is the ball |x − xo | < M (T − t). The uniqueness theorem is a consequence of
the following
Proposition 14.1 For all xo ∈ RN and for almost all 0 < τ < t < T
Z
Z
|u − v|(x, t)dx ≤
|u − v|(x, τ )dx.
(14.5)
|x−xo |<M(T −t)
|x−xo |<M(T −τ )
Proof. Assume xo = 0 and in (14.2) take
Z t−τ
Z ∞
ϕ(x, t) =
Jε (s)ds
t−t1
Jε (s)ds
(14.6)
|x|−M(T −t)+ε
where ε < τ < t1 < T − ε are arbitrary but fixed. Such a ϕ is admissible since
it is non-negative, is in C ∞ (ST ), and vanishes outside the truncated backward
cone
|x| < M (T − t) × τ − ε < t < t1 + ε .
Compute
ϕt = [Jε (t − τ ) − Jε (t − t1 )]
Z
∞
Jε (s)ds
|x|−M(T −t)+ε
Z t−τ
− Jε (|x| − M (T − t) + ε)M
Dϕ = −Jε (|x| − M (T − t) + ε)
x
|x|
Jε (s)ds
t−t1
Z t−τ
Put this in (14.2) and change the sign to obtain
t−t1
Jε (s)ds.
272
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
ZZ
ST
≤
[Jε (t − t1 ) − Jε (t − τ )]|u − v|
ZZ
ST
Z
t−τ
t−t1
Z
∞
Jε (s)dsdxdt
|x|−M(T −t)+ε
Jε (s)dsJε (|x| − M (T − t) + ε)
· {|F(u) − F(v)| − M |u − v|}dxdt.
By virtue of (14.1), the right-hand side is nonpositive. Letting ε → 0, by the
properties of the mollifiers, the left-hand side converges to
Z
Z
|u − v|(x, τ )dx.
|u − v|(x, t1 )dx −
|x|<M(T −t1 )
|x|<M(T −τ )
14.3 Stability in L1 (RN )
Let u and v be entropy solutions of (13.2) defined in the whole of S∞ . Fix
T > 0 and rewrite (14.5) as
Z
Z
|u − v|(xo − y, t)dy ≤
|u − v|(xo − y, τ )dy.
|y|<M(T −τ )
|y|<M(T −t)
Integrating this in dxo over RN gives
ku − vk
1,RN
(t) ≤
T −τ
T −t
N
ku − vk1,RN (τ ).
Since the solutions u and v are global in time, let T → ∞, and deduce that
the function t → ku − vk1,RN (t) is nondecreasing.
Theorem 14.2. Let u and v be, global-in-time, entropy solutions of (13.2)
originating from initial data uo and vo in L1 (RN ) ∩ L∞ (RN ), and let (14.1)
hold. Then
ku − vk1,RN (t) ≤ kuo − vo k1,RN
for a.e. t > 0.
15 The Maximum Principle for Entropy Solutions
Proposition 15.1 Let u and v be any two weak entropy solutions of (13.2).
Then for all xo ∈ RN
Z
Z
(u − v)+ (x, t)dx ≤
(u − v)+ (x, τ )dx
|x−xo |<M(T −t)
for a.e. 0 < τ < t < T .
|x−xo |<M(T −τ )
15 The Maximum Principle for Entropy Solutions
273
Proof. Since u and v are weak solutions of (13.2) in the sense of (5.4)–(5.5),
starting from these, we may arrive at an analogue of (14.4) with equality and
without the extra factor sign[u(x, t) − v(y, τ )]. Precisely, starting from (5.4)
written for
u − v(y, τ ) and [F(u(x, t)) − F(v(y, τ ))]
choose ϕ and λ as in (14.3) and proceed as before to arrive at
ZZ ZZ
{[u(x, t) − v(y, τ )](λt + λτ )
ST
ST
+ [F(u(x, t)) − F(v(y, τ ))] · (Dx λ + Dy λ)}dxdtdydτ = 0.
Add this to (14.4) and observe that
|u − v| + (u − v) = 2(u − v)+
to obtain
ZZ ZZ
ST
ST
2[u(x, t) − v(y, τ )]+ (λt + λτ ) + 1 + sign[u(x, t) − v(y, τ )]
× [F(u(x, t)) − F(v(y, τ ))] · (Dx λ + Dy λ) dxdtdydτ ≥ 0.
Proceeding as in the proof of Lemma 14.1, we arrive at the inequality
ZZ
{2(u − v)+ ϕt + (1 + sign [u − v])[F(u) − F(v)] · Dϕ}dxdt ≥ 0
ST
for all non-negative ϕ ∈ Co∞ (ST ). Take ϕ as in (14.6) and let ε → 0.
Corollary 15.1 Let u and v be two entropy solutions of (13.2), and let (14.1)
hold. Then uo ≥ vo implies u ≥ v in ST .
Corollary 15.2 Let u be an entropy solution of (13.2), and let (14.1) hold.
Then
ku(·, t)k∞,RN ≤ kuo k∞,RN
for a.e. 0 < t < T.
Corollary 15.3 Assume that uo ∈ L∞ (RN ) and let F ∈ C 1 (R). Then there
exists at most one bounded, weak entropy solution to the initial value problem
(13.2).
274
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
Problems and Complements
3c Solving the Cauchy Problem
3.1. Solve x · ∇u = α with Cauchy data h(·) on the hyperplane xN = 1, that
is
Γ = {ξi = si }, i = 1, . . . , N − 1
ξN = 1, ζ(s) = h(s) ∈ C 1 (RN −1 ).
Denote by x̄ = (x1 , . . . , xN −1 ) points in RN −1 and by (x̄, xN ), points in
RN . An integral surface is given by
x̄(s, t) = set ,
xN (s, t) = et ,
z(s, t) = h(s) + αt.
For xN > 0, we have s = x̄/xN , and the solution is given by
x̄
u(x) = h
+ ln xα
N.
xN
3.2. Let u solve the linear equation ai (x)uxi = γu for some γ ∈ R. Show that
the general solution is given by x → uo (x)u(x), where uo is a solution of
the associated homogeneous equation.
3.3. Show that the characteristic projections of
yux + xuy = γu,
γ>0
(3.1c)
are the curves
x(t) = xo cosh t + yo sinh t
y(t) = xo sinh t + yo cosh t
(xo , yo ) ∈ R2
and observe in particular that the lines x = ±y are characteristic.
3.4. Show that the general solution of the homogeneous equation associated
with (3.1c) is f (x2 − y 2 ), for any f ∈ C 1 (R).
3.5. Solve (3.1c) for Cauchy data u(·, 0) = h ∈ C 1 (R). The integral surfaces
are
x(s, t) = s cosh t, y(s, t) = s sinh t, z(s, t) = h(s)eγt .
The solution exists in the sector |x| > |y|, and it is given by
γ/2
p
x+y
2
2
.
u(x, t) = h( x − y )
x−y
This is discontinuous at x = y and continuous but not of class C 1 at
x = −y. Explain in terms of characteristics.
6c Explicit Solutions to the Burgers Equation
275
3.6. Consider (3.1c) with data on the characteristic x = y, that is
u(x, x) = h(x) ∈ C 1 (R).
In general, the problem is not solvable. Following the method of Section 3,
we find the integral surfaces
x(s, t) = s(cosh t + sinh t)
y(s, t) = s(cosh t + sinh t)
z(s, t) = h(s)eγt .
From these compute
set =
x+y
,
2
u(x, y) =
x+y
2
γ
h(s)
.
sγ
Therefore the problem is solvable only if h(s) = Csγ . It follows from 3.4
that C = f (x2 − y 2 ), for any f ∈ C 1 (R).
3.7. Show that the characteristic projections of yux − xuy = γu, are the
curves
x(t) = xo cos t + yo sin t
(xo , yo ) ∈ R2 .
y(t) = yo cos t − xo sin t
Solve the Cauchy problem with data u(x, 0) = h(x). Show that if γ = 0
then the Cauchy problem is globally solvable only if h(·) is symmetric.
6c Explicit Solutions to the Burgers Equation
6.1. Verify that for λ > 0, the following are families of weak solutions to the
Burgers equations in R × R+ .

for x < 0

 0x
√
+
for 0 ≤ x ≤ 2λt
(6.1c)
W (x, t) =

√
t
0
for x > 2λt.
W − (x, t) =


 0x
√
for x < − 2λt
√
for − 2λt ≤ x ≤ 0

t
0
for

0





x
U (x, t) = xt − λ





0 t
(6.2c)
x > 0.
for x < 0
λ
for 0 ≤ x ≤
2
λ
for
<x≤λ
2
for x > λ.
(6.3c)
276
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
6.2c Invariance of Burgers Equations by Some Transformation of
Variables
Let ϕ be a solution of Burgers equation in R×R+ . Verify that for all a, b, c ∈ R
the following transformed functions are also solutions of Burgers equation:
u(x, t) = ϕ(x + a, t + b) for t > −b
u(x, t) = a + ϕ(x − at, t)
(i)
(ii)
u(x, t) = aϕ(bx, abt) = Ta,b ϕ
x a bx
ab u(x, t) = + ϕ
,c−
t
t
t
t
(iii)
def
for t >
ab
.
c
(iv)
6.2. Assume that a weak solution ϕ is known of the initial value problem
1
ϕt + (ϕ2 )x = 0 in R × R+ ,
2
ϕ(·, 0) = ϕo .
where ϕo is subject to proper assumptions that would ensure existence of
such a ϕ. Find a solutions of the initial value problems
1
ut + (u2 )x = 0 in R × R+
2
u(·, 0) = a + ϕo ;
1
ut + (u2 )x = 0 in R × R+
2
u(·, 0) = γx + ϕo
A solution of the first is
u(x, t) = a + ϕ(x − at, t)
and a solution of the second is
u(x, t) =
x
1
t γx
.
+
ϕ
,
1 + γt 1 + γt 1 + γt 1 + γt
Note that the initial values of these solutions do not satisfy the assumptions (8.2).
6.3. Prove that those solutions of Burgers equations for which ϕ = T1,b ϕ,
are of the form f (x/t).
6.4. Prove that those
of Burgers equations for which ϕ = Tb,b ϕ,
psolutions
√
are of the form f ( x/t)/ t.
6.3c The Generalized Riemann Problem
Consider the initial value problem
1
ut + (u2 )x = 0 in R × R+
2
α + px for x < 0
u(x, 0) =
β + qx for x > 0
(6.4c)
13c The Entropy Condition
277
where α, β, p, q are given constants. Verify that if α ≤ β, then the solution to
(6.4c) is
 α + px


 1 + pt for x ≤ αt






x
for αt ≤ x ≤ βt
u(x, t) =
(α ≤ β)

t



 β + qx



for x ≥ βt
1 + qt
for all times 1 + (α ∧ β)t > 0. If α > β, the characteristics from the left of
x = 0 intersect the characteristics from the right. Let x = x(t) be the line of
discontinuity and verify that a weak solution is given by
 α + px

for x < x(t)


 1 + pt
u(x, t) =
(α > β)


β + qx


for x > x(t)
1 + qt
where x = x(t) satisfies the shock condition (4.7). Enforcing it gives
x′ (t) =
Solve this ODE to find
1 α + px(t) β + qx(t) .
+
2
1 + pt
1 + qt
x(t) =
√
√
α 1 + qt + β 1 + qt
√
√
t.
1 + pt + 1 + qt
13c The Entropy Condition
Solutions of (13.2) can be constructed by solving first the Cauchy problems
uε,t − ε∆uε + div F(uε ) = 0 in ST
uε (·, 0) = uo
and then letting ε → 0. Roughly speaking, as ε → 0, the term ε∆uε “disappears” and the solution is found as the limit, in a suitable topology, of the net
{uε }. The method can be made rigorous by estimating {uε }, uniformly in ε,
in the class of functions of bounded variation ([265]).
In what follows we assume that a priori estimates have been derived that
ensure that {uε } → u in L1loc (ST ). Let k ∈ R and write the PDE as
∂
(uε − k) − ε∆(uε − k) + div[F(uε ) − F(k)] = 0.
∂t
278
7 QUASI-LINEAR EQUATIONS OF FIRST ORDER
Let hδ (·) be the approximation to the Heaviside function introduced in (14.2)
of Chapter 5. Multiply the PDE by hδ (uε − k)ϕ, where ϕ ∈ Co∞ (ST ) is nonnegative and integrate by parts over ST to obtain
ZZ
ST
n ∂ Z
∂t
uε −k
0
hδ (s)ds ϕdxdt + εh′δ (uε − k)|Duε |2 ϕ
+ εhδ (uε − k)D(uε − k) · Dϕ
+ hδ (uε − k)[F(uε ) − F(k)] · Dϕ
o
+ h′δ (uε − k)[F(uε ) − F(k)] · D(uε − k)ϕ dxdt = 0.
First let δ → 0 and then let ε → 0. The various terms are transformed and
estimated as follows.
ZZ
Z
∂ uε −k
lim lim
hδ (s)ds ϕdxdt
ε→0 δ→0
ST ∂t
0
Z Z Z uε −k
= − lim lim
hδ (s)ds ϕt dxdt
ε→0 δ→0
ST
0
ZZ
|u − k|ϕt dxdt.
=−
ST
The second term on the left-hand side is non-negative and is discarded. Next
ZZ
lim lim
εhδ (uε − k)D(uε − k) · Dϕdxdt
ε→0 δ→0
ST
= lim lim
ε→0 δ→0
ZZ
= − lim lim
ε→0 δ→0
lim lim
ε→0 δ→0
ZZ
ST
=
ZZ
εD
ST
ZZ
ST
Z
uε −k
0
ε
Z
0
uε −k
hδ (s)ds · Dϕdxdt
hδ (s)ds ∆ϕdxdt = 0.
hδ (uε − k)[F(uε ) − F(k)] · Dϕdxdt
ST
sign(u − k)[F(u) − F(k)] · Dϕdxdt.
The last term is transformed and estimated as
ZZ
Z uε
div
h′δ (s − k)[F(s) − F(k)]ds ϕdxdt
ST
0
Z Z Z uε
=−
h′δ (s − k)[F(s) − F(k)]ds · Dϕdxdt.
ST
For ε > 0 fixed
0
14c The Kruzhkov Uniqueness Theorem
lim hδ′ (s − k)[F(s) − F(k)] = 0
279
a.e. s ∈ (0, uε ).
δ→0
Moreover, by (14.1)
0 ≤ hδ′ (s − k)[F(s) − F(k)] ≤ M.
Therefore by dominated convergence
ZZ
lim
hδ′ (uε − k)[F(uε ) − F(k)] · D(uε − k)ϕdxdt = 0.
δ→0
ST
Combining these remarks yields (13.3).
14c The Kruzhkov Uniqueness Theorem
The theorem of Kruzhkov holds for the following general initial value problem
ut − div F(x, t, u) = g(x, t, u) in ST
(14.1c)
1
(RN ).
u(·, 0) = uo ∈ Lloc
∞
(ST ) is an entropy solution of (14.1c) if for all k ∈ R
A function u ∈ Lloc
ZZ
sign(u − k) (u − k)ϕt + [F(x, t, u) − F(x, t, k)] · Dϕ
(14.2c)
ST
+ [Fi,xi (x, t, u) + g(x, t, u)]ϕ dxdt ≥ 0
provided the various integrals are well defined. Assume
g, Fi ∈ C 1 (ST × R) i = 1, . . . , N.
Moreover
F(x, t, u) − F(x, t, v)
u−v
N
X
Fi,xi (x, t, u) − Fi,xi (x, t, v)
u−v
i=1
∞,ST ×R
∞,ST ×R
(14.3c)
≤ Mo
≤ M1
(14.4c)
F(x, t, u) − F(x, t, v)
≤ M2
u−v
∞,ST ×R
g(x, t, u) − g(x, t, v)
≤ M3
u−v
∞,ST ×R
for given positive constants Mi , i = 0, 1, 2, 3. The initial datum is taken in the
sense of L1loc (RN ). Set M = max{Mo , M1 , M2 , M3 }.
Theorem 14.1c. Let u and v be two entropy solutions of (14.1c) and let
(14.3c)–(14.4c) hold. There exists a constant γ dependent only on N and the
numbers Mi , i = 0, 1, 2, 3, such that for all T > 0 and all xo ∈ RN
Z
Z
|u − v|(x, t)dx ≤ eγt
|uo − vo |dx
|x−xo |<M(T −t)
for a.e. 0 < t < T .
|x−xo |<MT
8
NONLINEAR EQUATIONS OF FIRST
ORDER
1 Integral Surfaces and Monge’s Cones
A first-order nonlinear PDE is an expression of the form
F (x, u, ∇u) = 0
(1.1)
where x ranges over a given region E ⊂ RN , the function u is in C 1 (E) and
F is a given smooth real-valued function of its arguments. If u is a solution
of (1.1), then its graph Σ(u) is an integral surface for (1.1). Conversely, a
surface Σ is an integral surface for (1.1) if it is the graph of a smooth function
u solution of (1.1). For a fixed (x, z) ∈ E × R, consider the associated equation
F (x, z, p) = 0 and introduce the set
P(x, z) = {the set of all p ∈ RN satisfying F (x, z, p) = 0}.
If Σ(u) is an integral surface for (1.1), then for every (x, z) ∈ Σ(u)
z = u(x)
and
p = ∇u(x).
(1.2)
Therefore solving (1.1) amounts to finding a function u ∈ C 1 (E) such that for
all x ∈ E, among the pairs (x, z) there is one for which (1.2) holds. Let Σ be
an integral surface for (1.1). For (xo , zo ) ∈ Σ consider the family of planes
z − zo = p · (x − xo ),
p ∈ P(x, z).
(1.3)
Since Σ is an integral surface, among these there must be one tangent Σ at
(xo , zo ). The envelope of such a family of planes is a cone C(xo , zo ), called
Monge’s cone with vertex at (xo , zo ). Thus the integral surface Σ is tangent,
at each of its points, to the Monge’s cone with vertex at that point.1
1
Gaspard Monge, Beaune, France 1746–Paris 1818, combined equally well his
scientific vocation with his political aspirations. He took part in the French Rev© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_9
281
282
8 NONLINEAR EQUATIONS OF FIRST ORDER
1.1 Constructing Monge’s Cones
The envelope of the family of planes in (1.3) is that surface S tangent, at each
of its points, to one of the planes of the family (1.3). Thus for each (x, z) ∈ S,
there exists p = p(x) such that the corresponding plane in (1.3), for such a
choice of p, has the same normal as S. These remarks imply that the equation
of S is
z − zo = p(x) · (x − xo ).
(1.4)
The tangency requirement can be written as
pj (x) + pi,xj (xi − xo,i )
|
{z
}
jth component of the
normal to S at x
This gives the N equations
pxj · (x − xo ) = 0,
=
|
pj (x)
{z
}
.
jth component of the normal
to the tangent plane at x
i = 1, . . . , N.
Since p(x) ∈ Po , the vector-valued function x → p(x) must also satisfy
j = 1, . . . , N
Dp F xo , zo , p(x) · pxj = 0,
where Dp F = (Fp1 , . . . , FpN ). It follows that for each x fixed in a neighborhood
of xo , the vectors Dp F and x − xo are parallel, and there exists λ(x) such that
Dp F xo , zo , p(x) = λ(x)(x − xo )
(1.5)
F xo , zo , p(x) = 0.
This is a nonlinear system of N + 1 equations in the N + 1 unknowns
p1 (x), . . . , pN (x), λ(x). Solving it and putting the functions x → p(x) so obtained in (1.4) gives the equation of the envelope.
1.2 The Symmetric Equation of Monge’s Cones
Eliminating λ from (1.4) and the first of (1.5) gives the symmetric equation
of the cone
xi − xo,i
z − zo
=
,
i = 1, . . . , N.
(1.6)
p · Dp F
Fpi
This implies the Cartesian form of the Monge’s cone C(xo , zo )
2
p · Dp F
2
|z − zo | =
|x − xo |2 .
|Dp F |
(1.7)
olution and became minister of the navy in the Robespierre government (1792).
Mathematician and physicist of diverse interests, he contributed with Lavoisier to
the chemical synthesis of water (1785), and with Bertholet and Vandermonde in
identifying various metallurgical states of iron (1794). The indicated construction
is in Feuilles d’Analyse appliquée à la Géométrie, lectures delivered at the École
Polytechnique in 1801, and published by J. Liouville in 1850.
2 Characteristic Curves and Characteristic Strips
283
Remark 1.1 If F (x, z, p) is such that the “coefficient” of |x−xo |2 is constant,
then the cone in (1.7) is circular and its axis is normal to the hyperplane z = 0.
This is occurs for the first-order nonlinear PDE |∇u| = const, which arises in
geometric optics.
We stress however that the indicated “coefficient” depends on x via the
functions x → pj (x), and therefore C(xo , zo ) is not, in general, a circular cone,
nor is its axis normal to the hyperplane z = 0.
2 Characteristic Curves and Characteristic Strips
Let ℓ denote the line of intersection between the cone in (1.6) and the hyperplane tangent to the integral surface Σ at (xo , zo ). For (x, z) ∈ ℓ, the
vector p(x) remains constant. Therefore, for infinitesimal increments dz and
dxi , along ℓ
dz
dx1
dxN
=
= ··· =
p · Dp F
Fp1
FpN
where p and Fpi are computed at p(x), constant along ℓ. We conclude that ℓ
has directions
(Dp F (xo , zo , p(x)), p · Dp F (xo , zo , p(x)))
where p(x) is computed on ℓ. Following these directions, starting from (xo , zo ),
trace a curve on the surface Σ. Such a curve, described in terms of a parameter
t ∈ (−δ, δ), for some δ > 0, takes the form
ẋ(t) = Dp F x(t), z(t), p(t) x(0) = xo
(2.1)
ż(t) = p · Dp F x(t), z(t), p(t)
z(0) = zo .
Here p(t) is the solution of the system (1.5) with xo and zo replaced by x(t)
and z(t), and computed at points x on the tangencyline of the integral surface
Σ with the Monge’s cone with vertex at x(t), z(t) . The system (2.1) is not
well defined, because the functions t → pi (t) are in general not known. For
quasi-linear equations, F (x, z, p) = ai (x, z)pi . In such a case Fpi = ai (x, z) are
independent of p and (2.1) are the characteristics originating at Po (Section 1
of Chapter 7). Because of this analogy, we call the curves (2.1), characteristics.
To render such a system well defined, observe that if Σ is an integral surface,
then pi = uxi (x). From this and the first of (2.1)
ṗi = uxi xj ẋj = uxi xj Fpj .
Also, from the PDE (1.1), by differentiation
Fxi + Fu uxi + Fpj uxi xj = 0.
Therefore
ṗi = −Fxi − Fu uxi ,
i = 1, . . . , N.
284
8 NONLINEAR EQUATIONS OF FIRST ORDER
Thus, the characteristics for the nonlinear equation (1.1) are the curves

ẋ(t) = Dp F x(t), z(t), p(t) 


ż(t) = p · Dp F x(t), z(t), p(t)
(−δ, δ) ∋ t → Γ (t) =
(2.2)

 ṗ(t) = −Dx F x(t), z(t), p(t)

−Fz x(t), z(t), p(t) p(t)
where Dx F = (Fx1 , . . . , FxN ). For every choice of “initial” data
x(0), z(0), p(0) = (xo , zo , po ) ∈ E × R × RN
the system (2.2) has a unique solution, local it t, with the interval of existence
depending, in general, on the initial datum. To simplify the presentation we
assume that the interval of unique solvability is (−δ, δ), for every choice of
data (xo , zo , po ).
2.1 Characteristic Strips
A solution of (2.2) can be thought of as a curve t → x(t), z(t) ∈ RN +1 whose
points are associated to an infinitesimal portion of the hyperplane troughthem
and normal p(t). Putting together these portions along t → x(t), z(t) , the
function t → Γ (t) can be regarded as a strip of infinitesimal width, called a
characteristic strip. These remarks suggest that integral surfaces are union of
characteristic strips. Let Σ be a hypersurface in RN +1 given as the graph of
z = u(x) ∈ C 1 (E), and for (xo , zo ) ∈ Σ, let t → Γ(xo ,zo ) (t) be the characteristic strip originating at (xo , zo ), that is, the unique solution of (2.2) with data
z(0) = zo = u(xo ),
x(0) = xo ,
p(0) = ∇u(xo ).
(2.3)
for all t ∈ (−δ, δ).
(2.4)
The surface Σ is a union of characteristic strips if for every (xo , zo ) ∈ Σ, the
strip t → Γ(xo ,zo ) (t) is contained in Σ, in the sense that
z(t) = u(x(t))
and
p(t) = ∇u(x(t))
Proposition 2.1 An integral surface for (1.1) is union of characteristic
strips.
Proof. Let Σ be the graph of a solution u ∈ C 1 (E) of (1.1). Having fixed
xo ∈ E, let t → x(t) be the unique solution of
x(0) = xo .
ẋ(t) = Dp F x(t), u(x(t)), ∇u(x(t)) ,
One verifies that the 2N + 1 functions
(−δ, δ) ∋ t →
x(t),
z(t) = u(x(t)),
p(x(t)) = ∇u(x(t))
solve (2.2), with initial data (2.3). These are then characteristic strips.
Remark 2.1 Unlike the case of quasi-linear equations, the converse does not
hold, as (2.4) are not sufficient for one to conclude that Σ is an integral
surface. Indeed, even though F is constant along t → x(t), z(t) , the PDE
(1.1) need not hold identically.
3 The Cauchy Problem
285
3 The Cauchy Problem
Let s = (s1 , . . . , sN −1 ) be an (N − 1)-dimensional parameter ranging over the
cube Qδ = (−δ, δ)N −1 . The Cauchy problem associated with (1.1) consists
in assigning an (N − 1)-dimensional hypersurface Γ ⊂ RN +1 of parametric
equations
x = ξ(s) = ξ1 (s), . . . ,ξN (s)
(3.1)
Qδ ∋ s → Γ (s) =
z = ζ(s),
ξ(s), ζ(s) ∈ E × R
and seeking a function u ∈ C 1 (E) such that ζ(s) = u(ξ(s)) for s ∈ Qδ and
such that the graph z = u(x) is an integral surface of (1.1).
An integral surface for the Cauchy problem, must be a union of characteristic strips, and it must contain Γ . Therefore, one might attempt to construct it
by drawing, from each point (ξ(s), ζ(s)) ∈ Γ , a characteristic strip, a solution
of (2.2), starting from the initial data
x(0) = ξ(s),
z(0) = ζ(s),
p(0, s) = p(s) ∈ RN .
However, a surface that is a union of characteristic strips need not be an
integral surface. Moreover, starting from a point on Γ , one may construct
∞N characteristic strips, each corresponding to a choice of the initial vector
p(0, s) = p(s). A geometric construction of a solution to the Cauchy problem
for (1.1), hinges on a criterion that would identify, for each (ξ(s), ζ(s)) ∈ Γ ,
those initial data p(0, s) for which the union of the corresponding characteristic
strips, is indeed an integral surface.
3.1 Identifying the Initial Data p(0, s)
Set (ξ(0), ζ(0)) = (ξo , ζo ) ∈ Γ , and assume that there exists a vector po such
that
(3.2)
Dζ(0) = po · ∇ξ(0)
F (ξo , ζo , po ) = 0,
and in addition2
det
∇ξ(0)
Dp F (ξo , ζo , po )
6= 0.
Consider now the N -valued function
Qδ × RN ∋ (s, p) → Ψ (s, p) =
∇ζ(s) − p · ∇ξ(s)
.
F ξ(s), ζ(s), p
By (3.2) such a function vanishes for (s, p) = (0, po ). More generally, for
s ∈ Qδ , we seek those vectors p(s) for which Ψ vanishes, that is Ψ (s, p(s)) = 0.
By the implicit function theorem, this defines, locally, a smooth N -valued
function Qδ ∋ s → p(s) such that
2
See Section 2.2 of Chapter 7 for symbolism and motivation.
286
8 NONLINEAR EQUATIONS OF FIRST ORDER
∇ζ(s) = p(s) · ∇ξ(s)
F ξ(s), ζ(s), p(s) = 0
for all s ∈ Qδ .
(3.3)
Such a representation holds locally in a neighborhood of s = 0, which might be
taken as Qδ by possibly reducing δ. The vector p(s), so identified, is the set of
initial data p(0, s) = p(s) to be taken in the construction of the characteristic
strips.
3.2 Constructing the Characteristic Strips
The characteristic strips may now be constructed as the solutions of the system
of ODE’s
d
x(t, s) = Dp F x(t, s), z(t, s), p(t, s)
dt
d
z(t, s) = p(t, s) · Dp F x(t, s), z(t, s), p(t, s)
dt
(3.4)
d
p(t, s) = −Dx F x(t, s), z(t, s), p(t, s)
dt
−Fz x(t, s), z(t, s), p(t, s) p(t, s)
with initial data given at each s ∈ Qδ
x(0, s) = ξ(s),
z(0, s) = ζ(s),
p(0, s) = p(s).
(3.5)
The solution of (3.4)–(3.5) is local in t, that is, it exists in a time interval
that depends on the initial data, or equivalently on the parameter s ∈ Qδ .
By further reducing δ if needed, we may assume that (3.4)–(3.5) is uniquely
solvable for (t, s) ∈ (−δ, δ) × Qδ . Having solved such a system, consider the
map
(−δ, δ) × Qδ ∋ (t, s) → x(t, s), z(t, s) .
This represents a surface Σ ⊂ RN +1 , which by construction contains a local
portion of Γ about (ξo , ζo ).
Proposition 3.1 The surface Σ is an integral surface for the Cauchy problem
(1.1), (3.1).
4 Solving the Cauchy Problem
To prove the proposition, we construct a function x → u(x) whose graph is
Σ and that solves (1.1) in a neighborhood of (ξo , ζo ). Observe first that by
continuity, (3.2) continues to hold in a neighborhood of s = 0, i.e.,
∇ξ(s)
6= 0
for s ∈ Qδ
det
Dp F ξ(s), ζ(s), p(s)
4 Solving the Cauchy Problem
287
where δ is further reduced if needed. Next consider the map
M : (−δ, δ) × Qδ ∋ (t, s) → x(t, s).
From (3.4)–(3.5) and the previous remarks
∇ξ(s)
Ds x(0, s)
6= 0
det
= det
xt (0, s)
Dp F ξ(s), ζ(s), p(s)
for all s ∈ Qδ . By continuity this continues to hold for t ∈ (−δ, δ), where δ is
further reduced if necessary. Therefore
xt (t, s)
det
(4.1)
6= 0 for all (t, s) ∈ (−δ, δ) × Qδ .
Ds x(t, s)
Therefore M is locally invertible in a neighborhood of ξo . In particular, there
exist ε = ε(δ), a cube Qε (ξo ), and smooth functions T and S, defined in
Qε (ξo ), such that t = T (x) and s = S(x) for x ∈ Qε (ξo ).
The function x → u(x) is constructed by setting
for x ∈ Qε (ξo ).
u(x) = z T (x), S(x)
By construction, u ξ(s) = ζ(s) and t → F x(t, s), z(t, s), p(t, s) is constant
for all s ∈ Qδ . Moreover, by (3.3), F is also constant along Γ . Therefore
F x(t, s), z(t, s), p(t, s) = 0 for all (t, s) ∈ (−δ, δ) × Qδ .
(4.2)
It remains to prove that
p T (x), S(x) = ∇u(x)
for all x ∈ Qε (ξo ).
(4.3)
Ds z(t, s) = ∇u · Ds x(t, s).
(4.4)
From the definition of u(·)
zt (t, s) = ∇u · xt (t, s),
These and the equations of the characteristic strips, yield
[∇u − p(t, s)] · xt (t, s) = 0
for all (t, s) ∈ (−δ, δ) × Qδ .
(4.5)
4.1 Verifying (4.3)
Lemma 4.1 The relation (4.3) would follow from
Ds z(t, s) = p(t, s) · Ds x(t, s)
for all (t, s) ∈ (−δ, δ) × Qδ .
Proof. Assuming (4.6) holds true, rewrite it as
∇u · Ds x(t, s) = p(t, s)Ds x(t, s)
(4.6)
288
8 NONLINEAR EQUATIONS OF FIRST ORDER
which follows by making use of the second of (4.4). Combining this with
(4.5) gives the following linear homogeneous algebraic system in the unknowns
∇u − p(t, s)
∇u − p(t, s) · xt (t, s) = 0
∇u − p(t, s) · Ds x(t, s) = 0.
By (4.1), this admits only the trivial solution for all (t, s) ∈ (−δ, δ) × Qδ .
To establish (4.6), set
M (t, s) = Ds z(t, s) − p(t, s) · Ds x(t, s)
and verify that by the first of (3.3), M (0, s) = 0 for all s ∈ Qδ . From (4.2)
Dp F · Ds p + Dx F · Ds x = −Fz Ds z.
Using this identity in s and the equations (3.4) of the characteristic strips,
compute
Mt = Ds zt − pt · Ds x − p · Ds xt
= Ds pDp F + p · Dp Ds F + Dx F · Ds x + Fz p · Ds x − p · Dp Ds F
= Fz p · Ds x − Fz Ds z
= −Fz (Ds z − p · Ds x) = −Fz M.
This has the explicit integral
Z t
M (t, s) = M (0, s) exp −
Fz dτ
0
and gives (t, s) → M (t, s) = 0, since M (0, s) = 0.
4.2 A Quasi-Linear Example in R2
Denote by (x, y) the coordinates in R2 , and given two positive numbers A and
ρ, consider the Cauchy problem
xuux − Auy = 0,
u(ρ, y) = y.
The surface Γ in (3.1) is the line z = y in the plane x = ρ, which can be
written in the parametric form
ξ1 (s) = ρ,
ξ2 (s) = s,
ζ(s) = s,
s ∈ R.
We solve the Cauchy problem in a neighborhood of (ρ, 0, 0). The vector po
satisfying (3.2) is po = (0, 1). The system (3.4) takes the form
5 The Cauchy Problem for the Equation of Geometrical Optics
289
xt (t, s) = x(t, s)z(t, s)
yt (t, s) = −A
zt (t, s) = x(t, s)z(t, s)p1 (t, s) − Ap2 (t, s)
p1,t (t, s) = −z(t, s)p1 (t, s) − x(t, s)p21 (t, s)
p2,t (t, s) = x(t, s)p1 (t, s)p2 (t, s)
with initial conditions
x(0, s) = ρ,
y(0, s) = s,
p1 (0, s) = 0,
z(0, s) = s,
p2 (0, s) = 1.
The solution is
z(t, s) = s,
y(t, s) = −At + s,
ln
x
= st.
ρ
Eliminate the parameters s and t to obtain the solution in implicit form
x
u2 (x, t) − yu(x, y) = A ln .
ρ
The solution is analytic in the region y 2 + 4A ln(x/ρ) > 0.
5 The Cauchy Problem for the Equation of Geometrical
Optics
Let Φo be a surface in RN with parametric equations x = ξ(s) where s is a
(N − 1)-parameter ranging over some cube Qδ ⊂ RN −1 . Consider the Cauchy
problem for the eikonal equation ([51] Chapter 9, Section 8)
|∇u| = 1
u
Φo
= 0.
(5.1)
The function x → u(x) is the time it takes a light ray to reach x starting from
a point source at the origin. The level sets Φt = [u = t] are the wave fronts
of the light propagation, and the light rays are normal to these fronts. Thus
Φo is an initial wave front, and the Cauchy problem seeks to determine the
fronts Φt at later times t. The Monge’s cones are circular, with vertical axis
and their equation is (Section 1.2)
|z − t| = |x − y|
for every y ∈ Φt .
The characteristic strips are constructed from (3.4)–(3.5) as
xt (t, s) = p(t, s)
zt (t, s) = 1
pt (t, s) = 0
x(0, s) = ξ(s)
z(0, t) = 0
p(0, s) = p(s).
Computing the initial vectors p(s) from (3.3) gives
s ∈ Qδ
t ∈ (−δ, δ)
(5.2)
290
8 NONLINEAR EQUATIONS OF FIRST ORDER
p(s) · ∇ξ(s) = 0,
|p(s)| = 1,
for all s ∈ Qδ .
Thus p(s) is a unit vector normal to the front Φo . By the third of (5.2) such a
vector is constant along characteristics, and the characteristic system has the
explicit integral
x(t, s) = tp(s) + ξ(s),
z(t, s) = t,
p(t, s) = p(s).
(5.3)
Therefore after a time t, the front Φo evolves into the front Φt , obtained by
transporting each point ξ(s) ∈ Φo , along the normal p(s) with unitary speed,
for a time t.
5.1 Wave Fronts, Light Rays, Local Solutions and Caustics
For a fixed s ∈ Qδ the first of (5.2) are the parametric equations of a straight
line in RN , which we denote by ℓ(t; s). Since p(s) is normal to the front Φo , such
a line can be identified with the light ray through ξ(s) ∈ Φo . By construction
such a ray is always normal to the wave front Φt that it crosses.
This geometrical interpretation is suggestive on the one hand of the underlying physics, and on the other, it highlights the local nature of the Cauchy
problem. Indeed the solution, as constructed, becomes meaningless if two of
these rays, say for example ℓ(t; s1 ) and ℓ(t; s2 ), intersect at some point, for
such a point would have to belong to two distinct wave fronts. To avoid such
an occurrence, the number δ that limits the range of the parameters s and t
has to be taken sufficiently small.
The possible intersection of the light rays ℓ(s; t) might depend also on the
initial front. If Φo is an (N − 1)-dimensional hyperplane, then all rays are
parallel and normal to Φo . In such a case the solution exists for all s ∈ RN −1
and all t ∈ R. If Φo is an (N − 1)-dimensional sphere of radius R centered at
the origin of RN , all rays ℓ(t; s) intersect at the origin after a time t = R. The
solution exists for all times, and the integral surfaces are right circular cones
with vertex at the origin.
The envelope of the family ℓ(t; s) as s ranges over Qδ , if it exists, is called
a caustic or focal curve. By definition of envelope, the caustic is tangent in
any of its points to at least one light ray. Therefore, such a tangency point is
instantaneously illuminated, and the caustic can be regarded as a light tracer
following the parameter t.
If Φo is a hyperplane the caustic does not exists, and if Φo is a sphere, the
caustic degenerates into its origin.
6 The Initial Value Problem for Hamilton–Jacobi
Equations
Denote by (x; xN +1 ) points in RN +1 , and for a smooth function u defined in a
domain of RN +1 , set ∇u = (Dx u, uxN +1 ). Given a smooth nonlinear function
6 The Initial Value Problem for Hamilton–Jacobi Equations
291
(x, xN +1 , p) → H(x, p; xN +1 )
defined in a domain of RN +1 × RN , consider the first-order equation
F (x; xN +1 , u, Dx u, uxN +1 ) = uxN +1 + H(x, Dx u; xN +1 ) = 0.
(6.1)
The Cauchy problem for (6.1) consists in giving an N -dimensional surface
Φ and a smooth function uo defined on Φ, and seeking a smooth function u
that solves (6.1) in a neighborhood of Φ and equals uo on Φ. If the surface
Φ is the hyperplane xN +1 = 0, it has parametric equations x = s and the
characteristic system (3.4) takes the form
xt (t, s) = Dp H(x(t, s), p(t, s); xN +1 (t, s))
xN +1,t (t, s) = 1
zt (x, t) = p(t, s) · Dp H(x(t, s), p(t, s); xN +1 (t, s)) + pN +1 (t, s)
pt (t, s) = −Dx H(x(t, s), p(t, s), xN +1 (t, s))
pN +1,t (t, s) = −DxN +1 H(x(t, s), p(t, s); xN +1 (t, s))
with the initial conditions
x(0, s) = s,
xN +1 (0, s) = 0,
p(0, s) = p(s),
z(0, s) = uo (s)
pN +1 (0, s) = pN +1 (s).
The second of these and the corresponding initial datum imply xN +1 = t.
Therefore the (N + 1)st coordinate may be identified with time, and the
Cauchy problem for the surface [t = 0] is the initial value problem for the
Hamilton–Jacobi equation (6.1). The characteristic system can be written
concisely as
xt (t, s) = Dp H(x(t, s), p(t, s); t)
pt (t, s) = −Dx H(x(t, s), p(t, s); t)
x(0, s) = s
p(0, s) = p(s)
(6.2)
where the initial data (p(s), pN +1 (s)) are determined from (3.3) as
p(s) = Ds uo (s),
pN +1 (0, s) = −H(s, p(s); 0).
(6.3)
Moreover, the functions (t, s) → pN +1 (t, s), z(t, s), satisfy
pN +1,t (t, s) = −H(x(t, s), p(t, s); t)
pN +1 (0, s) = −H(s, p(s); 0)
zt (x, t) = p(t, s) · Dp H(x(t, s), p(t, s); t) + pN +1 (t, s)
z(0, s) = uo (s).
It is apparent that (6.2) is independent of (6.3), and the latter can be integrated as soon as one determines the functions (t, s) → x(t, s), p(t, s), solutions
292
8 NONLINEAR EQUATIONS OF FIRST ORDER
of (6.2). Therefore (6.2) is the characteristic system associated with the initial
value problem for (6.1).
Consider now a mechanical system with N degrees of freedom governed
by a Hamiltonian H. The system (6.2) is precisely the canonical Hamiltonian system that describes the motion of the system, through its Lagrangian
coordinates t → x(t, s) and the kinetic momenta t → p(t, s), starting from
its initial configuration. Therefore the characteristics associated with the initial value problem for the Hamilton–Jacobi equation (6.1) are the dynamic
trajectories, in phase space, of the underlying mechanical system.
From now on we will restrict the theory to the case H(x, t, p) = H(p), that
is, the Hamiltonian depends only on the kinetic momenta p. In such a case
the initial value problem takes the form
ut + H(Dx u) = 0,
u(·, 0) = uo
(6.4)
where uo is a bounded continuous function in RN . The characteristic curves
and initial data are
xt (t, s) = Dp H(p(t, s)
x(0, s) = s
p(0, s) = Ds uo (s)
pN +1 (0, s) = −H(Ds uo (s)).
pt (t, s) = 0
pN +1,t (t, s) = 0,
Moreover
zt (t, s) = p(t, s) · Dp H(p(t, s)) + pN +1 (t, s)
z(0, s) = uo (s).
(6.5)
(6.6)
7 The Cauchy Problem in Terms of the Lagrangian
Assume that p → H(p) is convex and coercive, that is3
lim
|p|→∞
H(p)
= ∞.
kpk
The Lagrangian q → L(q), corresponding to the Hamiltonian H, is given by
the Legendre transform of H, that is4
L(q) = sup [q · p − H(p)] .
p∈RN
By the coercivity of H the supremum is achieved at a vector p satisfying
q = Dp H(p)
and
L(q) = q · p − H(p).
(7.1)
3
This occurs, for example, for H(p) = |p|1+α , for all α > 0. It does not hold
for the Hamiltonian H(p) = |p| corresponding to the eikonal equation. The Cauchy
problem for such noncoercive Hamiltonians is investigated in [142, 143, 144].
4
[51] Chapter 6 Section 5, and [50], Section 13 of the Complements of Chapter IV.
8 The Hopf Variational Solution
293
Moreover, q → L(q) is itself convex and coercive, and the Hamiltonian H is
the Legendre transform of the Lagrangian L, that is
H(p) = sup [p · q − L(q)] .
q∈RN
Since L is coercive, the supremum is achieved at a vector q, satisfying
p = Dq L(q)
H(p) = q · p − L(q).
and
(7.2)
The equations for the characteristic curves (6.5)–(6.6) can be written in terms
of the Lagrangian as follows. The equations in (7.1), written for q = xt (t, s),
and the first of (6.5) imply that the vector p(s, t) for which the supremum in
the Legendre transform of H is achieved is the solution of the second of (6.5).
Therefore
(7.3)
L(xt (t, s)) = xt (t, s) · p(t, s) − H(p(t, s)).
Taking the gradient of L with respect to xt and then the derivative with
respect to time t, gives
Dẋ L(ẋ) = p(t, s)
and
d ∂L(ẋ)
= 0,
dt ∂ ẋh
h = 1, . . . , N.
These are the Lagrange equations of motion for a mechanical system of Hamiltonian H.
8 The Hopf Variational Solution
Let u be a smooth solution in RN × R+ of the Cauchy problem (6.4) for a
smooth initial datum uo . Then for every x ∈ RN and every time t > 0, there
exists some s ∈ RN such that x = x(t, s), that is the position x is reached in
time t by the characteristic ℓ = {x(t, s)} originating at s. Therefore
u(x, t) = u(x(t, s), t)
and
Dx u(x(t, s), t) = p(t, s).
Equivalently, taking into account that u is a solution of (6.4)
Z
∂u
u(x, t) − uo (s) =
dℓ
ℓ ∂ℓ
Z t
Dx u(x(τ, s), τ ) · xt (τ, s) + ut (x(τ, s), τ ) dτ
=
0
Z t
p(τ, s) · xt (τ, s) − H(p(τ, s)) dτ.
=
0
Using now (7.1), this implies
u(x, t) =
Z
0
t
L(xτ (τ, s))dτ + uo (s).
(8.1)
294
8 NONLINEAR EQUATIONS OF FIRST ORDER
8.1 The First Hopf Variational Formula
The integral on the right-hand side is the Hamiltonian action of a mechanical
system with N degrees of freedom, governed by a Lagrangian L, in its motion
from a Lagrangian configuration s at time t = 0 to a Lagrangian configuration
x at time t. Introduce the class of all smooth synchronous variations
the collection of all smooth paths q(·)
s
=
.
Ksync
in RN such that q(0) = s and q(t) = x
By the least action principle ([51], Chapter IX, Section 2)
Z
0
t
L(xt (τ, s))dτ = min
s
q∈Ksync
Z
0
t
L(q̇(τ ))dτ.
Therefore
u(x, t) = min
s
q∈Ksync
≥ inf
Z
t
0
L(q̇(τ ))dτ + uo (s)
infy
y∈RN q∈Ksync
Z
0
t
L(q̇(τ ))dτ + uo (y) .
Such a formula actually holds with the equality sign, since if u(x, t) is known,
by (8.1), for each fixed x ∈ RN and t > 0 there exist some s ∈ RN and a
smooth curve τ → x(τ, s) of extremities s and x such that the infimum is
actually achieved. This establishes the first Hopf variational formula, that is,
if (x, t) → u(x, t) is a solution of the Cauchy problem (6.4), then
Z t
u(x, t) = min min
(8.2)
L(q̇(τ ))dτ + uo (y) .
y
y∈RN q∈Ksync
0
8.2 The Second Hopf Variational Formula
A drawback of the first Hopf variational formula is that, given x and t, it
y
for all y ∈ RN . The next variational
requires the knowledge of the classes Ksync
formula dispenses with such classes ([124, 125]).
Proposition 8.1 let (x, t) → u(x, t) be a solution of the minimum problem
(8.2). Then for all x ∈ RN and all t > 0
i
h x − y
+ uo (y) .
(8.3)
u(x, t) = min tL
t
y∈RN
Proof. For s ∈ RN consider the curve
τ → q(τ ) = s +
τ
(x − s)
t
τ ∈ [0, t].
9 Semigroup Property of Hopf Variational Solutions
295
If u(x, t) is a solution of (8.2)
u(x, t) ≤
Z
0
t
x − s
L(q̇(τ ))dτ + uo (s) = tL
+ uo (s)
t
and since s is arbitrary
h x − y
i
tL
+ uo (y) .
t
y∈RN
u(x, t) ≤ inf
s
Now let q ∈ Ksync
for some s ∈ RN . Since L(·) is convex, by Jensen’s inequality
Z t
Z
x − s
1
1 t
L
=L
q̇(τ )dτ ≤
L(q̇(τ ))dτ.
t
t 0
t 0
From this
Z t
x − s
tL
+ uo (s) ≤
L(q̇(τ ))dτ + uo (s).
t
0
Since s ∈ RN is arbitrary, by (8.2)
Z t
i
h x − y
+ uo (y) ≤ min min
L(q̇(τ ))dτ + uo (y)
inf tL
y
t
y∈RN q∈Ksync
y∈RN
0
= u(x, t).
9 Semigroup Property of Hopf Variational Solutions
Proposition 9.1 ([7, 23]) Let (x, t) → u(x, t) be a solution of the variational problem (8.3). Then for all x ∈ RN and every pair 0 ≤ τ < t
h
x − y
i
u(x, t) = min (t − τ )L
+ u(y, τ ) .
(9.1)
t−τ
y∈RN
Proof. Write (8.3) for x = η at time τ and let ξ ∈ RN be a point where the
minimum is achieved. Thus
η − ξ u(η, τ ) = τ L
+ uo (ξ).
τ
Since L(·) is convex
Therefore
x − ξ τ x − η τ η − ξ L
= 1−
L
+ L
.
t
t
t−τ
t
τ
296
8 NONLINEAR EQUATIONS OF FIRST ORDER
h x − y
i
x − ξ u(x, t) = min tL
+ uo (y) ≤ tL
+ uo (ξ)
t
t
y∈RN
x − η
η−ξ
≤ (t − τ )L
+ τL
+ uo (ξ)
t−τ
τ
x − η
+ u(η, τ )
= (t − τ )L
t−τ
h
x − η
i
≤ min (t − τ )L
+ u(η, τ ) .
t−τ
η∈RN
Now let ξ ∈ RN be a point for which the minimum in (8.3) is achieved, i.e.,
x − ξ u(x, t) = tL
+ uo (ξ).
t
For τ ∈ (0, t) write
η=
τ
τ
x+ 1−
ξ
t
t
=⇒
x−ξ
η−ξ
x−η
=
=
.
t−τ
t
τ
Moreover, by (8.3)
η − ξ + uo (ξ).
u(η, τ ) ≤ τ L
τ
Combining these remarks
x − η
x − η
η − ξ
(t − τ )L
+ u(η, τ ) ≤ (t − τ )L
+ τL
+ uo (ξ)
t−τ
t−τ
τ
x − ξ = tL
+ uo (ξ) = u(x, t).
t
From this
h
x − η
i
u(x, t) ≥ min (t − τ )L
+ u(η, τ ) .
t−τ
η∈RN
10 Regularity of Hopf Variational Solutions
For (x, t) → u(x, t) to be a solution of the Cauchy problem (6.4), it would have
to be differentiable. While this is in general not the case, the next proposition
asserts that if the initial datum uo is Lipschitz continuous, the corresponding
Hopf variational solution is Lipschitz continuous. Assume then that there is
a positive constant Co such that
|uo (x) − uo (y)| ≤ Co |x − y|
for all x, y ∈ RN .
(10.1)
Proposition 10.1 Let (x, t) → u(x, t) be a solution of (8.3) for an initial
datum uo satisfying (10.1). Then there exists a positive constant C depending
only on Co and H such that for all x, y ∈ RN and all t, τ ∈ R+
|u(x, t) − u(y, τ )| ≤ C(|x − y| + |t − τ |).
(10.2)
11 Hopf Variational Solutions (8.3) Are Weak Solutions of the Problem (6.4)
297
Proof. For a fixed t > 0, let ξ ∈ RN be a vector for which the minimum in
(8.3) is achieved. Then for all y ∈ RN ,
h y − η i
x − ξ u(y, t) − u(x, t) = inf tL
+ uo (η) − tL
− uo (ξ)
t
t
η∈RN
y − (y − (x − ξ)) ≤ tL
+ uo (y − (x − ξ))
t
x − ξ − uo (ξ)
− tL
t
= uo (y − (x − ξ)) − uo (ξ) ≤ Co |y − x|.
Interchanging the role of x and y gives
|u(x, t) − u(y, t)| ≤ Co |x − y|
for all x, y ∈ RN .
This establishes the Lipschitz continuity of u in the space variables uniformly
in time. The variational formula (8.3) implies
u(x, t) ≤ tL(0) + uo (x)
and
for all x ∈ RN
h x − y i
u(x, t) = min tL
+ uo (y) − uo (x) + uo (x)
t
y∈RN
i
h x − y − Co |x − y|
≥ uo (x) + min tL
t
y∈RN
≥ uo (x) − max [Co t|q| − tL(q)]
q∈RN
≥ uo (x) − t max max [p · q − L(q)]
|p|<Co q∈RN
= uo (x) − t max H(p).
|p|<Co
Therefore
|u(x, t) − uo (x)| ≤ C̄t,
where C̄ = L(0) ∧ max H(p).
|p|<Co
(10.3)
from this
|u(x, t) − u(x, τ )| ≤ C̄|t − τ |
for all t, τ ∈ R+ .
Remark 10.1 By the Rademacher theorem (x, t) → u(x, t) is a.e. differentiable in RN × R+ ([50], Chapter VII, Section 23).
11 Hopf Variational Solutions (8.3) Are Weak Solutions
of the Cauchy Problem (6.4)
Assume that uo is Lipschitz continuous as in (10.1). Then by (10.3), a solution
(x, t) → u(x, t) of the corresponding variational problem (8.3) takes the initial
298
8 NONLINEAR EQUATIONS OF FIRST ORDER
datum uo in the classical sense. The next proposition asserts that such a
variational solution satisfies the Hamilton–Jacobi equation (6.4) at each point
(x, t) where it is differentiable.
Proposition 11.1 Let (x, t) → u(x, t) be a solution of the minimum problem
(8.3). If u is differentiable at (x, t) ∈ RN × R+ , then
ut + H(Dx u) = 0
at (x, t).
Proof. Fix η ∈ RN and h > 0. By the semigroup property
h
x + hη − y i
u(x + hη, t + h) = min (t + h)L
+ uo (y)
t+h
y∈RN
i
h x + hη − y + u(y, t) ≤ hL(η) + u(x, t).
= min hL
h
y∈RN
From this
u(x + hη, t + h) − u(x, t)
≤ L(η)
h
and letting h → 0
η · Dx u(x, t) + ut (x, t) ≤ L(η).
Recalling that H is the Legendre transform of L
ut (x, t) + H(Dx u(x, t)) = ut (x, t) + max [q · Dx u(x, t) − L(q)] ≤ 0.
q∈RN
Let ξ ∈ RN be a point for which the minimum in (8.3) is achieved. Fix
0 < h < t, set τ = t − h, and let
x−ξ
τ
η−ξ
h
τ
ξ =⇒
=
; η = x − (x − ξ).
η = x+ 1−
t
t
t
τ
t
Using these definitions, compute
h η − y
i
x − ξ + uo (ξ) − min τ L
+ uo (y)
u(x, t) − u(η, τ ) = tL
t
τ
y∈RN
x − ξ η − ξ
≥ tL
+ uo (ξ) − τ L
− uo (ξ)
t
τ
x − ξ x − ξ
= hL
.
= (t − τ )L
t
t
From this
i
x − ξ h
1h
u(x, t) − u x − (x − ξ), t − h ≥ L
h
t
t
and letting h → 0
ut (x, t) +
x − ξ x−ξ
.
· Dx u(x, t) ≥ L
t
t
12 Some Examples
299
Since H is the Legendre transform of the Lagrangian L
ut (x, t) + H(Dx u(x, t)) = ut (x, t) + max [q · Dx u(x, t) − L(q)]
q∈RN
≥ ut (x, t) +
x − ξ x−ξ
≥ 0.
· Dx u(x, t) − L
t
t
12 Some Examples
Proposition 11.1 ensures that the variational solutions (8.3), satisfy the
Hamilton–Jacobi equation in (6.4) only at points of differentiability, as shown
by the examples below.
12.1 Example I
ut + 12 |Dx u|2 = 0
in RN × R+ ,
u(x, 0) = |x|.
The Lagrangian L corresponding to the Hamiltonian H(p) =
L(q) = max p · q − 21 |p|2 = 21 |q|2
1
2
2 |p|
(12.1)
is
p∈RN
and the Hopf variational solution is
u(x, t) = min
y∈RN
|x − y|2
+ |y| .
2t
The minimum is computed by setting
|x − y|2
y−x
y
Dy
+ |y| =
+
=0
2t
t
|y|
x
=⇒ =
t
From this compute, for |x| > t
y =x−
If |x| ≤ t
xt
|x|
=⇒
u(x, t) = |x| − 12 t
1
1
y.
+
t
|y|
for |x| > t.
|x − y|2
|x|2
x · y |y|2
+ |y| =
−
+
+ |y|
2t
2t
t
2t
|x|
|y|2
|x|2 + |y|2
|x|2
−
|y| +
+ |y| ≥
.
≥
2t
t
2t
2t
This holds for all y ∈ RN , and equality holds for y = 0. Therefore if |x| ≤ t
the minimum is achieved for y = 0, and

 |x|2
for |x| ≤ t
u(x, t) =
2t 1
 |x|
− 2t
for |x| > t.
One verifies that u satisfies the Hamilton–Jacobi equation in (12.1) in RN ×R+
except at the cone |x| = t.
300
8 NONLINEAR EQUATIONS OF FIRST ORDER
Remark 12.1 For fixed t > 0 the graph of x → u(x, t) is convex for |x| < t
and concave for |x| > t. In the region of convexity, the Hessian matrix of u is
I/t. Therefore for all ξ ∈ RN
uxi xj ξi ξj ≤
|ξ|2
.
t
(12.2)
In the region of concavity |x| > t
uxi xj ξi ξj = (|x|2 δij − xi xj )
ξi ξj
|ξ|2
≤
.
3
|x|
t
Therefore (12.2) holds in the whole of RN × R+ except for |x| = t.
12.2 Example II
ut + 21 |Dx u|2 = 0
As before, L(q) =
1
2
2 |q| ,
in RN × R+ ,
u(x, 0) = −|x|.
(12.3)
and the Hopf variational solution is
|x − y|2
u(x, t) = min
− |y| .
2t
y∈RN
The minimum is computed by setting
y−x
y
|x − y|2
− |y| =
−
=0
Dy
2t
t
|y|
=⇒
y = (|x| + t)
x
.
|x|
Therefore
u(x, t) = −|x| − 21 t
in RN × R+ .
One verifies that this function satisfies the Hamilton–Jacobi equation in (12.3)
for all |x| > 0.
Remark 12.2 For fixed t > 0, the graph of x → u(x, t) is concave, and
uxi xj ξi ξj ≤ 0
for all ξ ∈ RN
in RN × R+ − {|x| = 0}.
(12.4)
12.3 Example III
The Cauchy problem
ut + (ux )2 = 0
in R × R+ ,
u(x, 0) = 0
has the identically zero solution. However the function

for
|x| ≥ t
 0
x − t for 0 ≤ x ≤ t
u(x, t) =

−x − t for −t ≤ x ≤ 0.
(12.5)
is Lipschitz continuous in R×R+ , and it satisfies the equation (12.5) in R×R+
except on the half-lines x = ±t.
13 Uniqueness
301
Remark 12.3 For fixed t > 0, the graph of x → u(x, t) is convex for |x| < t
and concave for |x| > t. In the region of concavity, uxx = 0, whereas in the
region of convexity, u(·, t) is not of class C 2 , and whenever it does exist, the
second derivative does not satisfy an upper bound of the type of (12.2). This
lack of control on the convex part of the graph of u(·, t) is responsible for the
lack of uniqueness of the solution of (12.5).
This example raises the issue of identifying a class of solutions of the Cauchy
problem (6.4) within which uniqueness holds.
13 Uniqueness
Denote by Co the class of solutions (x, t) → u(x, t) to the Cauchy problem
(6.4), of class C 2 (RN × R+ ), uniformly Lipschitz continuous in RN × R+ and
such that the graph of x → u(x, t) is concave for all t > 0, that is

 u ∈ C 2 (RN × R+ )
for all ξ ∈ RN
in RN × R+
(13.1)
Co = uxi xj ξi ξj ≤ 0

|∇u| ≤ C
for some C > 0 in RN × R+ .
Proposition 13.1 Let u1 and u2 be two solutions of the Cauchy problem
(6.4) in the class Co . Then u1 = u2 .
Proof. Setting w = u1 − u2 , compute
Z 1
d
wt = H(Dx u2 ) − H(Dx u1 ) =
H(sDx u2 + (1 − s)Dx u1 )ds
ds
0
Z 1
=−
Hpj (sDx u2 + (1 − s)Dx u1 )ds wxj = −V · Dx w
(13.2)
0
where
V=
Z
0
1
Dp H(sDx u2 + (1 − s)Dx u1 )ds.
(13.3)
Multiplying (13.2) by 2w gives
wt2 = −V · Dx w2 = − div(Vw2 ) + w2 div V.
Lemma 13.1 div V ≤ 0.
Proof. Fix (x, t) ∈ RN × R+ and s ∈ (0, 1), set
p = sDx u2 (x, t) + (1 − s)Dx u1 (x, t)
zij = su2,xi xj (x, t) + (1 − s)u1,xi xj (x, t)
and compute
(13.4)
8 NONLINEAR EQUATIONS OF FIRST ORDER
302
div V =
Z
0
1
Hpi pj (p)zji ds.
The integrand is the trace of the product matrix (Hpi pj )(zij ). Since H is
convex, (Hpi pj ) is symmetric and positive semi-definite, and its eigenvalues λh = λh (x, t, s) for h = 1, . . . , N are non-negative. Since both matrices
(uℓ,xi xj ) for ℓ = 1, 2 are negative semi-definite, the same is true for the convex
combination
(zij ) = s(u2,xi xj ) + (1 − s)(u1,xi xj ).
In particular, its eigenvalues µh = µh (x, t, s) for h = 1, . . . , N are nonpositive.
Therefore
Hpi pj zji = trace(Hpi pj )(zij ) = λh µh ≤ 0.
This in (13.4) gives
wt2 + div(Vw2 ) ≤ 0
in RN × R+ .
(13.5)
Fix xo ∈ RN and T > 0, and introduce the backward characteristic cone with
vertex at (xo , T )
CM = |x − xo | ≤ M (T − τ ); 0 ≤ τ ≤ T
(13.6)
where M > 0 is to be chosen. The exterior unit normal to the lateral surface
of CM is
(x/|x|, M )
= (νx , νt ).
ν= √
1 + M2
For t ∈ (0, T ) introduce also the backward truncated characteristic cone
t
CM
= |x − xo | ≤ M (T − τ ); 0 ≤ τ ≤ t
(13.7)
Integrating (13.5) over such a truncated cone gives
Z
2
|x−xo |<M (T −t)
≤
Z
2
|x−xo |<M T
M
1 + M2
Z tZ
w (x, t)dx + √
w (x, 0)dx −
0
Z
t
0
Z
w2 dσ(τ )dτ
|x−xo |=M (T −τ )
|x−xo |=M (T −τ )
(13.8)
w2 V · νx dσ(τ )dτ
where dσ(τ ) is the surface measure on the sphere [|x − xo | = M (T − τ )].
Using the constant C in (13.1) and the definition (13.3) of Vi, choose M from
|V · νx | ≤ sup |Dp H(p)| = δM
|p|<C
for some δ ∈ (0, 1).
(13.9)
This choice of M in (13.8) gives
Z
Z Z
(1 − δ)M t
w2 (x, t)dx + √
w2 dσ(τ )dτ
1 + M 2 0 |x−xo |=M(T −τ )
|x−xo |<M(T −t)
Z
w2 (x, 0)dx.
≤
|x−xo |<MT
14 More on Uniqueness and Stability
Thus
Z
|x−xo |<M(T −t)
w2 (x, t)dx ≤
Z
w2 (x, 0)dx.
303
(13.10)
|x−xo |<MT
14 More on Uniqueness and Stability
Multiply (13.2) by f ′ (w), for some non-negative f ∈ C 1 (R) and then use
Lemma 13.1 to get
in RN × R+ .
f (w)t + div(Vf (w)) ≤ 0
By similar arguments
Z
|x−xo |<M(T −t)
f (w(x, t))dx ≤
Z
f (w(x, 0))dx.
(14.1)
|x−xo |<MT
By the change of variables x − xo = y
Z
Z
f (w(xo + y, t))dξ ≤
f (w(xo + y, 0))dy.
|y|<MT
|y|<M(T −t)
Integrating this in dxo over RN gives
N Z
Z
T
f (w(x, t))dx ≤
f (w(x, 0))dx
T −t
RN
RN
provided the integrals are convergent. Fix 0 < t < T and let T → ∞ to obtain
the stability estimate
Z
Z
f (w(x, t))dx ≤
f (w(x, 0))dx.
(14.2)
RN
RN
14.1 Stability in Lp (RN ) for All p ≥ 1
Proposition 14.1 Let u1 and u2 be solutions of (6.4) in the class Co introduced in (13.1). If both are in Lp (RN ) for some 1 ≤ p ≤ ∞, then
ku1 (·, t) − u2 (·, t)kp,RN ≤ ku1 (·, 0) − u2 (·, 0)kp,RN .
(14.3)
Proof. If 1 ≤ p < ∞, the conclusion follows from (14.2) for f (w) = |w|p . If
p = 1, take f (w) = sign(w), modulo an approximation process. If p = ∞ write
(14.1) with f (w) = |w|q for 1 < q < ∞, in the form
1/q
Z
N
q
|w(x,
t)|
dx
ωN [M (T − t)]N |x−xo |<M(T −t)
N/q 1/q
Z
N
T
q
|w(x,
0)|
dx
≤
T −t
ωN (M T )N |x−xo |<MT
304
8 NONLINEAR EQUATIONS OF FIRST ORDER
where ωN is the measure of the unit sphere in RN . Letting q → ∞ gives
kw(·, t)k∞,[|x−xo |<M(T −t)] ≤ kw(·, 0)k∞,[|x−xo |<MT ] .
This implies (14.3) for p = ∞ since xo is arbitrary.
14.2 Comparison Principle
Proposition 14.2 Let u1 and u2 be solutions of (6.4) in the class Co introduced in (13.1). If uo,1 ≤ uo,2 , then
u1 (·, t) ≤ u2 (·, t)
in RN
for all t > 0.
(14.4)
Proof. In (14.2) choose f (w) = w+ modulo an approximation process.
15 Semi-Concave Solutions of the Cauchy Problem
Let u be a solution of the Cauchy problem (6.4) of class C 2 (RN × R+ ) with no
requirement that the graph of u(·, t) be concave. We require, however, that in
those regions where such a graph is convex, the “convexity”, roughly speaking,
be controlled by some uniform bound of the second derivatives of u(·, t). In a
precise way it is assumed that there exists a positive constant γ such that
(uxi xj ) − γI ≤ 0
in RN × R+ .
Since this matrix inequality is invariant by rotations of the coordinate axes,
it is equivalent to
uνν ≤ γ
in RN × R+
for all |ν| = 1.
(15.1)
A solution of the Cauchy problem (6.4) satisfying such an inequality for all
unit vectors ν ∈ RN is called semi-concave.
15.1 Uniqueness of Semi-Concave Solutions
The example in Section 12.3 shows that initial data, however smooth, might
give rise to quasi-concave solutions and solutions for which (15.1) is violated.
Introduce the class
u ∈ C 2 (RN × R+ ) satisfies (15.1) and
C1 =
(15.2)
|∇u| ≤ C for some C > 0 in RN × R+ .
Proposition 15.1 Let u1 and u2 be two solutions of the Cauchy problem
(6.4) in the class C1 . Then u1 = u2 .
16 A Weak Notion of Semi-Concavity
305
Proof. Set w = u1 − u2 and proceed as in the proof of Proposition 13.1 to
arrive at (13.2). Multiplying the latter by sign w modulo an approximation
process gives
|w|t = − div(V|w|) + |w| div V
where V is defined in (13.3).
Lemma 15.1 There exists a constant γ̄ depending only on the constant γ in
(15.1) and C in (15.2) such that div V ≤ γ̄ in RN × R+ .
Proof. With the same notation as in Lemma 13.1
Hpi pj (p)zij = trace(Hpi pj )(zij ) = trace(Hpi pj )((zij ) − γI + γI)
≤ trace(Hpi pj )((zij ) − γI) + γtrace(Hpi pj (p)) ≤ γHpi pi (p).
Since the Hamiltonian is convex, Hpi pi (p) ≥ 0. Moreover, since the solutions
u1 and u2 are both uniformly Lipschitz in RN × R+
0 ≤ γHpi pi (p) ≤ γ sup Hpi pi (p) = γ̄.
|p|<C
Combining these estimates
|w|t + div(V|w|) ≤ γ̄|w|
in RN × R+ .
Introduce the backward characteristic cone CM and the truncated backward
t
characteristic cone CM
as in (13.6) and (13.7), where the constant M is chosen
as in (13.9). Similar calculations yield
Z tZ
Z
|w(x, τ )|dxdτ.
|w(x, t)|dx ≤ γ̄
|x−xo |<M(T −t)
0
|x−xo |<M(T −τ )
This implies w = 0 by Gronwall’s inequality.
16 A Weak Notion of Semi-Concavity
The most limiting requirement of the class C1 is that solutions have to be
of class C 2 (RN × R+ ). Such a requirement is not natural, since the equation
in (6.4) imposes no conditions on the second derivatives of it solutions. In
addition, Proposition 10.1 establishes only that variational solutions of (8.3)
are Lipschitz continuous, however smooth the initial datum might be. On the
other hand, the example of Section 12.3 shows that uniqueness fails if some
assumptions are not formulated on the graph of u(·, t) through the second
derivatives of solutions. A condition of semi-concavity can be imposed using
a discrete form of second derivatives. A solution of the Cauchy problem (6.4)
is weakly semi-concave if there exists a positive constant γ such that for every
unit vector ν ∈ RN and all h ∈ R
1 2
h .
(16.1)
u(x + hν, t) − 2u(x, t) + u(x − hν, t) ≤ γ 1 +
t
306
8 NONLINEAR EQUATIONS OF FIRST ORDER
Remark 16.1 The t-dependence on the right-hand side allows for nonsemiconcave initial data.
For ε > 0, let kε be a mollifying kernel in RN , and let uε (·, t) be the mollification of u(·, t) with respect to the space variables, i.e.,
Z
x → uε (x, t) =
kε (x − y)u(y, t)dy.
RN
Lemma 16.1 Let u(·, t) be weakly semi-concave in the sense of (16.1). Then
for every unit vector ν ∈ RN and all ε > 0,
1
uε,νν ≤ γ 1 +
in RN × R+ .
t
Proof. Fix ν ∈ RN and ε > 0 and compute
Z
kε,νν (x − y)u(y, t)dy
uε,νν (x, t) =
N
Z R
1
[kε (x + hν − y) − 2kε (x − y, t) + kε (x − hν − y)] u(y, t)dy
= lim 2
h→0 h
N
ZR
1
= lim 2
kε (x − η) [u(η + hν, t) − 2u(η, t) + u(η − hν, t)] dη
h→0 h
RN
Z
1
1
.
kε (x − η)dη = γ 1 +
≤γ 1+
t
t
RN
Corollary 16.1 Let u(·, t) be weakly semi-concave in the sense of (16.1).
Then for all ε > 0
1
I ≤ 0 in RN × R+ .
(uε,xi xj ) − γ 1 +
t
17 Semi-Concavity of Hopf Variational Solutions
The semi-concavity condition (16.1) naturally arises from the variational formula (8.3). Indeed, if uo is weakly semi-concave, the corresponding variational
solution is weakly semi-concave. Moreover, if the Hamiltonian p → H(p) is
strictly convex, then the corresponding variational solution is weakly semiconcave irrespective of whether the initial datum is weakly semi-concave. The
next two sections contain these results. Here we stress that they hold for the
variational solutions (8.3) and not necessarily for any solution of the Cauchy
problem (6.4).
17.1 Weak Semi-Concavity of Hopf Variational Solutions Induced
by the Initial Datum uo
Proposition 17.1 Let uo be weakly semi-concave, that is there exists a positive constant γo such that for every unit vector ν ∈ RN and all h ∈ R
17 Semi-Concavity of Hopf Variational Solutions
uo (x + hν) − 2uo (x) + u(x − hν) ≤ γo h2
307
in R.
Then x → u(x, t) is weakly semi-concave, uniformly in t, for the same constant
γo .
Proof. Let ξ ∈ RN be a vector where the minimum in (8.3) is achieved. Then
h x ± hν − y i
x − ξ u(x ± hν, t) = min tL
+ uo (y) ≤ tL
+ uo (ξ ± hν).
t
t
y∈RN
From this
u(x + hν, t)−2u(x, t) + u(x − hν, t)
x − ξ x − ξ + uo (ξ + hν) − 2tL
≤ tL
t
t
x − ξ − 2uo (ξ) + tL
+ uo (ξ − hν)
t
= uo (ξ + hν) − 2uo (ξ) + uo (ξ − hν) ≤ γo h2 .
17.2 Strictly Convex Hamiltonian
The Hamiltonian p → H(p) is strictly convex, if there exists a positive constant
co such that (Hpi pj ) ≥ co I in RN .
Lemma 17.1 Let p → H(p) be strictly convex. then for all p1 , p2 ∈ RN
p + p 1
1
co
1
2
≤ H(p1 ) + H(p2 ) − |p1 − p2 |2 .
H
2
2
2
8
Moreover, if L is the Lagrangian corresponding to H, then for all q1 , q2 ∈ RN
q + q 1
1
1
1
2
|q1 − q2 |2 .
L(q1 ) + L(q2 ) ≤ L
+
2
2
2
8co
Proof. Fix p1 6= p2 in RN and set
p̄ = 12 (p1 + p2 ),
ℓ = |p2 − p1 |,
ν = (p2 − p1 )/2ℓ.
Consider the two segments (p1 , p̄) and (p̄, p2 ) with parametric equations
(p1 , p̄) = {y(σ) = p̄ + σν; σ ∈ (0, − 21 ℓ)}
(p̄, p2 ) = {y(σ) = p̄ + σν; σ ∈ (0, 12 ℓ)}
and compute
H(p̄) = H(p1 ) +
H(p̄) = H(p2 ) −
Z
ℓ/2
Dσ H(p̄ − σν) · νdσ
0
Z
0
ℓ/2
Dσ H(p̄ + σν) · νdσ.
308
8 NONLINEAR EQUATIONS OF FIRST ORDER
Adding them up
2H(p̄) = H(p1 ) + H(p2 ) −
Z
0
ℓ/2
[Dσ H(p̄ + σν) − Dσ H(p̄ − σν)] · νdσ.
By the mean value theorem, there exists some σ ′ ∈ (0, 21 ℓ) such that
[Dσ H(p̄ + σν) − Dσ H(p̄ − σν)] · ν = Hpi pj (p̄ − σ ′ ν)νi νj 2σ ≥ co 2σ.
Combining these calculations
2H(p̄) ≤ H(p1 ) + H(p2 ) − co
Z
ℓ/2
2σdσ.
0
This proves the first statement. To prove the second, recall that L is the
Legendre transform of H. Therefore
L(q1 ) ≤ q1 · p1 − H(p1 ),
L(q2 ) ≤ q2 · p2 − H(p2 )
for all p1 , p2 ∈ RN . From this
1
1
1
1
1
L(q1 ) + L(q2 ) ≤ (q1 · p1 + q2 · p2 ) −
H(p1 ) + H(p2 )
2
2
2
2
2
p + p c
1
o
1
2
− |p1 − p2 |2 .
≤ (q1 · p1 + q2 · p2 ) − H
2
2
8
Transform
q + q p + p 1
1
1
2
1
2
+ (q1 − q2 ) · (p1 − p2 )
(q1 · p1 + q2 · p2 ) =
2
2
2
4
and combine with the previous inequality to obtain
p + p q + q p + p 1
1
1
2
1
2
1
2
−H
L(q1 ) + L(q2 ) ≤
2
2
2
2
2
c
1
1
1
o
−
|q1 − q2 |2 +
|q1 − q2 |2
|p1 − p2 |2 − (q1 − q2 ) · (p1 − p2 ) +
8
4
8co
8co
i2
h q + q i hr c
1
1
2
o
(q1 − q2 )
≤ max
· p − H(p) −
(p1 − p2 ) − √
2
8
p∈RN
8co
1
+
|q1 − q2 |2
8co
q + q 1
1
2
+
|q1 − q2 |2 .
≤L
2
8co
Proposition 17.2 Let H be strictly convex. Then every variational solution
of (8.3) is weakly semi-concave for all t > 0, in the sense of (16.1), for a
constant γ independent of uo .
18 Uniqueness of Weakly Semi-Concave Variational Hopf Solutions
309
Proof. Let ξ ∈ RN be a vector where the minimum in (8.3) is achieved. Then
h x + hν − y i
u(x + hν, t) − 2u(x, t) + u(x − hν, t) = min tL
+ uo (y)
t
y∈RN
h x − hν − y h x − ξ i
i
− 2 tL
+ uo (ξ) + min tL
+ uo (y)
t
t
y∈RN
h x + hν − ξ i
h x − ξ i
≤ tL
+ uo (ξ) − 2 tL
+ uo (ξ)
t
t
i
h x − hν − ξ + uo (ξ)
+ tL
t
x − ξ i
h 1 x + hν − ξ 1 x − hν − ξ + L
−L
= 2t L
2
t
2
t
t
|2hν/t|2
h2
≤ 2t
=
.
8co
tco
18 Uniqueness of Weakly Semi-Concave Variational
Hopf Solutions
Introduce the class C2 of solutions

 u is a variational solution of (8.3)
C2 = u(·, t) is weakly semi-concave in the sense of (16.1)

|∇u| ≤ C for some C > 0 in RN × R+ .
(18.1)
Theorem 18.1. The Cauchy problem has at most one solution within the
class C2 .
Proof. Let u1 and u2 be two solutions in C2 and set w = u1 − u2 . Proceeding
as in the proof of Proposition 13.1, we arrive at an analogue of (13.2), which
in this context holds a.e. in RN × R+ . From the latter, we derive
f (w)t = −V · Dx f (w)
a.e. in RN × R+
for any f ∈ C 1 (R), where V is defined in (13.3). For ε > 0, let uε,1 and uε,2
be the mollifications of u1 and u2 as in Section 16, and set
Z 1
Vε =
Dp H(sDx uε,2 + (1 − s)Dx uε,1 )ds.
0
With this notation
f (w)t = − div(Vε f (w)) + f (w) div Vε + (Vε − V) · Dx f (w).
Lemma 18.1 There exists a positive constant γ̄, independent of ε, such that
1
in RN × R+ .
div Vε ≤ γ̄ 1 +
t
310
8 NONLINEAR EQUATIONS OF FIRST ORDER
Proof. Same as in Lemma 15.1 with the proper minor modifications.
Putting this in the previous expression of f (w)t , and assuming that f (·) is
non-negative, gives
1
f (w) + (Vε − V) · Dx f (w).
(18.2)
f (w)t ≤ − div(Vε f (w)) + γ̄ 1 +
t
Introduce the backward characteristic cone CM and the truncated backward
t
characteristic cone CM
as in (13.6) and (13.7), where the constant M is chosen
as in (13.9). For a fixed 0 < σ < t < T , introduce also the truncated cone
t,σ
CM
= |x − xo | < M (T − τ ); 0 < σ < τ < t < T .
Now integrate (18.2) over such a cone. Proceeding as in the proof of Proposition 13.1, and taking into account the choice (13.9) of M , yields
Z
Z
f (w(x, σ))dx
f (w(x, t))dx ≤
|x−xo |<M(T −t)
+ γ̄
+
Z
Z t
σ
tZ
σ
1
1+
t
|x−xo |<M(T −σ)
Z
f (w(x, τ ))dxdτ
(18.3)
|x−xo |<M(T −τ )
|x−xo |<M(T −τ )
(Vε − V) · Dx f (w)dxdτ.
By the properties of the class C2 , |Dx w|, |Dx u1 |, and |Dx u2 | are a.e. bounded
in RN × R+ , independent of ε. Therefore
Z tZ
lim
(Vε − V) · Dx wdxdτ = 0.
ε→0
σ
|x−xo |<M(T −τ )
Observe first that (18.3) continues to hold for non-negative functions f (·),
uniformly Lipschitz continuous in R. Choose fδ (w) = (|w| − δ)+ , for some
fixed δ ∈ (0, 1). There exists σ > 0 such that fδ (w(·, τ )) = 0 for all τ ∈ (0, σ].
Indeed, by virtue of (10.3), for all τ ∈ (0, σ]
(|w(x, τ )| − δ)+ ≤ (|u1 (x, τ ) − uo (x)| + |u2 (x, τ ) − uo (x)| − δ)+
≤ (2C̄σ − δ)+ = 0
provided σ < δ/2C̄. These remarks in (18.3) yield
Z
fδ (w(x, t))dx
|x−xo |<M(T −t)
Z t
Z
1
≤ γ̄
1+
fδ (w(x, τ ))dxdτ
t
σ
|x−xo |<M(T −τ )
Z Z
1 t
fδ (w(x, τ ))dxdτ.
≤ γ̄ 1 +
σ σ |x−xo |<M(T −τ )
18 Uniqueness of Weakly Semi-Concave Variational Hopf Solutions
Setting
ϕδ (t) =
Z tZ
σ
311
fδ (w(x, τ ))dxdτ
|x−xo |<M(T −τ )
the previous inequality reads as
1
ϕ′δ (t) ≤ γ̄ 1 +
ϕδ (t)
σ
and ϕδ (σ) = 0.
This implies that ϕδ (τ ) = 0 for all τ ∈ (0, t), and since (xo , T ) is arbitrary,
|w| ≤ δ in RN × R+ , for all δ > 0.
9
LINEAR ELLIPTIC EQUATIONS WITH
MEASURABLE COEFFICIENTS
1 Weak Formulations and Weak Derivatives
Let E be a bounded domain in RN with boundary ∂E of class C 1 . Denote by
(aij ) an N × N symmetric matrix with entries aij ∈ L∞ (E), and satisfying
the ellipticity condition
λ|ξ|2 ≤ aij (x)ξi ξj ≤ Λ|ξ|2
(1.1)
for all ξ ∈ RN and all x ∈ E, for some 0 < λ ≤ Λ. The number Λ is the
least upper bound of the eigenvalues of (aij ) in E, and λ is their greatest
lower bound. A vector-valued function f = (f1 , . . . , fN ) : E → RN is said to
be in Lploc (E), for some p ≥ 1, if all the components fj ∈ Lploc (E). Given a
scalar function f ∈ L1loc (E) and a vector-valued function f ∈ L1loc (E), consider
the formal partial differential equation in divergence form (Section 3.1 of the
Preliminaries)
− aij uxi xj = div f − f in E.
(1.2)
Expanding formally the indicated derivatives gives a PDE of the type of (3.1)
of Chapter 1, which, in view of the ellipticity condition (1.1), does not admit
real characteristic surfaces (Section 3 of Chapter 1). In this formal sense, (1.2)
is a second-order elliptic equation.
Multiply (1.2) formally by a function v ∈ Co∞ (E) and formally integrate
by parts in E to obtain
Z
aij uxi vxj + fj vxj + f v dx = 0.
(1.3)
E
This is well defined for all v ∈ Co∞ (E), provided ∇u ∈ Lploc (E), for some p ≥ 1.
In such a case (1.3) is the weak formulation of (1.2), and u is a weak solution.
Such a weak notion of solution coincides with the classical one whenever the
various terms in (1.3) are sufficiently regular. Indeed, assume that f ∈ C(E)
and aij , f ∈ C 1 (E); if a function u ∈ C 2 (E) satisfies (1.3) for all v ∈ Co∞ (E),
integrating by parts gives
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_10
313
314
9 LINEAR ELLIPTIC EQUATIONS
Z
E
aij uxi
xj
+ div f + f vdx = 0
for all v ∈ Co∞ (E).
Thus u satisfies (1.2) in the classical sense. It remains to clarify the meaning
p
of ∇u ∈ Lloc
(E) for some p ≥ 1.
1.1 Weak Derivatives
p
(E), for some p ≥ 1, has a weak partial derivative in
A function u ∈ Lloc
p
p
(E)
Lloc (E) with respect to the variable xj if there exists a function wj ∈ Lloc
such that
Z
Z
(1.4)
uvxj dx = −
wj v dx for all v ∈ Co∞ (E).
E
E
If u ∈ C 1 (E), then wj = uxj in the classical sense. There are functions
admitting weak and not classical derivatives. As an example, u(x) = |x| for
x ∈ (−1, 1) does not have a derivative at x = 0; however it admits the weak
derivative
−1 in (−1, 0)
w=
as an element of L1 (−1, 1).
1 in (0, 1)
With a perhaps improper but suggestive symbolism we set wj = uxj , warning
that in general, uxj need not be the limit of difference quotients along xj , and
it is meant only in the sense of (1.4). The derivatives uxj in (1.3) are meant in
this weak sense, and solutions of (1.1) are sought as functions in the Sobolev
space ([241])
W 1,p (E) = {the set of u ∈ Lp (E) such that ∇u ∈ Lp (E)} .
(1.5)
A norm in W 1,p (E) is
kuk1,p = kukp + k∇ukp .
(1.6)
Proposition 1.1 (Meyers and Serrin [182]) W 1,p (E) is a Banach space
for the norm (1.6). Moreover, C ∞ (E) is dense in W 1,p (E).
Introduce also the two spaces
Wo1,p (E) = {the closure of Co∞ (E) in the norm (1.6)} .
Z
o
n
u dx = 0 .
W̃ 1,p (E) = all u ∈ W 1,p (E) such that
(1.7)
(1.8)
E
Proposition 1.2 Wo1,p (E) and W̃ 1,p (E) equipped with the norm (1.6) are
Banach spaces.
Functions in W 1,p (E) are more “regular” than merely elements in Lp (E), on
several accounts. First, they are embedded in Lq (E) for some q > p. Second,
they form a compact subset of Lp (E). Third, they have boundary values
(traces) on ∂E, as elements of Lp (∂E).
2 Embeddings of W 1,p (E)
315
2 Embeddings of W 1,p (E)
Since ∂E is of class C 1 , there is a circular spherical cone C of height h and solid
angle ω such that by putting its vertex at any point of ∂E, it can be properly
swung, by a rigid rotation, to remain in E. This is the cone condition of ∂E.
Denote by γ = γ(N, p) a constant depending on N and p and independent of
E and ∂E.
Theorem 2.1 (Sobolev–Nikol’skii [242]). If 1 < p < N then W 1,p (E) ֒→
∗
p
Lp (E), where p∗ = NN−p
, and there exists γ = γ(N, p), such that
kukp∗ ≤
o
γ n1
kukp + k∇ukp ,
ω h
p∗ =
Np
, for all u ∈ W 1,p (E). (2.1)
N −p
If p = 1 and |E| < ∞, then W 1,p (E) ֒→ Lq (E) for all 1 ≤ q < NN−1 and there
exists γ = γ(N, q) such that
o
n1
N −1
1
γ
for all u ∈ W 1,1 (E).
(2.2)
kukq ≤ |E| q − N
kuk1 + k∇uk1
ω
h
If p > N then W 1,p (E) ֒→ L∞ (E), and there exists γ = γ(N, p) such that
kuk∞ ≤
γ
(kukp + hk∇ukp )
ωhN/p
for all u ∈ W 1,p (E).
(2.3)
N
If p > N and in addition E is convex, then W 1,p (E) ֒→ C 1− p (Ē), and there
exists γ = γ(N, p) such that for every pair of points x, y ∈ Ē with |x − y| ≤ h
([184])
|u(x) − u(y)| ≤
N
γ
|x − y|1− p k∇ukp
ω
for all u ∈ W 1,p (E).
(2.4)
Remark 2.1 The constants γ(N, p) can be computed explicitly, and they
tend to infinity as p → N . This is expected as W 1,N (E) is not embedded in
L∞ (E). Indeed the function
[|x| < e−1 ] − {0} ∋ x → ln | ln |x|| ∈ W 1,N (|x| < e−1 )
is not essentially bounded about the origin. In this sense these embeddings
are sharp ([251]). In (2.2) the value q = 1∗ = NN−1 is not permitted, and the
corresponding constant γ(N, q) → ∞ as q → 1∗ . The limiting embedding for
p = N takes a special form ([50], Chapter IX, Section 13).
Remark 2.2 The structure of ∂E enters only through the solid angle ω and
the height h of the cone condition of ∂E. Therefore (2.1)–(2.4) continue to
hold for domains whose boundaries merely satisfy the cone condition.
Remark 2.3 If E is not convex, the estimate (2.4) can be applied locally.
Thus if p > N , a function u ∈ W 1,p (E) is locally Hölder continuous in E.
A proof of these embeddings is in Section 2c of the Complements.
316
9 LINEAR ELLIPTIC EQUATIONS
2.1 Compact Embeddings of W 1,p (E)
Theorem 2.2 (Rellich–Kondrachov [215, 139]). Let 1 ≤ p < N . Then
for all 1 ≤ q < p∗ , the embedding W 1,p (E) ֒→ Lq (E) is compact.
A proof is in Section 2.2c of the Complements.
Corollary 2.1 Let {un } be a bounded sequence in W 1,p (E). If 1 ≤ p < N ,
p
for each fixed 1 ≤ q < NN−p
there exist a subsequence {un′ } ⊂ {un }, and
1,p
u ∈ W (E) such that
{un′ } → u weakly in W 1,p (E)
{un′ } → u strongly in Lq (E).
and
3 Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E)
Theorem 3.1 (Gagliardo–Nirenberg [90, 193]). If 1 ≤ p < N , then
∗
p
Wo1,p (E) ֒→ Lp (E) for p∗ = NN−p
, and there exists γ = γ(N, p) such that
p∗ =
kukp∗ ≤ γk∇ukp
Np
N −p
for all u ∈ Wo1,p (E).
(3.1)
If p = N , then Wo1,p (E) ֒→ Lq (E) for all q > p, and there exists γ = γ(N, q)
such that
p
1− p
for all u ∈ Wo1,p (E).
(3.2)
kukq ≤ γk∇ukp q kukpq
If p > N , then Wo1,p (E) ֒→ L∞ (E), and there exists γ = γ(N, p) such that
N
1− N
p
kuk∞ ≤ γk∇ukpp kukp
for all u ∈ Wo1,p (E).
(3.3)
Remark 3.1 The constants γ(N, p) can be computed explicitly independent
of ∂E, and they tend to infinity as p → N . Unlike (2.2), the value 1∗ is
permitted in (3.1).
Functions in Wo1,p (E) are limits of functions in Co∞ (E) in the norm of
and in this sense they vanish on ∂E. This permits embedding inequalities such as (3.1)–(2.3) with constants γ independent of E and ∂E.
Inequalities of this kind would not be possible for functions u ∈ W 1,p (E).
For example, a constant nonzero function would not satisfy any of them. This
suggest that for them to hold some information is required on some values of
u. Let
Z
1
u dx
uE =
|E| E
Wo1,p (E),
denote the integral average of u over E. The multiplicative embeddings (3.1)–
(3.3) continue to hold for functions of zero average. Denote by γ = γ(N, E, ∂E)
a constant depending on N , |E|, and the C 1 –smoothness of ∂E, but invariant
under homothetic transformations of E.
3 Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E)
317
Theorem 3.2 (Golovkin–Poincarè [107]). If 1 < p < N , then W 1,p (E) ֒→
∗
p
Lp (E), where p∗ = NN−p
, and there exists γ = γ(N, p, E, ∂E) such that
p∗ =
ku − uE kp∗ ≤ γk∇ukp ,
Np
N −p
for all u ∈ W 1,p (E).
(3.4)
If p > N , then W 1,p (E) ֒→ L∞ (E), and there exists γ = γ(N, p) such that
N
1− N
p
ku − uE k∞ ≤ γk∇ukpp ku − uE kp
for all u ∈ W 1,p (E).
(3.5)
If p = N , then W 1,p (E) ֒→ Lq (E) for all q > p, and there exists γ = γ(N, q)
such that
1− p
q
ku − uE kq ≤ γk∇ukp
p
ku − uE kpq
for all u ∈ W 1,p (E).
(3.6)
Remark 3.2 When E is convex, a simple proof of Theorem 3.2 is due to
Poincarè and it is reported in Section 3.2c of the Complements. In such a case
the constant γ(N, E, ∂E) in (3.4) has the form
γ(N, E, ∂E) = C
(diam E)N
|E|
for some absolute constants C > 1 depending only on N .
3.1 Some Consequences of the Multiplicative Embedding
Inequalities
Corollary 3.1 The norm k·k1,p in Wo1,p (E), introduced in (1.6), is equivalent
to k∇ukp ; that is, there exists a positive constant γo = γo (N, p, E) such that
γo kuk1,p ≤ k∇ukp ≤ kuk1,p
for all u ∈ Wo1,p (E).
(3.7)
Corollary 3.2 The norm k · k1,p in W̃ 1,p (E), is equivalent to k∇ukp ; that is,
there exists a positive constant γ̃o = γ̃o (N, p, E) such that
γ̃o kuk1,p ≤ k∇ukp ≤ kuk1,p
for all u ∈ W̃ 1,p (E).
(3.8)
Corollary 3.3 Wo1,2 (E) and W̃ 1,2 (E) are Hilbert spaces with equivalent inner
products
hu, vi + h∇u, ∇vi and h∇u, ∇vi
(3.9)
where h·, ·i denotes the standard inner product in L2 (E).
318
9 LINEAR ELLIPTIC EQUATIONS
4 The Homogeneous Dirichlet Problem
Given f , f ∈ L∞ (E), consider the homogeneous Dirichlet problem
− aij uxi xj = div f − f
in E
on ∂E.
=0
u
(4.1)
∂E
The PDE is meant in the weak sense (1.3) by requiring that u ∈ W 1,p (E) for
some p ≥ 1. The homogeneous boundary datum is enforced, in a weak form,
by requiring that u be in the space Wo1,p (E) defined in (1.7). Seeking solutions
u ∈ Wo1,p (E) implies that in (1.3), by density, one may take v = u. Thus, by
taking into account the ellipticity condition (1.1)
Z
Z
λ
|∇u|2 dx ≤
(|f ||∇u| + |f ||u|)dx.
E
This forces p = 2 and identifies
of (4.1) should be sought.
E
Wo1,2 (E)
as the natural space where solutions
Theorem 4.1. The homogeneous Dirichlet problem (4.1) admits at most one
weak solution u ∈ Wo1,2 (E).
Proof. If u1 , u2 ∈ Wo1,2 (E) are weak solutions of (4.1)
Z
aij (u1 − u2 )xi vxj dx = 0
for all v ∈ Wo1,2 (E).
E
This and the ellipticity condition (1.1) imply k∇(u1 −u2 )k2 = 0. Thus u1 = u2
a.e. in E, by the embedding of Theorem 3.1.
5 Solving the Homogeneous Dirichlet Problem (4.1) by
the Riesz Representation Theorem
Regard (1.3) as made out of two pieces
Z
Z
aij uxi vxj dx and ℓ(v) = − (fj vxj + f v)dx
a(u, v) =
(5.1)
E
E
for all v ∈ Wo1,2 (E). Finding a solution to (4.1) amounts to finding u ∈
Wo1,2 (E) such that
a(u, v) = ℓ(v)
for all v ∈ Wo1,2 (E).
(5.2)
The first term in (5.1) is a bilinear form in Wo1,2 (E). By the ellipticity condition (1.1) and Corollary 3.1
6 Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods
319
2
2
≤ λk∇uk22 ≤ a(u, u) ≤ Λk∇uk22 ≤ Λkuk1,2
λγo2 kuk1,2
.
Therefore a(·, ·) is an inner product in Wo1,2 (E) equivalent to any one of the
inner products in (3.9). The second term ℓ(·) in (5.1) is a linear functional
in Wo1,2 (E), bounded in k · k1,2 , and thus bounded in the norm generated by
the inner product a(·, ·). Therefore by the Riesz representation theorem it is
represented as in (5.2) for a unique u ∈ Wo1,2 (E) ([162]).1
6 Solving the Homogeneous Dirichlet Problem (4.1) by
Variational Methods
Consider the nonlinear functional
def
Wo1,2 (E) ∋ u → J(u) =
Z
E
1
2 aij uxi uxj
+ fj uxj + f u dx.
(6.1)
One verifies that J(·) is strictly convex in Wo1,2 (E), that is
J tu + (1 − t)v < tJ(u) + (1 − t)J(v)
for every pair (u, v) of nontrivial elements of Wo1,2 (E), and all t ∈ (0, 1).
Assume momentarily N > 2, and let 2∗∗ be the Hölder conjugate of 2∗ , so
that
2N
1
1
2N
,
2∗∗ =
,
and
+ ∗∗ = 1.
2∗ =
∗
N −2
N +2
2
2
The functional J(u) is estimated above using the ellipticity condition (1.1),
Hölder’s inequality, and the embedding (3.1):
1
Λk∇uk22 + kf k2 k∇uk2 + kf k2∗∗ kuk2∗
2
1
≤ Λk∇uk22 + (kf k2 + γkf k2∗∗ ) k∇uk2
2
1
(kf k2 + γkf k2∗∗ )2 .
≤ Λk∇uk22 +
2Λ
J(u) ≤
Similarly, J(u) is estimated below by
J(u) ≥
1
1
2
λk∇uk22 − (kf k2 + γkf k2∗∗ ) .
4
λ
Therefore
−Fo + 14 λk∇uk22 ≤ J(u) ≤ Λk∇uk22 + Fo
(6.2)
for all u ∈ Wo1,2 (E), where
1
The Riesz representation theorem is in [50], Chapter VI Section 18. This solution
method is in [162] and it is referred to as the Lax–Milgram theorem.
320
9 LINEAR ELLIPTIC EQUATIONS
1
2
(kf k2 + γkf k2∗∗ ) .
λ
Here γ is the constant appearing in the embedding inequality (3.1). These
estimates imply that the convex functional J(·) is bounded below, and we
denote by Jo its infimum. A minimum can be sought by a minimizing sequence
{un } ⊂ Wo1,2 (E) such that
Fo =
J(un ) < J(un−1 ) < · · · < J(u1 )
and
lim J(un ) = Jo .
(6.3)
Wo1,2 (E).
By (6.2) and (6.3), the sequence {un } is bounded in
Therefore a
subsequence {un′ } ⊂ {un } can be selected such that {un′ } → u weakly in
Wo1,2 (E), and {J(un′ )} → Jo . Since the norm a(·, ·) introduced in (5.1) is
weakly lower semi-continuous
lim inf J(un′ ) = Jo ≥ J(u).
Thus J(u) = Jo and J(u) ≤ J(w) for all w ∈ Wo1,2 (E). Enforcing this last
condition for functions w = u + εv, for ε > 0 and v ∈ Co∞ (E), gives
Z
Z
ε2
J(u) ≤ J(u) +
aij uxi vxj + fj vxj + f v dx.
aij vxi vxj dx + ε
2 E
E
Divide by ε and let ε → 0 to get
Z
aij uxi vxj + fj vxj + f v dx ≥ 0
E
for all v ∈ Co∞ (E).
Changing v into −v shows that this inequality actually holds with equality
for all v ∈ Co∞ (E) and establishes that u is a weak solution of the Dirichlet
problem (4.1). In view of its uniqueness, as stated in Theorem 4.1 the whole
minimizing sequence {un } converges weakly to the unique minimizer of J(·).
Summarizing, the PDE in divergence form (4.1) is associated with a natural
functional J(·) in Wo1,2 (E), whose minimum is a solutions of the PDE in
Wo1,2 (E). Conversely, the functional J(·) in Wo1,2 (E) generates naturally a
PDE in divergence form whose solutions are the solutions of the homogeneous
Dirichlet problem (4.1).
6.1 The Case N = 2
The arguments are the same except for a different use of the embedding inequality (3.1) in estimating the term containing f u in (6.1). Pick 1 < p < 2,
define p∗ as in (3.1), and let p∗∗ be its Hölder conjugate, so that
p∗ =
Np
,
N −p
p∗∗ =
Np
,
N (p − 1) + p
1
1
+ ∗∗ = 1.
∗
p
p
One verifies that p∗∗ > 1 and estimates
Z
2−p
f u dx ≤ kf kp∗∗ kukp∗ ≤ γkf kp∗∗ k∇ukp ≤ γ|E| 2 kf kp∗∗ k∇uk2 .
E
The proof now proceeds as before except that the term γkf k2∗∗ is now replaced
by γ|E|2−p 2kf kp∗∗ .
7 Solving the Problem (4.1) by Galerkin Approximations
321
6.2 Gâteaux Derivative and The Euler Equation of J(·)
If u is a minimum for J(·) the function of one variable t → J(u + tv), for an
arbitrary but fixed v ∈ Co∞ (E), has a minimum for t = 0. Therefore
Z
d
0 = J(u + tv) t=0 =
aij uxi vxj + fj vxj + f v dx.
(6.4)
dt
E
This reinforces that solutions of the Dirichlet problem (4.1) are minima of
J(·) and vice versa. The procedure leading to (6.4), which is called the Euler
equation of J(·), has a broader scope. It can be used to connect stationary
points, not necessarily minima, of a functional J(·), for which no convexity
information is available, to solutions of its Euler equation.
The derivative of ε → J(u + εϕ) at ε = 0 is called the Gâteaux derivative
of J(·) at u. Relative variations from J(u) to J(v) are computed along the
“line” in R × Wo1,2 (E) originating at u and “slope” ϕ. In this sense Gâteaux
derivatives are directional derivatives. A more general notion of derivative of
J(·) is that of Fréchet derivative, where relative variations from J(u) to J(v)
are computed for any v however varying in a Wo1,2 (E)-neighborhood of u ([9],
Chapter I, Section 1,2).
7 Solving the Homogeneous Dirichlet Problem (4.1) by
Galerkin Approximations
Let {wn } be a countable, complete, orthonormal system for Wo1,2 (E). Such a
system exists, since Wo1,2 (E) is separable, and it can be constructed, for example, by the Gram–Schmidt procedure, for the natural inner product a(u, v)
introduced in (5.1). Thus in particular
Z
Ahk =
aij wh,xi wk,xj dx = δhk
(7.1)
E
where δhk is thePKronecker delta. Every v ∈ Wo1,2 (E) admits the representation as v =
vk wk for constants vk . The Galerkin method consists in
constructing the solution u of (4.1) as the weak Wo1,2 (E) limit, as n → ∞, of
the finite-dimensional approximations
un =
n
P
cn,h wh
h=1
where the coefficients cn,h are computed by enforcing an n-dimensional version
of the PDE in its weak form (1.3). Precisely, un is sought as the solution of
Z n
n
n
P
P
P
vk wk dx = 0
vk wk,xj + f
(7.2)
vk wk,xj + fj
aij un,xi
E
k=1
k=1
k=1
322
9 LINEAR ELLIPTIC EQUATIONS
for all v ∈ Wo1,2 (E). From this
Z
Z
nP
n
n
P
cn,h
aij wh,xi wk,xj dx +
vk
k=1
h=1
E
E
Setting
Ahk =
Z
aij wh,xi wk,xj dx,
E
φk = −
o
fj wk,xj + f wk dx = 0
Z
E
fj wk,xj + f wk dx
and taking into account that v ∈ Wo1,2 (E) is arbitrary, the coefficients cn,h
are computed as the unique solution of the linear algebraic system
n
P
cn,h Ahk = φk
for k = 1, . . . , n.
(7.3)
h=1
By (7.1), Ahk is the identity matrix, and therefore cn,h = φh for all n, h ∈ N.
With un so determined, put v = un in (7.2), and assuming momentarily that
N > 2, estimate
λk∇un k22 ≤ kf k2 k∇un k2 + kf k2∗∗ kun k2∗ ≤ (kf k2 + γkf k2∗∗ ) k∇un k2
where γ is the constant of the embedding inequality (3.1). Therefore {un }
is bounded in Wo1,2 (E), and a subsequence {un′ } ⊂ {un } can be selected,
converging weakly to some u ∈ Wo1,2 (E). Letting n → ∞ in (7.2) shows that
such a u is a solution of the homogeneous Dirichlet problem (4.1). In view of
its uniqueness, the whole sequence {un } of finite-dimensional approximations
converges weakly to u. The same arguments hold true for N = 2, by the minor
modifications indicated in Section 6.1.
7.1 On the Selection of an Orthonormal System in Wo1,2 (E)
The selection of the complete system {wn } is arbitrary. For example the
Gram–Schmidt procedure could be carried on starting from a countable collection of linearly independent elements of Wo1,2 (E), and using anyone of the
equivalent inner products in (3.9), or the construction could be completely
independent of Gram–Schmidt procedure. The Galerkin method continues to
hold, except that the coefficients Ahk defined in (7.1) are no longer identified
by the Kronecker symbol. This leads to the determination of the coefficients
cn,h as solutions of the linear algebraic system (7.3), whose leading n × n matrix (Ahk ) is still invertible, for all n, because of the ellipticity condition (1.1).
The corresponding unique solutions might depend on the nth approximating
truncation, and therefore are labeled by cn,h .
While the method is simple and elegant, it hinges on a suitable choice
of complete system in Wo1,2 (E). Such a choice is suggested by the specific
geometry of E and the structure of the matrix (aij ) ([34]).
8 Traces on ∂E of Functions in W 1,p (E)
323
7.2 Conditions on f and f for the Solvability of the Dirichlet
Problem (4.1)
Revisiting the proofs of these solvability methods shows that the only required
conditions on f and f are
if N > 2
q = 2∗∗
2
q
f ∈ L (E)
(7.4)
f ∈ L (E) where
any q > 1 if N = 2.
Theorem 7.1. Let (7.4) hold. Then the homogeneous Dirichlet problem (4.1)
admits a unique solution.
8 Traces on ∂E of Functions in W 1,p(E)
8.1 The Segment Property
The boundary ∂E has the segment property if there exist a locally finite open
covering of ∂E with balls {Bt (xj )} centered at xj ∈ ∂E with radius t, a
corresponding sequence of unit vectors nj , and a number t∗ ∈ (0, 1) such that
x ∈ Ē ∩ Bt (xj )
=⇒
x + tnj ∈ E
for all t ∈ (0, t∗ ).
(8.1)
Such a requirement forces, in some sense, the domain E to lie locally on one
side of its boundary. However no smoothness is required on ∂E. As an example
for x ∈ R let
(p
1
|x| sin2
for |x| > 0
h(x) =
and E = [y > h].
(8.2)
x
0
for x = 0.
The set E satisfies the segment property. The cone property does not imply the
segment property. For example, the unit disc from which a radius is removed
satisfies the cone property and does not satisfy the segment property. The
segment property does not imply the cone property. For example the set in
(8.2), does not satisfy the cone property. The segment property does not imply
that ∂E is of class C 1 . Conversely, ∂E of class C 1 does not imply the segment
property.
A remarkable fact about domains with the segment property is that functions in W 1,p (E) can be extended “outside” E to be in W 1,p (E ′ ), for a larger
open set E ′ containing E. A consequence of such an extension is that functions in W 1,p (E) can be approximated in the norm (1.6) by functions smooth
up to ∂E. Precisely
Proposition 8.1 Let E be a bounded open set in RN with boundary ∂E of
class C 1 and with the segment property. Then Co∞ (RN ) is dense in W 1,p (E).
Proof. Section 8.1c of the Complements.
324
9 LINEAR ELLIPTIC EQUATIONS
8.2 Defining Traces
Denote by γ = γ(N, p, ∂E) a constant that can be quantitatively determined
a priori in terms of N , p, and the structure of ∂E only.
Proposition 8.2 Let ∂E be of class C 1 and satisfy the segment property. If
1 ≤ p < N , there exists γ = γ(N, p, ∂E) such that for all ε > 0
1
kukp∗ N −1 ;∂E ≤ εk∇ukp + γ 1 +
kukp
N
ε
for all u ∈ Co∞ (RN ).
(8.3)
If p = N , then for all q ≥ 1, there exists γ = γ(N, q, ∂E) such that for all
ε>0
1
kukq;∂E ≤ εk∇ukp + γ 1 +
kukp for all u ∈ Co∞ (RN ).
(8.4)
ε
If p > N , there exists γ = γ(N, p, ∂E) such that for all ε > 0
1
kukp
kuk∞,∂E ≤ εk∇ukp + γ 1 +
ε
N
|u(x) − u(y)| ≤ γ|x − y|1− p kuk1,p for all x, y ∈ Ē.
(8.5)
Proof. Section 8.2c of the Complements.
Remark 8.1 The constants γ in (8.3) and (8.5) tend to infinity as p → N ,
and the constant γ in (8.4) tends to infinity as q → ∞.
Since u ∈ Co∞ (RN ), the values of u on ∂E are meant in the classical sense. By
Proposition 8.1, given u ∈ W 1,p (E) there exists a sequence {un } ⊂ Co∞ (RN )
such that {un } → u in W 1,p (E). In particular, {un } is Cauchy in W 1,p (E)
and from (8.3) and (8.4)
kun − um kp∗ N −1 ;∂E ≤ γkun − um k1,p
N
kun − um kq;∂E ≤ γkun − um k1,p
if 1 ≤ p < N
for fixed q ≥ 1 if p ≥ N > 1.
By the completeness of the spaces Lp (∂E), for p ≥ 1
∗ N −1
Lp N (∂E) if 1 ≤ p < N
{un ∂E } → tr(u) in
Lq (∂E)
for fixed q ≥ 1 if p ≥ N > 1.
(8.6)
One verifies that tr(u) is independent of the particular sequence {un }. Therefore given u ∈ W 1,p (E), this limiting process identifies its “boundary values”
tr(u), called the trace of u on ∂E, as an element of Lr (E), with r specified by (8.6). With perhaps an improper but suggestive symbolism we write
tr(u) = u|∂E .
9 The Inhomogeneous Dirichlet Problem
325
8.3 Characterizing the Traces on ∂E of Functions in W 1,p (E)
The trace of a function in W 1,p (E) is somewhat more regular that merely an
element in Lp (∂E) for some p ≥ 1. For v ∈ C ∞ (∂E) and s ∈ (0, 1), set
Z Z
|v(x) − v(y)|p
k|v|kps,p;∂E =
dσ(x)dσ(y) < ∞
(8.7)
(N −1)+sp
∂E ∂E |x − y|
where dσ(·) is the surface measure on ∂E. Denote by W s,p (∂E) the collections
of functions v in Lp (∂E) with finite norm
kvks,p;∂E = kvkp,∂E + k|v|ks,p;∂E .
(8.8)
The next theorem characterizes the traces of functions in W 1,p (E) in terms
of the spaces W s,p (∂E).
Theorem 8.1. Let ∂E be of class C 1 and satisfy the segment property. If
u ∈ W 1,p (E), then tr(u) ∈ W s,p (∂E), with s = 1 − p1 . Conversely, given
v ∈ W s,p (∂E), with s = 1 − p1 , there exists u ∈ W 1,p (E) such that tr(u) = v.
Proof. Section 8.3c of the Complements.
9 The Inhomogeneous Dirichlet Problem
Assume that ∂E is of class C 1 and satisfies the segment property. Given f
1
and f satisfying (7.4) and ϕ ∈ W 2 ,2 (∂E), consider the Dirichlet problem
− aij uxi x = div f − f
in E
j
(9.1)
on ∂E.
u
=ϕ
∂E
By Theorem 8.1 there exists v ∈ W 1,2 (E) such that tr(v) = ϕ. A solution of
(9.1) is sought of the form u = w + v, where w ∈ Wo1,2 (E) is the unique weak
solution of the auxiliary, homogeneous Dirichlet problem
− aij wxi xj = div f̃ − f in E
where f˜j = fj + aij vxi .
(9.2)
on ∂E
w ∂E = 0
Theorem 9.1. Assume that ∂E is of class C 1 and satisfies the segment prop1
erty. For every f and f satisfying (7.4) and ϕ ∈ W 2 ,2 (∂E), the Dirichlet
problem (9.1) has a unique weak solution u ∈ W 1,2 (E).
Remark 9.1 The class W 1,2 (E) where a weak solution is sought characterizes the boundary data ϕ on ∂E that ensure solvability.
326
9 LINEAR ELLIPTIC EQUATIONS
10 The Neumann Problem
Assume that ∂E is of class C 1 and satisfies the segment property. Given f
and f satisfying (7.4), consider the formal Neumann problem
− aij uxi xj = div f − f
in E
(10.1)
on ∂E
(aij uxi + fj ) nj = ψ
where n = (n1 , . . . , nN ) is the outward unit normal to ∂E and ψ ∈ Lp (∂E)
for some p ≥ 1. If aij = δij , f = f = 0, and ψ are sufficiently regular, this
is precisely the Neumann problem (1.3) of Chapter 2. Since aij ∈ L∞ (E)
and f ∈ L2 (E), neither the PDE nor the boundary condition in (10.1) are
well defined, and they have to be interpreted in some weak form. Multiply
formally the first of (10.1) by v ∈ Co∞ (RN ) and integrate by parts over E,
as if both the PDE and the boundary condition were satisfied in the classical
sense. This gives formally
Z
Z
aij uxi vxj + fj vxj + f v dx =
(10.2)
ψv dσ.
∂E
E
If v ∈ Co∞ (RN ) is constant in a neighborhood of E, this implies the necessary
condition of solvability2
Z
Z
ψ dσ.
f dx =
E
(10.3)
∂E
It turns out that this condition linking the data f and ψ is also sufficient
for the solvability of (10.1), provided a precise class for ψ is identified. By
Proposition 8.1, if (10.2) holds for all v ∈ Co∞ (RN ), it must hold for all
v ∈ W 1,2 (E), provided a solution u is sought in W 1,2 (E). In such a case, the
right-hand side is well defined if ψ is in the conjugate space of integrability
of the traces of functions in W 1,2 (E). Therefore the natural class for the
Neumann datum is

 q = 2(N − 1)

if N > 2
N
ψ ∈ Lq (∂E),
where
(10.4)


any q > 1
if N = 2.
Theorem 10.1. Let ∂E be of class C 1 and satisfy the segment property. Let
f and f satisfy (7.4) and ψ satisfy (10.4) and be linked by the compatibility
condition (10.3). Then the Neumann problem (10.1) admits a solution in the
weak form (10.2) for all v ∈ W 1,2 (E). The solution is unique up to a constant.
Proof. Consider the nonlinear functional in W 1,2 (E)
2
This is a version of the compatibility condition (1.4) of Chapter 2; see also
Theorem 6.1 of Chapter 3, and Section 6 of Chapter 4.
10 The Neumann Problem
def
J(u) =
Z
E
1
2 aij uxi uxj
+ fj uxj + f u dx −
Z
ψtr(u)dσ.
327
(10.5)
∂E
By the compatibility condition (10.3), J(u) = J(u − uE ), where uE is the
integral average of u over E. Therefore J(·) can be regarded as defined in the
space W̃ 1,2 (E) introduced in (1.8). One verifies that J(·) is strictly convex in
W̃ 1,2 (E). Assume momentarily that N > 2, let 2∗∗ be the Hölder conjugate
of 2∗ , and estimate
Z
fj uxj dx ≤ kf k2 k∇uk2
E
Z
f udx ≤ kf k2∗∗ kuk2∗ ≤ γkf k2∗∗ k∇uk2
E
where γ is the constant in the embedding inequality (3.4). Similarly, using the
trace inequality (8.3)
Z
ψtr(u)dσ ≤ kψkq;∂E ktr(u)k2∗ N −1 ;∂E
N
∂E
≤ γkψkq;∂E (kuk2 + k∇uk2 ) ≤ 2γ 2 kψkq;∂E k∇uk2
where γ is the largest of the constants in (3.4) and (8.3). Therefore
−F1 + 41 λk∇uk22 ≤ J(u) ≤ Λk∇uk22 + F1
(10.6)
for all u ∈ W̃ 1,2 (E), where
F1 =
2
1
kf k2 + γkf k2∗∗ + 2γ 2 kψkq;∂E .
λ
By Corollary 3.2, the kuk1,2 norm of W̃ 1,2 (E) is equivalent to k∇uk2 . With
these estimates in hand the proof can now be concluded by a minimization
process in W̃ 1,2 (E), similar to that of Section 6. The minimum u ∈ W̃ 1,2 (E)
satisfies (10.2), for all v ∈ W 1,2 (E) and the latter can be characterized as the
Euler equation of J(·).
Essentially the same arguments continue to hold for N = 2, modulo minor
variants that can be modeled after those in Section 6.1.
10.1 A Variant of (10.1)
The compatibility condition (10.3) has the role of estimating J(·) above and
below as in (10.6), via the multiplicative embeddings of Theorem 3.2. Consider
next the Neumann problem
− aij uxi xj + µu = div f − f
in E
(10.7)
on ∂E
(aij uxi + fj ) nj = ψ
328
9 LINEAR ELLIPTIC EQUATIONS
where µ > 0, f , f ∈ L2 (E), and ψ satisfies (10.4). The problem is meant in its
weak form
Z
Z
(10.8)
aij uxi vxj + fj vxj + µuv + f v dx =
ψv dσ
∂E
E
for all v ∈ W 1,2 (E). No compatibility conditions are needed on the data f
and ψ for a solution to exist, and in addition, the solution is unique.
Theorem 10.2. Let ∂E be of class C 1 and satisfy the segment property. Let
f , f ∈ L2 (E), and let ψ satisfy (10.4). Then the Neumann problem (10.7) with
µ > 0 admits a unique solution in the weak form (10.8).
Proof. If u1 and u2 are two solutions in W 1,2 (E), their difference w satisfies
Z
aij wxi wxj + µw2 dx = 0.
E
The nonlinear functional in W 1,2 (E)
Z
Z
def
2
1
1
J(u) =
a
u
u
+
µu
+
f
u
+
f
u
dx
−
j xj
2 ij xi xj
2
E
ψtr(u)dσ
(10.9)
∂E
is strictly convex. Then a solution can be constructed by the variational
method of Section 6, modulus establishing an estimate analogous to (6.2)
or (10.6), with k∇uk2 replaced by the norm kuk1,2 of W 1,2 (E). Estimate
Z
Z
f udx ≤ kf k2 kuk2
fj uxj dx ≤ kf k2 k∇uk2 ,
E
E
Z
ψtr(u)dσ ≤ γkψkq;∂E (kuk2 + k∇uk2 )
∂E
where γ is the constant of the trace inequality (8.3). Then the functional J(·)
is estimated above and below by
−Fµ +
1
4
min{λ; µ}kuk21,2 ≤ J(u) ≤
3
2
max{Λ; µ}kuk21,2 + Fµ
(10.10)
for all u ∈ W 1,2 (E), where
Fµ =
1
2
(kf k2 + kf k2 + γkψkq;∂E ) .
min{λ; µ}
11 The Eigenvalue Problem
Consider the problem of finding a nontrivial pair (µ, u) with µ ∈ R and
u ∈ Wo1,2 (E), a solution of
12 Constructing The Eigenvalues of (11.1)
− aij uxi
xj
= µu
in E
on ∂E.
u=0
This is meant in the weak sense
Z
aij uxi vxj − µuv dx = 0
E
for all v ∈ Wo1,2 (E).
329
(11.1)
(11.2)
If (µ, u) is a solution pair, µ is an eigenvalue and u is an eigenfunction of (11.1).
In principle, the pair (µ, u) is sought for µ ∈ C, and for u in the complexvalued Hilbert space Wo1,2 (E), with complex inner product as in Section 1 of
Chapter 4. However by considerations analogous to those of Proposition 7.1 of
that Chapter, eigenvalues of (11.1) are real, and eigenfunctions can be taken
to be real-valued. Moreover, any two distinct eigenfunctions corresponding to
two distinct eigenvalues are orthogonal in L2 (E).
Proposition 11.1 Eigenvalues of (11.1) are positive. Moreover, to each
eigenvalue µ there correspond at most finitely many eigenfunctions, linearly
independent, and orthonormal in L2 (E).
Proof. If µ ≤ 0, the functional
Wo1,2 (E)
∋u→
Z
E
(aij uxi uxj − µu2 ) dx
(11.3)
is strictly convex and bounded below by λk∇uk22 . Therefore it has a unique
minimum, which is the unique solution of its Euler equation (11.1). Since
u = 0 is a solution, it is the only one. Let {un } be a sequence of eigenfunctions
linearly independent in L2 (E), corresponding to µ. Without loss of generality
p
we may assume they are orthonormal. Then from (11.2), k∇un k2 ≤ µ/λ for
all n, and {un } is equi-bounded in Wo1,2 (E). If {un } is infinite, a subsequence
can be selected, and relabeled with n, such that {un } → u weakly in Wo1,2 (E)
2
and strongly in L2 (E).
√ However, {un } cannot be a Cauchy sequence in L (E),
since kun − um k2 = 2 for all n, m.
Let {uµ,1 , . . . , uµ,nµ } be the linearly independent eigenfunctions corresponding
to the eigenvalue µ. The number nµ is the multiplicity of µ. If nµ = 1, then µ
is said to be simple.
12 Constructing The Eigenvalues of (11.1)
Minimize the strictly convex functional a(·, ·) on the unit sphere S1 of L2 (E),
that is
Z
a(u, u) = min
min
aij uxi uxj dx
1,2
1,2
u∈Wo (E)
kuk2 =1
u∈Wo (E)
kuk2 =1
E
330
9 LINEAR ELLIPTIC EQUATIONS
and let µ1 ≥ 0 be its minimum value. A minimizing sequence {un } ⊂ S1
is bounded in Wo1,2 (E), and a subsequence {un′ } ⊂ {un } can be selected
such that {un′ } → w1 weakly in Wo1,2 (E) and strongly in L2 (E). Therefore
w1 ∈ S1 , it is nontrivial, µ1 > 0, and
µ1 = lim a(un′ , un′ ) ≥ lim a(w1 , w1 ) ≥ µ1 .
Thus
µ1 ≤
a(w1 + v, w1 + v)
kw1 + vk22
for all v ∈ Wo1,2 (E).
(12.1)
(12.2)
It follows that the functional
Wo1,2 (E) ∋ v → Iµ1 (v) =
Z
aij w1,xi vxj − µ1 w1 v dx
E
Z
1
+
aij vxi vxj − µ1 v 2 dx
2 E
is non-negative, its minimum is zero, and the minimum is achieved for v = 0.
Thus
d
Iµ (tv) t=0 = 0 for all v ∈ Wo1,2 (E).
dt 1
The latter is precisely (11.2) for the pair (µ1 , w1 ). While this process identifies µ1 uniquely, the minimizer w1 ∈ S1 depends on the choice of subsequence
{un′ } ⊂ {un }. Thus a priori, to µ1 there might correspond several eigenfunctions in Wo1,2 (E) ∩ S1 .
If µn and the set of its linearly independent eigenfunctions have been
found, set
En = {span of the eigenfunctions of µn }
Wn1,2 (E)
= Wo1,2 (E) ∩ [E1 ∪ · · · ∪ En ]⊥
consider the minimization problem
min
1,2
u∈Wn (E)
kuk2 =1
a(u, u) =
min
1,2
u∈Wn (E)
kuk2 =1
Z
aij uxi uxj dx
E
and let µn+1 > µn be its minimum value. A minimizing sequence {un } ⊂ S1
is bounded in Wn1,2 (E), and a subsequence {un′ } ⊂ {un } can be selected
such that {un′ } → wn+1 weakly in Wn1,2 (E) and strongly in L2 (E). Therefore
wn+1 ∈ S1 is nontrivial, and
µn+1 = lim a(un′ , un′ ) ≥ lim a(wn+1 , wn+1 ) ≥ µn+1 .
Thus
µn+1 ≤
a(wn+1 + v, wn+1 + v)
kwn+1 + vk22
for all v ∈ Wn1,2 (E).
13 The Sequence of Eigenvalues and Eigenfunctions
331
It follows that the functional
Wn1,2 (E) ∋ v → Iµn+1 (v) =
Z
aij wn+1,xi vxj − µn+1 wn+1 v dx
E
Z
1
+
aij vxi vxj − µn+1 v 2 dx
2 E
is non-negative, its minimum is zero, and the minimum is achieved for v = 0.
Thus
d
Iµ (tv) t=0 = 0 for all v ∈ Wn1,2 (E).
dt n+1
This implies
Z
aij wn+1,xi vxj − µn+1 wn+1 v dx = 0
(12.3)
E
for all v ∈ Wn1,2 (E). The latter coincides with (11.2), except that the test
functions v are taken out of Wn1,2 (E) instead of the entire Wo1,2 (E). Any
v ∈ Wo1,2 (E) can be written as v = v ⊥ + vo , where v ⊥ ∈ Wn1,2 (E) and vo has
the form
kn
P
vo =
vj wj
j=1
where vj are constants, and wj are eigenfunctions of (11.1) corresponding to
eigenvalues µj , for j ≤ n. By construction
Z
aij wn+1,xi vo,xj − µn+1 wn+1 vo dx = 0.
E
Hence (11.2) holds for all v ∈ Wo1,2 (E), and (µn+1 , wn+1 ) is a nontrivial solution pair of (11.1). While this process identifies µn+1 uniquely, the minimizer
wn+1 ∈ S1 depends on the choice of subsequence {un′ } ⊂ {un }. Thus a priori,
to µn+1 there might correspond several eigenfunctions.
13 The Sequence of Eigenvalues and Eigenfunctions
This process generates a sequence of eigenvalues µn < µn+1 each with its own
multiplicity. The linearly independent eigenfunctions {wµn ,1 , . . . , wµn ,nµn }
corresponding to µn can be chosen to be orthonormal. The eigenfunctions
are relabeled with n to form an orthonormal sequence {wn }, and each is associated with its own eigenvalue, which in this reordering remains the same as
the index of the corresponding eigenfunctions ranges over its own multiplicity.
We then write
µ1 ≤ µ2 ≤ · · · ≤ µn ≤ · · ·
(13.1)
w1 w2 · · · wn · · ·
332
9 LINEAR ELLIPTIC EQUATIONS
Proposition 13.1 Let {µn } and {wn } be as in (13.1). Then {µn } → ∞ as
n → ∞. The orthonormal system {wn } is complete in L2 (E). The system
√
{ µn wn } is orthonormal and complete in Wo1,2 (E) with respect to the inner
product a(·, ·).
Proof. If {µn } → µ∞ < ∞, the sequence {wn } will be bounded in Wo1,2 (E),
and by compactness, a subsequence {wn′ } ⊂ {wn } can be selected such that
{wn′ } → w strongly in L2 (E). Since {wn } is orthonormal in L2 (E)
2=
lim
n′ ,m′ →∞
kwn′ − wm′ k2 → 0.
Let f ∈ L2 (E) be nonzero and orthogonal to the L2 (E)-closure of {wn }. Let
uf ∈ Wo1,2 (E) be the unique solution of the homogeneous Dirichlet problem
(4.1) with f = 0, for such a given f . Since f 6= 0, the solution uf 6= 0 can be
renormalized so that kuf k2 = 1. Then, for all n ∈ N
µn =
inf
1,2
u∈Wn (E)
kuk2 =1
a(u, u) ≤ a(uf , uf ) ≤ kf k2 .
√
It is apparent that { µn wn } is an orthogonal system in Wo1,2 (E) with respect
to the inner product a(·, ·). To establish its completeness in Wo1,2 (E) it suffices
to verify that a(wn , u) = 0 for all wn implies u = 0. This in turn follows from
the completeness of {wn } in L2 (E).
Proposition 13.2 µ1 is simple and w1 > 0 in E.
Proof. Let (µ1 , w) be a solution pair for (11.1) for the first eigenvalue µ1 .
Since w ∈ Wo1,2 (E), also w± ∈ Wo1,2 (E), and either of these can be taken as a
test function in the corresponding weak form (11.2) for the pair (µ1 , w). This
gives
a(w± , w± ) = µ1 kw± k22 .
Therefore in view of the minimum problem (12.2), the two functions w± are
both non-negative solutions of
− aij wx±i x = µ1 w±
weakly in E
j
(13.2)
±
on ∂E.
w =0
Lemma 13.1 The functions w± are Hölder continuous in E, and if w+ (xo ) >
0 (w− (xo ) > 0), for some xo ∈ E, then w+ > 0 (w− > 0) in E.
Assuming the lemma for the moment, either w+ ≡ 0 or w− ≡ 0 in E. Therefore, since w = w+ − w− , the eigenfunction w can be chosen to be strictly
positive in E. If v and w are two linearly independent eigenfunctions corresponding to µ1 , they can be selected to be both positive in E and thus cannot
be orthogonal. Thus v = γw for some γ ∈ R, and µ1 is simple.
The proof of Lemma 13.1 will follow from the Harnack Inequality of Section 8
of Chapter 11.
14 A Priori L∞ (E) Estimates for Solutions of the Dirichlet Problem (9.1)
333
14 A Priori L∞ (E) Estimates for Solutions of the
Dirichlet Problem (9.1)
A weak sub(super)-solution of the Dirichlet problem (9.1) is a function u ∈
W 1,2 (E), whose trace on ∂E satisfies tr(u) ≤ (≥)ϕ and such that
Z
(14.1)
aij uxi vxj + fj vxj + f v dx ≤ (≥)0
E
for all non-negative v ∈ Wo1,2 (E). A function u ∈ W 1,2 (E) is a weak solution
of the Dirichlet problem (9.1), if and only if is both a weak sub- and supersolution of that problem.
Proposition 14.1 Let u ∈ W 1,2 (E) be a weak sub-solution of (9.1) for N ≥
2. Assume
f ∈ LN +ε (E),
ϕ+ ∈ L∞ (∂E),
f+ ∈ L
N +ε
2
(E)
(14.2)+
for some ε > 0. Then u+ ∈ L∞ (E) and there exists a constant Cε that can
be determined a priori only in terms of λ, Λ, N , ε, and the constant γ in the
Sobolev embedding (3.1)–(3.2), such that
ess sup u+ ≤ max ess sup ϕ+ ; Cε kf kN +ε ; |E|δ kf+ k N +ε |E|δ
(14.3)+
E
2
∂E
where
δ=
ε
.
N (N + ε)
(14.4)
A similar statement holds for supersolutions. Precisely
Proposition 14.2 Let u ∈ W 1,2 (E) be a weak super-solution of (9.1) for
N ≥ 2. Assume
ϕ− ∈ L∞ (∂E),
f ∈ LN +ε (E),
f− ∈ L
N +ε
2
(E)
for some ε > 0. Then u− ∈ L∞ (E) and
ess sup u− ≤ max ess sup ϕ− ; Cε kf kN +ε ; |E|δ kf− k N +ε |E|δ
E
∂E
2
(14.2)−
(14.3)−
for the same constants Cε and δ.
Remark 14.1 The constant Cε in (14.3)± is “stable” as ε → ∞, in the sense
that if f and f± are in L∞ (E), then u± ∈ L∞ (E) and there exists a constant
C∞ depending on the indicated quantities except ε, such that
n
o
1
1
ess sup u± ≤ max ess sup ϕ± ; C∞ kf k∞ ; |E| N kf± k∞ |E| N .
(14.5)
E
∂E
334
9 LINEAR ELLIPTIC EQUATIONS
Remark 14.2 The constant Cε tends to infinity as ε → 0. Indeed, the propositions are false for ε = 0, as shown by the following example. For N > 2, the
two equations
N −2
1
−
2
2
|x| ln |x| |x| ln2 |x|
xj
where fj =
|x|2 ln |x|
where f =
∆u = f
∆u = fj,xj
are both solved, in a neighborhood E of the origin by u(x) = ln ln |x| . One
verifies that
N +ε
N
f ∈ L 2 (E)
f ∈ LN (E)
and f ∈
/ L 2 (E) for any ε > 0
and f ∈
/ LN +ε (E) for any ε > 0.
Remark 14.3 The propositions can be regarded as a weak form of the maximum principle (Section 4.1 of Chapter 2). Indeed, if u is a weak sub(super)solution of the Dirichlet problem (9.1), with f = f = 0, then u+ ≤ tr(u)+
(u− ≤ tr(u)− ).
15 Proof of Propositions 14.1–14.2
It suffices to establish Proposition 14.1. Let u ∈ W 1,2 (E) be a weak subsolution of the Dirichlet problem (9.1), in the sense of (14.1) for all nonnegative v ∈ Wo1,2 (E). Let k ≥ kϕ+ k∞,∂E to be chosen, and set
kn = k 2 −
2
1 ,
n−1
An = [u > kn ],
n = 1, 2, . . . .
(15.1)
Then (u−kn )+ ∈ Wo1,2 (E) for all n ∈ N, and it can be taken as a test function
in the weak formulation (14.1) to yield
Z
[aij uxi + fj ] (u − kn )+xj + f+ (u − kn )+ dx ≤ 0.
E
From this, estimate
λk∇(u −
kn )+ k22
≤ kf χAn k2 k∇(u − kn )+ k2 +
Z
f+ (u − kn )+ dx
Z
λ
1
≤ k∇(u − kn )+ k22 + kf χAn k22 +
f+ (u − kn )+ dx
4
λ
E
2
1
λ
≤ k∇(u − kn )+ k22 + kf k2N +ε |An |1− N +ε
4Z
λ
+
E
E
f+ (u − kn )+ dx.
The last term is estimated by Hölder’s inequality as
15 Proof of Propositions 14.1–14.2
Z
E
where
f+ (u − kn )+ dx ≤ kf+ k
p∗
N +ε
=
p∗ − 1
2
and
p∗
p∗ −1
335
k(u − kn )+ kp∗
p∗ =
Np
.
N −p
(15.2)
For these choices one verifies that 1 < p < 2 for all N ≥ 2. Therefore by (3.1)
of the embedding of Theorem 3.1
Z
f+ (u − kn )+ dx ≤ γkf+ k N +ε k∇(u − kn )+ kp
2
E
1
p
≤ γk∇(u − kn )+ k2 kf+ k N +ε |An | p (1− 2 )
2
2
λ
γ2
≤ k∇(u − kn )+ k22 + kf+ k2N +ε |An | p −1 .
2
4
λ
Combining these estimates gives
2
k∇(u − kn )+ k22 ≤ Co2 |An |1− N +2δ
(15.3)
where
2γ 2
max kf k2N +ε ; |E|2δ kf+ k2N +ε
2
2
λ
and δ is defined in (14.4).
Co2 =
15.1 An Auxiliary Lemma on Fast Geometric Convergence
Lemma 15.1 Let {Yn } for n = 1, 2, . . . , be a sequence of positive numbers
linked by the recursive inequalities
Yn+1 ≤ bn KYn1+σ
(15.4)
for some b > 1, K > 0, and σ > 0. If
2
Y1 ≤ b−1/σ K −1/σ
(15.5)
Then {Yn } → 0 as n → ∞.
Proof. By direct verification by applying (15.5) recursively.
15.2 Proof of Proposition 14.1 for N > 2
By the embedding inequality (3.1) for Wo1,2 (E) and (15.3)
Z
k2
(u − kn )2+ χ[u>kn+1 ] dx ≤ k(u − kn )+ k22
|A
|
≤
n+1
4n
E
2
2
≤ k(u − kn )+ k22∗ |An | N ≤ γ 2 k∇(u − kn )+ k22 |An | N
≤ γ 2 Co2 |An |1+2δ .
(15.6)
336
9 LINEAR ELLIPTIC EQUATIONS
From this
4n γ 2 Co2
|An |1+2δ
for all n ∈ N
(15.7)
k2
where γ is the constant of the embedding inequality (3.1). If {|An |} → 0 as
n → ∞, then u ≤ 2k a.e. in E. By Lemma 15.1, this occurs if
|An+1 | ≤
2
|A1 | ≤ |E| ≤ 2−1/2δ Co−1/δ k 1/δ .
This in turn is satisfied if k is chosen from k = 21/2δ Co |E|δ .
15.3 Proof of Proposition 14.1 for N = 2
The main difference is in the application of the embedding inequality in (15.6),
leading to the recursive inequalities (15.7). Let q > 2 to be chosen, and modify
(15.6) by applying the embedding inequality (3.2) of Theorem 3.1, as follows.
First
1
2
k(u − kn )+ k2 ≤ k(u − kn )+ kq |An | 2 (1− q )
1− 2
2
1
2
≤ γ(q)k∇(u − kn )+ k2 q k(u − kn )+ k2q |An | 2 (1− q )
1
1
≤ k(u − kn )+ k2 + γ(q)k∇(u − kn )+ k2 |An | 2 .
2
Therefore
k2
|An+1 | ≤
4n
Z
E
(u − kn )2+ χ[u>kn+1 ] dx
≤ k(u − kn )+ k22 ≤ 2γ(q)Co2 |An |1+2δ .
16 A Priori L∞ (E) Estimates for Solutions of the
Neumann Problem (10.1)
A weak sub(super)-solution of the Neumann problem (10.1) is a function
u ∈ W 1,2 (E) satisfying
Z
Z
ψv dσ
(16.1)
aij uxi vxj + fj vxj + f v dx ≤ (≥)
∂E
E
for all non-negative test functions v ∈ W 1,2 (E). A function u ∈ W 1,2 (E) is a
weak solution of the Neumann problem (10.1), if and only if is both a weak
sub- and super-solution of that problem.
Proposition 16.1 Let ∂E be of class C 1 and satisfying the segment property.
Let u ∈ W 1,2 (E) be a weak sub-solution of (10.1) for N ≥ 2, and assume that
ψ+ ∈ LN −1+σ (∂E),
f ∈ LN +ε (E),
f+ ∈ L
N +ε
2
(E)
(16.2)+
16 A Priori L∞ (E) Estimates for Solutions of the Neumann Problem (10.1)
337
for some σ > 0 and ε > 0. Then u+ ∈ L∞ (E), and there exists a positive
constant Cε that can be determined quantitatively a priori only in terms of
the set of parameters {N, λ, Λ, ε, σ}, the constant γ in the embeddings of Theorem 2.1, the constant γ of the trace inequality of Proposition 8.2, and the
structure of ∂E through the parameters h and ω of its cone condition such
that
ess sup u+ ≤ Cε max ku+ k2 ; kψ+ kq;∂E ; kf kN +ε ; |E|δ kf+ k N +ε
(16.3)+
2
E
where
q = N − 1 + σ,
σ=ε
N −1
,
N
and
δ=
ε
.
N (N + ε)
(16.4)
Proposition 16.2 Let ∂E be of class C 1 and satisfying the segment property.
Let u ∈ W 1,2 (E) be a weak super-solution of (10.1) for N ≥ 2, and assume
that
ψ− ∈ LN −1+σ (∂E),
f ∈ LN +ε (E),
f− ∈ L
N +ε
2
(E)
(16.2)−
for some σ > 0 and ε > 0. Then u− ∈ L∞ (E)
ess sup u− ≤ Cε max ku− k2 ; kψ− kq;∂E ; kf kN +ε ; |E|δ kf− k N +ε
2
E
(16.3)−
where the parameters q, σ, δ and Cε are the same as in (16.3)+ and (16.4).
Remark 16.1 The dependence on some norm of u, for example ku± k2 , is
expected, since the solutions of (10.1) are unique up to constants.
Remark 16.2 The constant Cε in (16.3)± is “stable” as ε → ∞, in the sense
that if
ψ± ∈ L∞ (∂E), f ∈ L∞ (E), f± ∈ L∞ (E)
(16.5)
then u± ∈ L∞ (E) and there exists a constant C∞ depending on the indicated
quantities except ε and σ such that
1
(16.6)
ess sup u± ≤ C∞ max ku± k2 ; ess sup ψ± ; kf k∞ ; |E| N kf± k∞ .
E
∂E
Remark 16.3 The constant Cε in (16.3)± tends to infinity as ε → 0. The
order of integrability of f and f required in (16.2)± is optimal for u± to be in
L∞ (E). This can be established by the same local solutions in Remark 14.2.
Also, the order of integrability of ψ± is optimal for u± to be in L∞ (E), even
if f = f = 0. Indeed, the propositions are false for σ = 0 and f = f = 0,
as shown by the following counterexample. Consider the family of functions
parametrized by η > 0 (Section 8 of Chapter 2)
R × R+ = E ∋ (x, y) → Fη (x, y) =
p
1
ln x2 + (y + η)2 .
2π
338
9 LINEAR ELLIPTIC EQUATIONS
One verifies that Fη are harmonic in E, and on the boundary y = 0 of E
Fη,y
One also verifies that
Z
y=0
=
Fη,y (x, 0)dx =
R
η
.
2π(x2 + η 2 )
1
2
for all η > 0.
Therefore if an estimate of the type of (16.3)± were to exist for σ = 0, and
with C independent of σ, we would have, for all (x, y) in a neighborhood Eo
of the origin
|Fη (x, y)| ≤ C(1 + kFη,y (·, 0)k1,∂E ) = 23 C
for all η > 0.
Letting η → 0 gives a contradiction. While the counterexample is set in R×R+ ,
it generates a contradiction in a subset of Eo about the origin of R2 .
Remark 16.4 The constants Cε in (16.3)± and C∞ in (16.6) depend on the
embedding constants of Theorem 2.1. As such, they depend on the structure
of ∂E through the parameters h and ω of its cone condition. Because of this
dependence, Cε and C∞ tend to ∞ as either h → 0 or ω → 0.
Remark 16.5 The propositions are a priori estimates, which assume that a
sub(super)-solution exists. Sufficient conditions for the existence of a solution
require that f and ψ must be linked by the compatibility condition (10.3).
Such a requirement, however, plays no role in the a priori L∞ (E) estimates.
17 Proof of Propositions 16.1–16.2
It suffices to establish Proposition 16.1. Let u ∈ W 1,2 (E) be a sub-solution
of the Neumann problem (10.1), in the sense of (16.1), and for k > 0 to be
chosen, define kn and An as in (15.1). In the weak formulation (16.1) take
v = (u − kn+1 )+ ∈ W 1,2 (E), to obtain
Z
Z
[(aij uxi + fj )(u − kn+1 )+xj + f+ (u − kn+1 )+ ]dx ≤
ψ+ (u − kn+1 )+ dσ.
E
∂E
Estimate the various terms by making use of the embedding (2.1), as follows
Z
fj (u − kn+1 )+xj dx ≤ k∇(u − kn+1 )+ k2 kf χAn+1 k2
E
≤
2
λ
1
k∇(u − kn+1 )+ k22 + kf k2N +ε |An+1 |1− N +ε .
4
λ
Next, for the same choices of p∗ as in (15.2), by the embedding (2.1) of Theorem 2.1
17 Proof of Propositions 16.1–16.2
Z
E
339
f+ (u − kn+1 )+ dx ≤ kf+ k N +ε k(u − kn+1 )+ kp∗
2
1
γ
kf+ k N +ε
k(u − kn+1 )+ kp + k∇(u − kn+1 )+ kp
2
ω
h
λ
≤ k∇(u − kn+1 )+ k22 + k(u − kn+1 )+ k22
4
1 δ
2
γ2 1
+
|E| kf+ k2N +ε |An+1 |1− N +2δ
+ 2
2
ω λ 4h2
≤
where γ is the constant of the embedding inequality (2.1), and δ is defined in
(14.4). Setting
F 2 = kf k2N +ε + |E|δ kf+ k2N +ε ,
2
the previous remarks imply
C1 =
1 γ2 1
+ 2
2
ω λ 4h
2
λ
k∇(u − kn+1 )+ k22 ≤ k(u − kn+1 )+ k22 + C1 F 2 |An+1 |1− N +2δ
2
Z
+
∂E
ψ+ (u − kn+1 )+ dσ .
The last integral is estimated by means of the trace inequality (8.3). Let
N + ε
q = N − 1 + σ = (N − 1)
N
be the order of integrability of ψ on ∂E and determine p∗ and p from
1−
1
1 N
= ∗
,
q
p N −1
p∗ =
Np
,
N −p
1
p−1 N
=
.
q
p N −1
One verifies that for these choices, 1 < p < 2 < N , and the trace inequality
(8.3) can be applied. Therefore
Z
ψ+ (u − kn+1 )+ dσ ≤ kψkq;∂E k(u − kn+1 )+ kp∗ N −1 ;∂E
N
∂E
≤kψ+ kq;∂E k∇(u − kn+1 )+ kp + 2γk(u − kn+1 )+ kp
1
1
≤kψ+ kq;∂E k∇(u − kn+1 )+ k2 + 2γk(u − kn+1 )+ k2 |An+1 | p − 2
λ
≤ k∇(u − kn+1 )+ k22 + k(u − kn+1 )+ k22
4
2
1
+ γ2 +
kψ+ k2q;∂E |An+1 | p −1 .
λ
Denote by Cℓ , ℓ = 1, 2, . . . generic positive constants that can be determined
quantitatively a priori, only in terms of the set of parameters {N, λ, Λ}, the
constant γ in the embeddings of Theorem 2.1, the constant γ of the trace
340
9 LINEAR ELLIPTIC EQUATIONS
inequality of Proposition 8.2, and the structure of ∂E through the parameters
h and ω of the cone condition. Then combining the previous estimates yields
the existence of constants C2 and C3 such that
2
k∇(u − kn+1 )+ k22 ≤ C2 k(u − kn+1 )+ k22 + C3 F∗2 |An+1 |1− N +2δ
(17.1)
where we have set
F∗2 = max kψk2q;∂E ; kf k2N +ε ; |E|2δ kf+ k2N +ε .
2
17.1 Proof of Proposition 16.1 for N > 2
By the embedding inequality (2.1) of Theorem 2.1 for W 1,2 (E) and (17.1)
2
k(u − kn+1 )+ k22 ≤ k(u − kn+1 )+ k22∗ |An+1 | N
2
2γ 2
1
k∇(u − kn+1 )+ k22 + 2 k(u − kn+1 )+ k22 |An+1 | N
2
ω
h
2
≤ C4 k(u − kn )+ k22 |An+1 | N + C5 F∗2 |An+1 |1+2δ .
≤
For all n ∈ N
def
Yn =
Z
E
(u − kn )2+ dx ≥
Z
An+1
(u − kn+1 )2+ dx ≥
(17.2)
k2
|An+1 |.
4n
Therefore the previous inequality yields
Z
N2+ε
4n C4
F 2 42n C5 1+2δ
1
2
Yn+1 ≤ 4δ Yn1+2δ
+ ∗2
Y
.
u
dx
2
k
k E
k k 4δ n
Take k ≥ max kuk2 ; F∗ }, so that
Z
1
F∗2
≤ 1.
u2 dx ≤ 1
and
2
k E
k2
This choice leads to the recursive inequalities
42n C6 1+2δ
Y
(17.3)
k 4δ n
for a constant C6 that can be determined a priori only in terms of {N, λ, Λ},
the constants γ in the embedding inequalities of Theorem 2.1, the trace inequalities of Proposition 8.2, the smoothness of ∂E through the parameters ω
and h of its cone condition, and is otherwise independent of f , f , and ψ. By
the fast geometric convergence Lemma 15.1, {Yn } → 0 as n → ∞, provided
Yn+1 ≤
2
−1/2δ 2
Y1 ≤ 2−1/δ C6
k .
We conclude that by choosing
2
1/4δ
k = 21/2δ C6
max{kuk2 ; F∗ }
then Y∞ = k(u − 2k)+ k2 = 0, and therefore u ≤ 2k in E.
18 Miscellaneous Remarks on Further Regularity
341
17.2 Proof of Proposition 16.1 for N = 2
The only differences occur in the application of the embedding inequalities of
Theorem 2.1, in the inequalities (17.2), leading to the recursive inequalities
(17.3). Inequality (17.2) is modified by fixing 1 < p < 2 and applying the
embedding inequality (2.1) of Theorem 2.1 for 1 < p < N . This gives
k(u − kn+1 )+ k22 ≤ k(u − kn+1 )+ k2p∗ |An+1 |2
p−1
p
p−1
≤ γ(p, h, ω) k∇(u − kn+1 )+ k2p + k(u − kn+1 )+ k2p |An+1 |2 p
≤ γ(p, h, ω) k∇(u − kn+1 )+ k22 + k(u − kn+1 )+ k22 |An+1 |
≤ γk(u − kn )+ k22 |An+1 | + C5 F∗2 |An+1 |1+2δ .
18 Miscellaneous Remarks on Further Regularity
1,2
A function u ∈ Wloc
(E) is a local weak solution of (1.2), irrespective of possible
boundary data, if it satisfies (1.3) for all v ∈ Wo1,2 (Eo ) for all open sets Eo
such that Ēo ⊂ E. On the data f and f assume
f ∈ LN +ε (E),
f± ∈ L
N +ε
2
(E),
for some ε > 0.
(18.1)
The set of parameters {N, λ, Λ, ε, kf kN +ε, kf k N +ε } are the data, and we say
2
that a constant C, γ, . . . depends on the data if it can be quantitatively determined a priori in terms of only these quantities. Continue to assume that the
boundary ∂E is of class C 1 and with the segment property. For a compact
set K ⊂ RN and η ∈ (0, 1) continue to denote by k| ·|kη;K the Hölder norms
introduced in (8.3) of Chapter 2.
1,2
Theorem 18.1. Let u ∈ Wloc
(E) be a local weak solution of (1.2) and let
(18.1) hold. Then u is locally bounded and locally Hölder continuous in E,
and for every compact set K ⊂ E, there exist positive constants γK , and CK
depending upon the data and dist{K; ∂E}, and α ∈ (0, 1) depending only on
the data and independent of dist{K; ∂E}, such that
k|u|kα;K ≤ γK (data, dist{K; ∂E}).
(18.2)
Theorem 18.2. Let u ∈ W 1,2 (E) be a solution of the Dirichlet problem (9.1),
with f and f satisfying (18.1) and ϕ ∈ C ǫ (∂E) for some ǫ ∈ (0, 1). Then u
is Hölder continuous in Ē and there exist constants γ > 1 and α ∈ (0, 1), depending upon the data, the C 1 structure of ∂E, and the Hölder norm k|ϕ|kǫ;∂E ,
such that
k|u|kα,Ē ≤ γ(data, ϕ, ∂E).
(18.3)
Theorem 18.3. Let u ∈ W 1,2 (E) be a solution of the Neumann problem
(10.1), with f and f satisfying (16.1) and ψ ∈ LN −1+σ (∂E) for some σ ∈
342
9 LINEAR ELLIPTIC EQUATIONS
(0, 1). Then u is Hölder continuous in Ē, and there exist constants γ and
α ∈ (0, 1), depending on the data, the C 1 structure of ∂E, and kψkN −1+σ;∂E ,
such that
(18.4)
k|u|kα,Ē ≤ γ(data, ψ, ∂E).
The precise structure of these estimates in terms of the Dirichlet data ϕ or
the Neumann ψ, as well as the dependence on the structure of ∂E is specified
in more general theorems for functions in the DeGiorgi classes (Theorem 7.1
and Theorem 8.1 of the next Chapter). These are the key, seminal facts in the
theory of regularity of solutions of elliptic equations. They can be used, by
boot-strap arguments, to establish further regularity on the solutions, whenever further regularity is assumed on the data.
Problems and Complements
1c Weak Formulations and Weak Derivatives
1.1c The Chain Rule in W 1,p (E)
Proposition 1.1c Let u ∈ W 1,p (E) for some p ≥ 1, and let f ∈ C 1 (R)
satisfy sup |f ′ | ≤ M , for some positive constant M . Then f (u) ∈ W 1,p (E)
and ∇f (u) = f ′ (u)∇u.
Proposition 1.2c Let u ∈ W 1,p (E) for some p ≥ 1. Then u± ∈ W 1,p (E)
and
sign(u)∇u
a.e. in [u± > 0]
∇u± =
0
a.e. in [u = 0].
Proof (Hint). To prove the statement for u+ , for ε > 0, apply the previous
proposition with
√
u 2 + ε2 − ε
for u > 0
fε (u) =
0
for u ≤ 0.
Then let ε → 0.
Corollary 1.1c Let u ∈ W 1,p (E) for some p ≥ 1. Then |u − k| ∈ W 1,p (E),
for all k ∈ R, and ∇u = 0 a.e. on any level set of u.
2c Embeddings of W 1,p (E)
343
Corollary 1.2c Let f, g ∈ W 1,p (E) for some p ≥ 1. Then f ∧ g and f ∨ g
are in W 1,p (E) and

 ∇f a.e. in [f > g]
∇f ∧ g = ∇g a.e. in [f < g]

0
a.e. in [f = g].
A similar formula holds for f ∨ g.
1.2. Prove Propostion 1.2 and the first part of Propostion 1.1.
In the same way as done in (1.5), one can introduce Sobolev spaces of
higher order. Indeed, for k ∈ N and p ≥ 1, one defines
W k,p (E) = {u ∈ Lp (E) : Dα u ∈ Lp (E) for |α| ≤ k}.
For 1 ≤ p < ∞, a norm for W k,p (E) is given by
 p1

X
kDα ukpp  ,
kukk,p = 
|α|≤k
whereas for p = ∞ we have
kukk,∞ = sup kDα uk∞ .
|α|≤k
The spaces W k,p (E) are Banach spaces when endowed with their norms. Moreover, we have
Wok,p (E) = {the closure of Co∞ (E) in the norm of W k,p (E)}.
2c Embeddings of W 1,p (E)
It suffices to prove the various assertions for u ∈ C ∞ (E). Fix x ∈ E and let
Cx ⊂ Ē be a cone congruent to the cone C of the cone property. Let n be the
unit vector exterior to Cx , ranging over its same solid angle, and compute
Z h
Z h
Z
ρ
1 h
∂ 1−
u(ρn)dρ ≤
|∇u(ρn)|dρ +
|u(x)| =
|u(ρn)|dρ.
h
h 0
0
0 ∂ρ
Integrating over the solid angle of Cx gives
Z
Z
|∇u(y)|
1
|u(y)|
ω|u(x)| ≤
dy +
dy
N
−1
h Cx |x − y|N −1
C |x − y|
Z x
Z
1
|∇u(y)|
|u(y)|
≤
dy +
dy.
N
−1
h E |x − y|N −1
E |x − y|
(2.1c)
The right-hand side is the sum of two Riesz potentials of the form (10.1) of
Chapter 2. The embeddings (2.1)–(2.3) are now established from this and the
estimates of Riesz potentials (10.2) of Proposition 10.1 of Chapter 2. Complete
the estimates and compute the constants γ explicitly.
344
9 LINEAR ELLIPTIC EQUATIONS
2.1c Proof of (2.4)
Let Cx,ρ be the cone of vertex at x, radius 0 < ρ ≤ h, coaxial with Cx and
with the same solid angle ω. Denote by (u)x,ρ the integral average of u over
Cx,ρ .
Lemma 2.1c For every pair x, y ∈ E such that |x − y| = ρ ≤ h
|u(y) − (u)x,ρ | ≤
Proof. For all ξ ∈ Cx,ρ
|u(y) − u(ξ)| =
Z
γ(N, p) 1− Np
ρ
k∇ukp .
ω
1
∂
u y + t(ξ − y) dt .
∂t
0
Integrate in dξ over Cx,ρ , and then in the resulting integral perform the change
of variables y + t(ξ − y) = η. The Jacobian is t−N , and the new domain of
integration is transformed into those η for which |y − η| = t|ξ − y| as ξ ranges
over Cx,ρ . Such a transformed domain is contained in the ball B2ρt (y). These
operations give
Z 1 Z
ω N
|ξ − y| |∇u(y + t(ξ − y))|dξ dt
ρ |u(y) − (u)x,ρ | ≤
N
Cx,ρ
0
Z 1
Z
≤
t−(N +1)
|η − y||∇u(η)|dη dt
0
≤ γ(N, p)
Z
0
E∩B2ρt (y)
1
1
t−(N +1) (2ρt)N (1− p )+1 k∇ukp t.
To conclude the proof of (2.4), fix x, y ∈ E, let z = 12 (x + y), ρ = 21 |x − y|,
and estimate
|u(x) − u(y)| ≤ |u(x) − (u)z,ρ | + |u(y) − (u)z,ρ| ≤
N
γ(N, p)
|x − y|1− p k∇ukp .
ω
2.2c Compact Embeddings of W 1,p (E)
The proof consists in verifying that a bounded subset of W 1,p (E) satisfies the
conditions for a subset of Lq (E) to be compact ([50], Chapter V). For δ > 0
let
Eδ = x ∈ E dist{x; ∂E} > δ .
For q ∈ [1, p∗ ) and u ∈ W 1,p (E)
1
1
kukq,E−Eδ ≤ kukp∗ |E − Eδ | q − p∗ .
Next, for h ∈ RN of length |h| < δ compute
3c Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E)
Z
Eδ
|u(x + h) − u(x)|dx ≤
Z
Eδ
≤ |h|
Z
Z
1
0
1
0
d
u(x + th) dtdx
dt
Z
Eδ
|∇u(x + th)|dx dt ≤ |h||E|
Therefore for all σ ∈ (0, 1q )
Z
Z
|Th u − u|qσ+q(1−σ) dx
|Th u − u|q dx =
Eδ
Eδ
≤
Z
Eδ
qσ Z
|Th u − u|dx
Eδ
Choose σ so that
q(1 − σ)
= p∗ ,
1 − qσ
345
that is,
|Th u − u|
σq =
q(1−σ)
1−qσ
p−1
p
k∇ukp .
1−qσ
dx
.
p∗ − q
.
p∗ − 1
Such a choice is possible if 1 < q < p∗ . Applying the embedding Theorem 2.1
gives
−q
Z
pp∗∗ −1
Z
(1−σ)q
q
(1−σ)q
|Th u − u| dx ≤ γ
kuk1,p
|Th u − u|dx
Eδ
Eδ
for a constant γ depending only on N, p and the geometry of the cone property
of E. Combining these estimates
kTh u − ukq,Eδ ≤ γ1 |h|σ kuk1,p.
3c Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E)
3.1c Proof of Theorem 3.1 for 1 ≤ p < N
Lemma 3.1c Let u ∈ Co∞ (E) and N > 1. Then
kuk
N
N −1
≤
N
Q
j=1
1/N
kuxj k1
.
Proof. If N = 2
ZZ
ZZ
2
u(x1 , x2 )u(x1 , x2 )dx1 dx2
u (x1 , x2 )dx1 dx2 =
E
Z ZE
max u(x1 , x2 ) max u(x1 , x2 )dx1 dx2
≤
x2
x1
Z E
Z
max u(x1 , x2 )dx1
max u(x1 , x2 )dx2
=
x2
R x1
ZRZ
ZZ
≤
|ux1 |dx
|ux2 |dx.
E
E
346
9 LINEAR ELLIPTIC EQUATIONS
Thus the lemma holds for N = 2. Assuming that it does hold for N , set
x = (x1 , . . . , xN )
and
x = (x̄, xN +1 ).
By repeated application of Hölder’s inequality and the induction
Z Z
N +1
N +1
N
kuk N +1 =
|u(x, xN +1 )| N dxdxN +1
N
N
ZR R
Z
1
=
dxN +1
|u(x, xN +1 )||u(x, xN +1 )| N dx
RN
R
≤
≤
≤
Z
dxN +1
Z
|u(x, xN +1 )|dx
RN
R
Z
E
Z
E
N1 Z N Z
Q
|uxN +1 |dx
Np
N −p
RN
|u(x, xN +1 )|
|uxj (x̄, xN +1 )|dx̄
RN
R j=1
N1
N
N −1
dx
NN−1
dxN +1
N +1 Z
N1
N1
N1 N Z
Q
Q
=
|uxj |dx
|uxj |dx
|uxN +1 |dx
j=1
Next, for 1 ≤ p < N write
kuk
N1 Z
=
Z
w
j=1
E
N −p
NN−1 p(N
−1)
dx
N
N −1
E
E
where w = |u|
p(N −1)
N −p
and apply Lemma 3.1c to the function w. This gives
kuk
Np
N −p
≤
N
Q
j=1
Z
= γ(N, p)
E
N −p
N1 p(N
−1)
|wxj |dx
N
Q
j=1
where
γ=
Z
E
|u|
p(N −1)
N −p −1
p(N − 1)
N −p
N −p
N p(N
−1)
|uxj |dx
N −p
p(N
−1)
.
Now for all j = 1, . . . , N , by Hölder’s inequality
Z
E
|u|
Therefore
Z
N
Q
j=1
p(N −1)
N −p −1
E
|u|
|uxj |dx ≤
p(N −1)
N −p −1
Z
E
p−1
p1 Z
p
Np
N
−p
.
|u|
dx
|uxi | dx
p
E
N −p
N p(N
N −p
p−1
−1)
N
Q
−1)
kuxi kpN p(N −1) kuk p(N
=
|uxj |dx
Np
N −p
j=1
N −p
p(N −1)
≤ k∇ukp
kuk
p−1 N
p N −1
Np
N −p
.
3c Multiplicative Embeddings of Wo1,p (E) and W̃ 1,p (E)
347
3.2c Proof of Theorem 3.1 for p ≥ N > 1
Let F (x; y) be the fundamental solution of the Laplacean. Then for u ∈
Co∞ (E), by the Stokes formula (2.3)–(2.4) of Chapter 2
Z
Z
u(x) = −
∇u(y) · ∇y F (x; y)dy
F (x; y)∆u(y)dy =
N
RN
Z
Z R
∇u(y) · ∇y F (x; y)dy.
∇u(y) · ∇y F (x; y)dy +
=
|x−y|>ρ
|x−y|<ρ
The last integral can be computed by an integration by parts, and equals
Z
Z
1
∇u(y) · ∇y F (x; y)dy =
u(y)dσ
ωN ρN −1 |x−y|=ρ
|x−y|>ρ
since F (x; ·) is harmonic in RN − {x}. Here dσ denotes the surface measure
on the sphere |x − y| = ρ. Put this in the previous expression of u(x), multiply
by N ωN ρN −1 , and integrate in dρ over (0, R), where R is a positive number
to be chosen later. This gives
Z RZ
|∇u(y)|
N
ωN R |u(x)| ≤ N
dy ρN −1 dρ
N −1
0
|x−y|<ρ |x − y|
Z RZ
+N
|u(y)|dσ dρ.
0
From this, for all x ∈ E
Z
ωN |u(x)| ≤
BR (x)
|x−y|=ρ
N
|∇u(y)|
dy + N
|x − y|N −1
R
= I1 (x, R) + N I2 (x, R).
Z
BR (x)
|u(y)|dy
3.2.1c Estimate of I1 (x, R)
Choose two positive numbers a, b < N such that
1
a
= N − 1.
+b 1−
q
p
Since p ≥ N , this choice is possible for the indicated range of q. Now write
p
1
1
|∇u|
1
|∇u| q
= |∇u|p( p − q )
a
1
N
−1
q
|x − y|
|x − y| |x − y|b(1− p )
and apply Hölder’s inequality with the conjugate exponents
1
1
1 1
+ + 1−
= 1.
−
p q
q
p
348
9 LINEAR ELLIPTIC EQUATIONS
This gives
1− p
q
I1 (x, R) ≤ k∇ukp
Z
BR (x)
|∇u(y)|p
dy
|x − y|a
1q Z
BR (x)
1
dy
|x − y|b
1− p1
.
Taking the qth power and integrating over E gives
1
+ q1
1− p
kI1 (R)kq ≤
ωN
1
1
1
RN ( N − p + q )
k∇ukp .
1
1
(N − a) q (N − b)1− p
3.2.2c Estimate of I2 (x, R)
I2 (x, R) ≤ R
≤
−N
ωN
N
Z
p
|x−y|<R
1− p1
|u(y)| dy
1− p
q
N
R− p kukp
p1 Z
1dy
|x−y|<R
Z
|ξ|<R
1− p1
|u(x + ξ)|p dξ
q1
.
Take the qth power and integrate in dx over RN to obtain
kI2 kq ≤
ω 1+ q1 − p1
N
N
1
1
R−N ( p − q ) kukp .
3.2.3c Proof of Theorem 3.1 for p ≥ N > 1 (Concluded)
Combining these estimates yields
kukq ≤ γ R1−δ k∇ukp + R−δ kukp ,
δ=N
1
p
−
1
q
for a constant γ(N, p, q, a, b). Minimizing the right-hand side with respect to
the parameter R proves the estimate.
3.3c Proof of Theorem 3.2 for 1 ≤ p < N and E Convex
Having fixed x, y ∈ E, let R(x, y) be the distance from x to ∂E along y − x
and write
Z R(x,y)
y−x
∂
u(x + ρn) dρ,
n=
.
|u(x) − u(y)| ≤
∂ρ
|y
− x|
0
Integrate in dy over E to obtain
|E||u(x) − uE | ≤
Z Z
E
0
R(x,y)
|∇u(x + ρn)|dρ dy.
5c Solving the Problem (4.1) by the Riesz Representation Theorem
349
The integral in dy is calculated by introducing polar coordinates with pole
at x. Therefore if n is the angular variable spanning the sphere |n| = 1, the
right-hand side is majorized by
Z R(x,y)
Z diam E Z
|∇u(x + ρn)|
N −1
(diam E)
ρN −1
dρdndr
|x − y|N −1
|n|=1 0
0
Z
|∇u|
≤ (diam E)N
dy.
N −1
E |x − y|
Therefore
|u(x) − uE | ≤
(diam E)N
|E|
Z
E
|∇u(y)|
dy.
|x − y|N −1
The proof is now concluded using the estimates of the Riesz potentials in
Section 10 of Chapter 2. The remaining cases for p ≥ N are left as an exercise
following similar arguments in the analogous multiplicative embeddings of
Wo1,p (E).
5c Solving the Homogeneous Dirichlet Problem (4.1) by
the Riesz Representation Theorem
Consider formally the linear operator with variable coefficients
L(u) = − aij uxi + aj u x + bi uxi + cu
j
and the associated, formal bilinear form
Z
aij uxi vxj + aj uvxj + bi uxi v + cuv dx.
a(u, v) =
(5.1c)
(5.2c)
E
Assuming that (aij ) satisfies the ellipticity condition (1.1), all the various
terms are well defined for u, v ∈ Wo1,2 (E), provided c ∈ L∞ (E) and
N
a = (a1 , . . . , aN )
L (E)
if N > 2
∈
(5.3c)
b = (b1 , . . . , bN )
Lq (E)
for some q > 2 if N = 2.
The homogeneous Dirichlet problem (4.1) takes the form
L(u) = div f − f in E,
and u
∂E
= 0 on ∂E.
(5.4c)
The latter is meant in the weak form of seeking u ∈ Wo1,2 (E) such that
Z
f · ∇v + f v dx
(5.5c)
a(u, v) = −
E
for all v ∈ Wo1,2 (E). The unique solvability of this problem can be established
almost verbatim by any one of the methods of Sections 5–7, provided the bilinear form a(·, ·) introduced in (5.2c) generates an inner product in Wo1,2 (E),
350
9 LINEAR ELLIPTIC EQUATIONS
equivalent to any one of the inner products in (3.9), precisely, if there are
constants 0 < λo < Λo such that
λo k∇uk22 ≤ a(u, u) ≤ Λo k∇uk22
for all u ∈ Wo1,2 (E).
This can be ensured by a number of conditions on a, b, and c, and on the size
of E. Let c = c+ −c− be partitioned into its positive and negative parts. Prove
that if N > 2, the following condition is sufficient for the unique solvability
of (5.4c):
γ kakN + kbkN + γkc− k∞ ≤ (1 − ε)λ
(5.6c)
for some ε ∈ (0, 1), where γ is the constant appearing in the embedding
inequality (3.1). The latter occurs, for example, if c ≥ 0, b ∈ Lq (E) for some
q > N , and |E| is sufficiently small. Prove that another sufficient condition is
c ≥ co > 0
1
(kak∞ + kbk∞ ) ≤ co .
4(1 − ε)λ
and
(5.7c)
6c Solving the Homogeneous Dirichlet Problem (4.1) by
Variational Methods
The homogeneous Dirichlet problem (5.4c), can also be solved by variational
methods. The corresponding functional is
Z
2J(u) = {[aij uxi + bi uxi + (aj + bj )u + 2fj ]uxj + (b · ∇u + cu + 2f )u}dx.
E
The same minimization procedure can be carried out, provided b and c satisfy
either (5.6c) or (5.7c).
6.1c More General Variational Problems
More generally one might consider minimizing functionals of the type
Z
1,p
F (x, u, ∇u)dx,
p>1
(6.1c)
Wo (E) ∋ u → J(u) =
E
where the function
E × R × RN ∋ (x, z, q) → F (x, z, q)
is measurable in x for a.e. (z, q) ∈ RN +1 , differentiable in z and q for a.e.
x ∈ E, and satisfies the structure condition
λ|q|p − f (x) ≤ F (x, z, q) ≤ Λ|q|p + f (x)
(6.2c)
1
for a given non-negative f ∈ L (E). On F impose also the convexity (ellipticity) condition, that is, F (x, z, ·) ∈ C 2 (RN ) for a.e. (x, z) ∈ E × R, and
Fqi qj ξi ξj ≥ λ|ξ|p
for all ξ ∈ RN for a.e. (x, z) ∈ E × R.
(6.3c)
6c Solving the Homogeneous Dirichlet Problem (4.1) by Variational Methods
351
A Prototype Example
Let (aij ) denote a symmetric N × N matrix with entries aij ∈ L∞ (E) and
satisfying the ellipticity condition (1.1), and consider the functional
Z
Wo1,p (E) ∋ u → pJ(u) =
(6.4c)
(|∇u|p−2 aij uxi uxj + pf u)dx
E
for a given f ∈ Lq (E), where q ≥ 1 satisfies
1 1
1
+ =
+1
p q
N
if 1 < p < N,
and q ≥ 1 if p ≥ N.
(6.5c)
Let v = (v1 , . . . , vN ) be a vector-valued function defined in E. Verify that the
map
Z
[Lp (E)]N ∋ v =
p
E
|v|p−2 aij vi vj dx
N
defines a norm in [L (E)] equivalent to kvkp . Since the norm is weakly lower
semi-continuous, for every sequence {vn } ⊂ [Lp (E)]N weakly convergent to
some v ∈ [Lp (E)]N
Z
Z
|v|p−2 aij vi vj dx.
lim inf
|vn |p−2 aij vi,n vj,n dx ≥
E
E
The convexity condition (6.3c), called also the Legendre condition, ensures
that a similar notion of semi-continuity holds for the functional J(·) in (6.1c)
(see[185]).
Lower Semi-Continuity
A functional J from a topological space X into R is lower semi-continuous if
[J > a] is open in X for all a ∈ R. Prove the following:
Proposition 6.1c Let X be a topological space satisfying the first axiom of
countability. A functional J : X → R is lower semi-continuous if and only if
for every sequence {un } ⊂ X convergent to some u ∈ X
lim inf J(un ) ≥ J(u).
6.3. The epigraph of J is the set
EJ = {(x, a) ∈ X × R J(x) ≤ a}.
Assume that X satisfies the first axiom of countability and prove that J
is lower semi-continuous if and only if its epigraph is closed.
6.4. Prove that J : X → R is convex if and only if its epigraph is convex.
6.5. Prove the following:
352
9 LINEAR ELLIPTIC EQUATIONS
Proposition 6.2c Let J : Wo1,p (E) → R be the functional in (6.1c) where
F satisfies (6.2c)–(6.3c). Then J is weakly lower semi-continuous.
Hint: Assume first that F is independent of x and z and depends only on
q. Then J may be regarded as a convex functional from J˜ : [Lp (E)]N → R.
Prove that its epigraph is (strongly and hence weakly) closed in [Lp (E)]N .
6.6. Prove the following:
Proposition 6.3c Let J : Wo1,p (E) → R be the functional in (6.1c) where
F satisfies (6.2c)–(6.3c). Then J has a minimum in Wo1,p (E).
Hint: Parallel the procedure of Section 6.
6.7. The minimum claimed by Proposition 6.3c need not be unique. Provide a counterexample. Formulate sufficient assumptions on F to ensure
uniqueness of the minimum.
6.8c Gâteaux Derivatives, Euler Equations and Quasi-Linear
Elliptic Equations
Let X be a Hausdorff space. A functional J : X → R is Gâteaux differentiable
at w ∈ X in the direction of some v ∈ X if there exists an element J ′ (w; v) ∈ R
such that
J(w + tv) − J(w)
lim
= J ′ (w; v).
t→0
t
The equation
J ′ (w; v) = 0 for all v ∈ X
is called the Euler equation of J. In particular, (6.4) is the Euler equation of
the functional in (6.1). The Euler equation of the functional in (6.4c) is
− |∇u|p−2 aij uxi xj = f.
(6.6c)
In the special case (aij ) = I this is the p-Laplacean equation
− div |∇u|p−2 ∇u = f.
(6.7c)
The Euler equation of the functional in (6.1c) is
− div A(x, u, ∇u) + B(x, u, ∇u) = 0,
u ∈ Wo1,p (E)
(6.8c)
where A = ∇q F and B = Fz . The equation is elliptic in the sense that
(aij ) = (Fqi qj ) satisfies (6.3c). Thus the functional in (6.1c) generates the
PDE in (6.8c) as its Euler equation, and minima of J are solutions of (6.8c).
8c Traces on ∂E of Functions in W 1,p (E)
353
6.8.1c Quasi-Linear Elliptic Equations
Consider now (6.8c) independently of its variational origin, where
A(x, z, q) ∈ RN
E × R × RN ∋ (x, z, q) →
B(x, z, q) ∈ R
(6.9c)
are continuous functions of their arguments and subject to the structure conditions

 A(x, z, q) · q ≥ λ|q|p − C p
|A(x, z, q)| ≤ Λ|q|p−1 + C p−1
(6.10c)

|B(x, z, q)| ≤ C|q|p−1 + C p
for all (x, z, q) ∈ E × R × RN , for given positive constants λ ≤ Λ and nonnegative constant C. A local solution of (6.8c)–(6.10c), irrespective of possible
1,p
(E) satisfying
prescribed boundary data, is a function u ∈ Wloc
Z
A(x, u, ∇u)∇v + B(x, u, ∇u)v dx = 0 for all v ∈ Wo1,p (Eo ) (6.11c)
E
for every open set Eo such that Ēo ⊂ E. In general, there is not a function F
satisfying (6.2c)–(6.3c) and a corresponding functional as in (6.1c) for which
(6.8c) is its Euler equation. It turns out, however, that local solutions of (6.8c)–
(6.10c), whenever they exist, possess the same local behavior, regardless of
their possible variational origin (Chapter 10).
6.8.2c Quasi-Minima
1,p
Let J : Wloc
(E) → R be given by (6.1c), where F satisfies (6.2c) but not
1,p
necessarily (6.3c). A function u ∈ Wloc
(E) is a Q-minimum for J if there is a
number Q ≥ 1 such that
J(u) ≤ QJ(u + v)
for all v ∈ Wo1,p (Eo )
for every open set Eo such that Ēo ⊂ E. The notion is of local nature. Minima
are Q-minima, but the converse is false. Every functional of the type (6.1c)–
(6.3c) generates a quasi-linear elliptic PDE of the type of (6.8c)–(6.10c). The
1,p
(E) of
converse is in general false. However every local solution u ∈ Wloc
(6.8c)–(6.10c) is a Q-minimum, in the sense that there exists some F satisfying
(6.2c), but not necessarily (6.3c), such that u is a Q-minimum for the function
J in (6.1c) for such a F ([98]).
8c Traces on ∂E of Functions in W 1,p (E)
8.1c Extending Functions in W 1,p (E)
Establish Proposition 8.1 by the following steps. Let RN
+ be the upperhalf space xN > 0 and denote its coordinates by x = (x̄, xN ), where
354
9 LINEAR ELLIPTIC EQUATIONS
N
so that ∂E is the hyperplane
x̄ = (x1 , . . . , xN −1 ). Assume first that E = R+
N
1,p
xN = 0. Given u ∈ W (R+ ), set ([172])
u(x̄, xN )
if xN > 0
ũ(x̄, xN ) =
−3u(x̄, −xN ) + 4u(x̄, − 21 xN )
if xN < 0.
Prove that ũ ∈ W 1,p (RN ), and that Co∞ (RN ) is dense in W 1,p (RN
+ ).
If ∂E is of class C 1 and has the segment property, it admits a finite covering
with balls Bt (xj ) for some t > 0, and xj ∈ ∂E for j = 1, . . . , m. Let then
U = {Bo , B2t (x1 ), . . . , B2t (xm )},
Bo = E −
m
S
B̄t (xj )
j=1
be an open covering of E, and let Φ be a partition of unity subordinate to U .
Set
ψj = {the sum of the ϕ ∈ Φ supported in B2t (xj )}
so that
u=
m
P
uj
where
j=1
uj =
uψj in E
0
otherwise.
By construction, uj ∈ W 1,p (B2t (xj )) with bounds depending on t. By choosing
t sufficiently small, the portion ∂E ∩B2t (xj ) can be mapped, in a local system
of coordinates, into a portion of the hyperplane xN = 0. Denote by Uj an
open ball containing the image of B2t (xj ) and set Uj+ = Uj ∩ [xN > 0].
The transformed functions ūj belong to W 1,p (Uj+ ). Perform the extension as
indicated earlier, return to the original coordinates and piece together the
various integrals each relative to the balls B2t (xj ) of the covering U . This
technique is refereed to as local “flattening of the boundary”.
8.2c The Trace Inequality
Proposition 8.1c Let u ∈ Co∞ (RN ). If 1 ≤ p < N , there exists a constant
γ = γ(N, p) such that
.
ku(·, 0)kp∗ N −1 ,RN −1 ≤ γk∇ukp,RN
+
N
(8.1c)
If p > N , there exist constants γ = γ(N, p) such that
1− N
N
p
ku(·, 0)k∞,RN −1 ≤ γkukp,RNp k∇ukp,R
N
+
N
|u(x̄, 0) − u(ȳ, 0)| ≤ γ|x̄ − ȳ|1− p k∇ukp,RN
+
for all x̄, ȳ ∈ RN −1 .
(8.2c)
+
(8.3c)
8c Traces on ∂E of Functions in W 1,p (E)
355
Proof. For all x̄ ∈ RN −1 and all r ≥ 1
Z ∞
r
|u(x̄, xN )|r−1 |uxN (x̄, xN )|dxN .
|u(x̄, 0)| ≤ r
0
Integrate both sides in dx̄ over RN −1 and apply Hölder’s inequality to the
resulting integral on the right-hand side to obtain
r−1
,
N kuk
ku(·, 0)krr,RN −1 ≤ rk∇ukp,R+
q,RN
where q =
+
p
(r − 1).
p−1
(8.4c)
Apply this with r = p∗ NN−1 , and use the embedding (3.1) of Theorem 3.1 to
get
1
1− 1
r
ku(·, 0)kp∗ N −1 ,RN −1 ≤ γkukp∗,Rr N k∇ukp,R
N ≤ γk∇ukp,RN .
+
N
+
+
The domain
satisfies the cone condition with cone C of solid angle 12 ωN
and height h ∈ (0, ∞). Then (8.3c) follows from (2.4) of Theorem 2.1, whereas
(8.2c) follows from (2.3) of the same theorem, by minimizing over h ∈ (0, ∞).
RN
+
Prove Proposition 8.2 by a local flattening of ∂E.
8.3c Characterizing the Traces on ∂E of Functions in W 1,p (E)
+1
+1
Set RN
= RN × R+ and denote the coordinates in RN
by (x, t) where
+
+
N
x ∈ R and t ≥ 0. Also set
∂
∂
∂ .
,...,
,
∇ = ∇N ,
∇N =
∂x1
∂xN
∂t
+1
Proposition 8.2c Let u ∈ Co∞ (RN
). Then
+
1
1− 1
p
p
k|u(·, 0)|k1− p1 ,p;RN ≤ γkut kp,R
N +1 k∇N uk
p,RN +1
+
(8.5c)
+
where γ = γ(p) depends only on p and γ(p) → ∞ as p → 1.
Proof. For every pair x, y ∈ RN , set 2ξ = x − y and consider the point
+1
z ∈ RN
of coordinates z = ( 12 (x + y), λ|ξ|), where λ is a positive parameter
+
to be chosen. Then
|u(x, 0) − u(y, 0)| ≤ |u(z) − u(x, 0)| + |u(z) − u(y, 0)|
Z 1
Z 1
≤ |ξ|
|∇N u(x − ρξ, λρ|ξ|)|dρ + |ξ|
|∇N u(y + ρξ, λρ|ξ|)|dρ
0
+ λ|ξ|
Z
0
1
0
|ut (x − ρξ, λρ|ξ|)|dρ + λ|ξ|
Z
0
1
|ut (y + ρξ, λρ|ξ|)|dρ.
356
9 LINEAR ELLIPTIC EQUATIONS
From this
|u(x, 0) − u(y, 0)|p
1
≤ p
N
+(p−1)
2
|x − y|
+
+
+
Z
1
|∇N u(x − ρξ, λρ|ξ|)|
|x − y|
0
1
2p
Z
1 p
λ
2p
1 p
λ
2p
1
|∇N u(y + ρξ, λρ|ξ|)|
0
Z
Z
N −1
p
dρ
1
|x − y|
dρ
|ut (x − ρξ, λρ|ξ|)|
dρ
|ut (y + ρξ, λρ|ξ|)|
dρ
|x − y|
0
1
N −1
p
p
|x − y|
0
N −1
p
N −1
p
p
p
p
.
Next integrate both sides over RN × RN . In the resulting inequality take the
1
p power and estimate the various integrals on the right-hand side by the
continuous version of Minkowski’s inequality. This gives
k|u(·, 0)|k1− p1 ,RN
Z
p1
|∇N u(x − ρξ, λρ|ξ|)|p
≤
dxdy dρ
|x − y|N −1
RN RN
0
p1
Z 1Z Z
|ut (x − ρξ, λρ|ξ|)|p
+λ
dxdy dρ.
|x − y|N −1
0
RN RN
Z
1
Z
Compute the first integral by integrating first in dy and perform such integration in polar coordinates with pole at x. Denoting by n the unit vector
spanning the unit sphere in RN and recalling that 2|ξ| = |x − y|, we obtain
Z Z
|∇N u(x − ρξ, λρ|ξ|)|p
dxdy
|x − y|N −1
RN RN
Z ∞
Z
Z
dn
d|ξ|
|∇N u(x + ρn|ξ|, λρ|ξ|)|p dx
=2
RN
|n|=1
0
Z
ωN
=2
|∇N u|p dx.
+1
λρ RN
+
Compute the second integral in a similar fashion and combine them into
Z 1
1
1
k|u(·, 0)|k1− p1 ,p;RN ≤ 21/p λ− p k∇N ukp,RN +1
ρ− p dρ
+
1
0
Z
1
1
ρ− p dρ
+ 21/p λ1− p kut kp,RN +1
+
0
1
1
−p
1/p p
λ k∇N ukp,RN +1 + λ1− p kut kp,RN +1 .
=2
+
+
p−1
The proof is completed by minimizing with respect to λ.
Prove Theorem 8.1 by the following steps:
9c The Inhomogeneous Dirichlet Problem
357
N +1
) has a trace on
8.4. Proposition 8.2c shows that a function in W 1,p (R+
1
1−
,p
RN = [xN +1 = 0] in W p (RN ). Prove the direct part of the theorem for
general ∂E of class C 1 and with the segment property by a local flattening
technique.
1
+1
8.5. Every v ∈ W 1− p ,p (RN ) admits an extension u ∈ W 1,p (RN
) such
+
that v = tr(u). To construct such an extension, assume first that v is
continuous and bounded in RN . Let Hv (x, t) be its harmonic extension in
+1
RN
constructed in Section 8 of Chapter 2, and in particular in (8.3),
+
and set
u(x, xN +1 ) = Hv (x, xN +1 )e−xN +1 .
+1
Verify that u ∈ W 1,p (RN
) and that tr(u) = v. Modify the construction
+
to remove the assumption that v is bounded and continuous in RN .
8.6. Prove that the Poisson kernel K(·; ·) in RN × R+ , constructed in (8.2)
of Section 8 of Chapter 2, is not in W 1,p (RN × R+ ) for any p ≥ 1. Argue
indirectly by examining its trace on xN +1 = 0.
8.7. Prove a similar fact for the kernel in the Poisson representation of harmonic functions in a ball BR (formula (3.9) of Section 3 of Chapter 2).
9c The Inhomogeneous Dirichlet Problem
9.1c The Lebesgue Spike
The segment property on ∂E is required to ensure an extension of ϕ into
E by a function v ∈ W 1,2 (E). Whence such an extension is achieved, the
structure of ∂E does not play any role. Indeed, the problem is recast into one
with homogeneous Dirichlet data on ∂E whose solvability by either methods
of Sections 5–7 use only the embeddings of Wo1,2 (E) of Theorem 3.1, whose
constants are independent of ∂E. Verify that the domain of Section 7.2 of
Chapter 2 does not satisfy the segment property. Nevertheless the Dirichlet
problem (7.3), while not admitting a classical solution, has a unique weak solution given by (7.2). Specify in what sense such a function is a weak solution.
9.2c Variational Integrals and Quasi-Linear Equations
Consider the quasi-linear Dirichlet problem
− div A(x, u, ∇u) + B(x, u, ∇u) = 0
u
∂E
=ϕ∈W
1
,p
1− p
(∂E)
in E
on ∂E
(9.1c)
where the functions A and B satisfy the structure condition (6.10c). Assume
moreover, that (9.1c) has a variational structure, that is, there exists a function F , as in Section6.1c, and satisfying (6.2c)–(6.3c), such that A = ∇q F
358
9 LINEAR ELLIPTIC EQUATIONS
and B = Fz . A weak solution is a function u ∈ W 1,p (E) such that tr(u) = ϕ
and satisfying (6.11c). Introduce the set
Kϕ = u ∈ W 1,p (E) such that tr(u) = ϕ
(9.2c)
and the functional
Kϕ ∋ u → J(u) =
Z
E
F (x, u, ∇u)dx
p > 1.
(9.3c)
Prove the following:
9.3. Kϕ is convex and weakly (and hence strongly) closed. Hint: Use the
trace inequalities (8.3)–(8.5).
9.4. Subsets of Kϕ , bounded in W 1,p (E) are weakly sequentially compact.
9.5. The functional J in (9.3c) has a minimum in Kϕ . Such a minimum is
a solution of (9.1c) and the latter is the Euler equation of J. Hint: Use
Proposition 6.2c.
9.6. Solve the inhomogeneous Dirichlet problem for the more general linear
operator in (5.1c).
9.7. Explain why the method of extending the boundary datum ϕ and recasting the problem as a homogeneous Dirichlet problem might not be
applicable for quasi-linear equations of the form (9.1c). Hint: Examine
the functionals in (6.4c) and their Euler equations (6.6c)–(6.7c).
10c The Neumann Problem
Consider the quasi-linear Neumann problem
− div A(x, u, ∇u) + B(x, u, ∇u) = 0
in E
A(x, u, ∇u) · n = ψ
on ∂E
(10.1c)
where n is the outward unit normal to ∂E and ψ satisfies (10.4). The functions A and B satisfy the structure condition (6.10c) and have a variational
structure in the sense of Section 9.2c. Introduce the functional
Z
Z
ψtr(u)dσ.
(10.2c)
F (x, u, ∇u)dx −
W 1,p (E) ∋ u → J(u) =
E
∂E
In dependence of various assumptions on F , identify the correct weakly closed
subspace of W 1,p (E), where the minimization of J should be set, and find such
a minimum, to coincide with a solution of (10.1c).
As a starting point, formulate sufficient conditions on the various parts of
the operators in (5.1c), (6.6c), and (6.7c) that would ensure solvability of the
corresponding Neumann problem. Discuss uniqueness.
14c A Priori L∞ (E) Estimates for Solutions of the Dirichlet Problem (9.1)
359
11c The Eigenvalue Problem
11.1. Formulate the eigenvalue problem for homogeneous Dirichlet data as
in (11.1) for the more general operator (5.1c). Formulate conditions on
the coefficients for an analogue of Proposition 11.1 to hold.
11.2. Formulate the eigenvalue problem for homogeneous Neumann data.
State and prove a proposition analogous to Proposition 11.1. Extend it to
the more general operator (5.1c).
12c Constructing the Eigenvalues
12.1. Set up the proper variational functionals to construct the eigenvalues
for homogeneous Dirichlet data for the more general operator (5.1c). Formulate conditions on the coefficients for such a variational problem to be
well posed.
12.2. Set up the proper variational functionals to construct the eigenvalues
for homogeneous Neumann data. Extend these variational integrals and
formulate sufficient conditions to include the more general operator (5.1c).
13c The Sequence of Eigenvalues and Eigenfunctions
13.1. It might seem that the arguments of Proposition 13.2 would apply to
all eigenvalues and eigenfunctions. Explain where the argument fails for
the eigenvalues following the first.
13.2. Formulate facts analogous to Proposition 13.1 for the sequence of eigenvalues and eigenfunctions for homogeneous Dirichlet data for the more
general operator (5.1c).
13.3. Formulate facts analogous to Proposition 13.1 for the sequence of eigenvalues and eigenfunctions for homogeneous Neumann data.
14c A Priori L∞ (E) Estimates for Solutions of the
Dirichlet Problem (9.1)
The proof of Propositions 14.1–14.2 shows that the L∞ (E)-estimate stems
only from the recursive inequalities (15.3), and a sup-bound would hold for
any function satisfying them. For these inequalities to hold the linearity of the
PDE in (9.1) is immaterial. As an example, consider the quasi-linear Dirichlet problem (9.1c) where B = 0 and A is subject to the structure condition
(6.10c). In particular the problem is not required to have a variational structure.
360
9 LINEAR ELLIPTIC EQUATIONS
14.1. Prove that weak solutions of such a quasi-linear Dirichlet problem satisfy recursive inequalities analogous to (15.3). Prove that they are essentially bounded with an upper of the form (14.5), with f = 0 and f = C p ,
where C is the constant in the structure conditions (6.10c).
14.2. Prove that the boundedness of u continues to hold, if A and B satisfy
the more general conditions
A(x, z, q) · q ≥ λ|q|p − f (x)
|B(x, z, q)| ≤ f (x)
for some f ∈ L
N +ε
p
(E).
Prove that an upper bound for kuk∞ has the same form as (14.3)± with
f = 0 and the same value of δ.
15c A Priori L∞ (E) Estimates for Solutions of the
Neumann Problem (10.1)
The estimates (16.3)± and (16.6) are a sole consequence of the recursive inequalities (17.3) and therefore continue to hold for weak solutions of equations
from which they can be derived.
15.1. Prove that they can be derived for weak solutions of the quasi-linear
Neumann problem (10.1c), where A and B satisfy the structure conditions
(6.10c) and are not required to be variational. Prove that the estimate
takes the form
1
kuk∞ ≤ Cσ max kuk2 ; kψkN −1+σ ; |E| N .
15.2. Prove that L∞ (E) estimates continue to hold if the constant C in
the structure conditions (6.10c) is replaced by a non-negative function
N +ε
f ∈ L p for some ε > 0. In such a case the estimate takes exactly the
form (16.6) with f = 0.
15.3. Establish L∞ (E) estimates for weak solutions to the Neumann problem
for the operator L(·) in (5.1c).
15.4. The estimates deteriorate if either the opening or the height of the
circular spherical cone of the cone condition of ∂E tend to zero (Remark 16.4). Generate examples of such occurrences for the Laplacean in
dimension N = 2.
15.1c Back to the Quasi-Linear Dirichlet Problem (9.1c)
The main difference between the estimates (14.3)± and (16.3)± is that the
right-hand side contains the norm ku± k2 of the solution. Having the proof
of Proposition 14.1 as a guideline, establish L∞ (E) bounds for solutions of
the quasi-linear Dirichlet (9.1c) where A and B satisfy the full quasi-linear
15c A Priori L∞ (E) Estimates for Solutions of the Neumann Problem (10.1)
361
structure (6.10c), where, in addition, C may be replaced by a non-negative
N +ε
function f ∈ L p for some ε > 0. Prove that the resulting estimate has the
form
kuk∞ ≤ max{kϕk∞,∂E ; Cε [kuk2 ; |E|pδ kf k N +ε ]}.
p
10
DEGIORGI CLASSES
1 Quasi-Linear Equations and DeGiorgi Classes
A quasi-linear elliptic equation in an open set E ⊂ RN is an expression of the
form
− div A(x, u, ∇u) + B(x, u, ∇u) = 0
(1.1)
1,p
where for u ∈ Wloc
(E), the functions
A x, u(x), ∇u(x) ∈ RN
E∋x→
B(x, u(x), ∇u(x) ∈ R
are measurable and satisfy the structure conditions
A x, u, ∇u · ∇u ≥ λ|∇u|p − f p
|A x, u, ∇u | ≤ Λ|∇u|p−1 + f p−1
|B x, u, ∇u | ≤ Λo |∇u|p−1 + fo
(1.2)
for given constants 0 < λ ≤ Λ and Λo > 0, and given non-negative functions
f ∈ LN +ε (E),
fo ∈ L
N +ε
p
(E),
for some ε > 0.
(1.3)
The Dirichlet and Neumann problems for these equations were introduced in
Sections 9.2c and 9c of the Complements of Chapter 9, their solvability was
established for a class of functions A and B, and L∞ (E) bounds were derived
for suitable data. Here we are interested in the local behavior of these solutions
1,p
irrespective of possible prescribed boundary data. A function u ∈ Wloc
(E) is
a local weak sub(super)-solution of (1.1), if
Z
[A(x, u, ∇u)∇v + B(x, u, ∇u)v] dx ≤ (≥)0
(1.4)
E
for all non-negative test functions v ∈ Wo1,p (Eo ), for every open set Eo such
1,p
that Ēo ⊂ E. A local weak solution to (1.1) is a function u ∈ Wloc
(E) satisfy1,p
ing (1.4) with the equality sign, for all v ∈ Wo (Eo ). No further requirements
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_11
363
364
10 DEGIORGI CLASSES
are placed on A and B other than the structure conditions (1.2). Specific
examples of these PDEs are those introduced in the previous chapter. In particular they include the class of linear equations (1.2), those in (5.1c)–(5.4c),
and the nonlinear p-Laplacian-type equations in (6.6c)–(6.7c) of the Complements of Chapter 9. In all these examples the coefficients of the principal
part are only measurable. Nevertheless local weak solutions of (1.1) are locally
Hölder continuous in E. If p > N , this follows from the embedding inequality
(2.4) of Theorem 2.1 of Chapter 9. If 1 < p ≤ N , this follows from their membership in more general classes of functions called DeGiorgi classes, which are
introduced next. Let Bρ (y) ⊂ E denote a ball of center y and radius ρ; if y
is the origin, write Bρ (0) = Bρ . For σ ∈ (0, 1), consider the concentric ball
Bσρ (y) and denote by ζ a non-negative, piecewise smooth cutoff function that
equals 1 on Bσρ (y), vanishes outside Bρ (y) and such that |∇ζ| ≤ [(1 − σ)ρ]−1 .
Let u be a local sub(super)-solution of (1.1). For k ∈ R, the localized truncations ±ζ p (u − k)± belong to Wo1,p (E) and can be taken as test functions v in
(1.4). Using the structure conditions (1.2) yields
Z
λ
|∇(u − k)± |p ζ p dx
Bρ (y)
≤
Z
Bρ (y)
+
Z
|∇(u − k)± |p−1 ζ p−1 (pΛ|∇ζ| + Λo ζ)(u − k)± dx
Bρ (y)
+
≤
λ
2
+
Z
f p ζ p χ[(u−k)± >0] + pf p−1 ζ p−1 (u − k)± |∇ζ|) dx
Bρ (y)
fo (u − k)± ζ p dx
Bρ (y)
|∇(u − k)± |p ζ p dx +
Z
Z
Bρ (y)
f p χ[(u−k)± >0] dx +
Z
γ(Λ, p)
(1 − σ)p ρp
Bρ (y)
Z
Bρ (y)
(u − k)p± dx
fo (u − k)± ζ p dx
where ρ has been taken so small that ρ ≤ max{1; Λo}−1 . Next estimate
Z
p
+pδ
1− N
f p χ[(u−k)± >0] dx ≤ kf kpN +ε |A±
k,ρ |
Bρ (y)
where we have assumed 1 < p ≤ N , and
A±
k,ρ = [(u − k)± > 0] ∩ Bρ (y)
and
δ=
ε
.
N (N + ε)
(1.5)
The term involving fo is estimated by Hölder’s inequality with conjugate
exponents
q∗
Nq
N +ε
= ∗
,
q∗ =
.
p
q −1
N −q
365
1 Quasi-Linear Equations and DeGiorgi Classes
Continuing to assume 1 < p ≤ N , one checks that 1 < q < p < N for all
N ≥ 2 and the Sobolev embedding of Theorem 3.1 of Chapter 9, can be
applied since (u − k)± ζ ∈ Wo1,q (Bρ (y)). Therefore
Z
fo (u − k)± ζ p dx ≤ kfo k N +ε k(u − k)± ζkq∗
p
Bρ (y)
≤ γ(N, p)kfo k N +ε k∇[(u − k)± ζ]kq
p
1
1
q−p
≤ γ(N, p)k∇[(u − k)± ζ]kp kfo k N +ε |A±
k,ρ |
p
Z
Z
λ
≤
|∇(u − k)± |p ζ p dx + (u − k)p± |∇ζ|p dx
4 E
E
p
p
p
± 1− N + p−1 pδ
+ γ(N, p, λ)kfo k p−1
.
N +ε |Ak,ρ |
p
Continue to assume that ρ ≤ max{1; Λo }−1 and combine these estimates to
conclude that there exists a constant γ = γ(N, p, λ, Λ) dependent only on the
indicated quantities and independent of ρ, y, k, and σ such that for 1 < p ≤ N
k∇(u − k)± kpp,Bσρ (y) ≤
γ
k(u − k)± kpp,Bρ (y)
(1 − σ)p ρp
p
(1.6)
1− N +pδ
+ γ∗p |A±
k,ρ |
where δ is given by (1.5) and
p
γ∗p = γ(N, p) kf kpN +ε + kfo k p−1
N +ε .
(1.7)
p
1.1 DeGiorgi Classes
Let E be an open subset of RN , let p ∈ (1, N ], and let γ, γ∗ , and δ be given
positive constants. The DeGiorgi class DG+ (E, p, γ, γ∗ , δ) is the collection of
1,p
all functions u ∈ Wloc
(E) such that (u − k)+ satisfy (1.6) for all k ∈ R, and
for all pair of balls Bσρ (y) ⊂ Bρ (y) ⊂ E. Local weak sub-solutions of (1.1)
belong to DG+ , for the constants γ, γ∗ and δ identified in (1.5)–(1.7). The
DeGiorgi class DG− (E, p, γ, γ∗ , δ) are defined similarly, with (u−k)+ replaced
by (u−k)− . Local weak super-solutions of (1.1) belong to DG− . The DeGiorgi
classes DG(E, p, γ, γ∗ , δ) are the intersection of DG+ ∩ DG− , or equivalently
1,p
the collection of all functions u ∈ Wloc
(E) satisfying (1.6) for all pair of balls
Bσρ (y) ⊂ Bρ (y) ⊂ E and all k ∈ R. We refer to these classes as homogeneous
if γ∗ = 0. In such a case the choice of the parameter δ is immaterial. The set
of parameters {N, p, γ} are the homogeneous data of the DG classes, whereas
γ∗ and δ are the inhomogeneous parameters. This terminology stems from the
structure of (1.6) versus the structure of the quasi-linear elliptic equations in
(1.1), and is evidenced by (1.7).
Functions in DG have remarkable properties, irrespective of their connection with the quasi-linear equations (1.1). In particular, they are locally
366
10 DEGIORGI CLASSES
bounded, and locally Hölder continuous in E. Even more striking is that
non-negative functions in DG satisfy the Harnack inequality of Section 5.1 of
Chapter 2, which is typical of non-negative harmonic functions.
2 Local Boundedness of Functions in the DeGiorgi
Classes
We say that constants C, γ, . . . depend only on the data, and are independent
of γ∗ and δ, if they can be quantitatively determined a priori only in terms
of the homogeneous parameters {N, p, γ}. The dependence on the inhomogeneous parameters {γ∗ , δ} will be traced, as a way to identify those additional
properties afforded by inhomogeneous structures.
Theorem 2.1 (DeGiorgi [47]). Let u ∈ DG± and τ ∈ (0, 1). There exists a
constant C depending only on the data such that for every pair of concentric
balls Bτ ρ (y) ⊂ Bρ (y) ⊂ E
ess sup u± ≤ max γ∗ ρ
Bτ ρ (y)
Nδ
;
C
1
(1 − τ ) δ
Z
Bρ
p1 .
up± dx
For homogeneous DG± classes, γ∗ = 0 and δ can be taken δ =
(2.1)
1
N.
Proof. Having fixed the pair of balls Bτ ρ (y) ⊂ Bρ (y) ⊂ E assume y = 0 and
consider the sequences of nested concentric balls {Bn } and {B̃n }, and the
sequences of increasing levels {kn }
1−τ
Bn = Bρn (0) where ρn = τ ρ + n−1 ρ
2
ρn + ρn+1
31−τ
B̃n = Bρ̃n (0) where ρ̃n =
= τρ +
ρ
2
2 2n
1
kn = k − n−1 k
2
(2.2)
where k > 0 is to be chosen. Introduce also non-negative piecewise smooth
cutoff functions

1
for x ∈ Bn+1


 ρ̃ − |x|
2n+1
n
(2.3)
ζn (x) =
=
(ρ̃n − |x|) for ρn+1 ≤ |x| ≤ ρ̃n

ρ̃ − ρn+1
(1 − τ )ρ

 n
0
for |x| ≥ ρ̃n
for which
|∇ζn | ≤
2n+1
.
(1 − τ )ρ
Write down the inequalities (1.6) for (u − kn+1 )+ , for the levels kn+1 over the
pair of balls B̃n ⊂ Bn for which (1 − σ) = 2−(n+1) (1 − τ ), to get
2 Local Boundedness of Functions in the DeGiorgi Classes
p
k∇(u − kn+1 )+ kp,
≤
B̃
n
367
2(n+1)p γ
k(u − kn+1 )+ kpp,Bn
(1 − τ )p ρp
p
1− N +pδ
+ γ∗p |A+
.
kn+1 ,ρn |
In the arguments below, γ is a positive constant depending only on the data
and that might be different in different contexts.
2.1 Proof of Theorem 2.1 for 1 < p < N
Apply the embedding inequality (3.1) of Theorem 3.1 of Chapter 9 to the
functions (u − kn+1 )+ ζn over the balls B̃n to get
k(u − kn+1 )+ kpp,Bn+1 ≤ k(u − kn+1 )+ ζn kpp,B̃
≤ k(u − kn+1 )+ ζn k
n
p
p∗
p∗ ,B̃n
p
N
|A+
kn+1 ,ρ̃n |
p
N
≤ γk∇[(u − kn+1 )+ ζn ]kpp,B̃ |A+
kn+1 ,ρ̃n |
n
2pn
k(u − kn+1 )+ kpp,Bn
≤γ
(1 − τ )p ρp
p
p
+
p
1− N
+pδ
N
+ γ∗ |Akn+1 ,ρn |
|A+
kn+1 ,ρn | .
Next
k(u − kn )+ kpp,Bn =
≥
Z
Bn
Z
(u − kn )p+ dx ≥
Bn ∩[u>kn+1 ]
Therefore
|A+
kn+1 ,ρn | ≤
Z
Bn ∩[u>kn+1 ]
(kn+1 − kn )p dx ≥
(u − kn )p+ dx
kp +
|A
|.
2np kn+1 ,ρn
2np
k(u − kn )+ kpp,Bn .
kp
(2.4)
(2.5)
(2.6)
Combining these estimates yields
N +p
k(u −
kn+1 )+ kpp,Bn+1
p
2np N
1
p(1+ N
)
≤γ
p k(u − kn )+ kp,B
n
p
p
p
N
(1 − τ ) ρ k
+
Set
Yn =
1
kp
Z
Bn
2np(1+pδ)
γγ∗p p(1+pδ) k(u
k
(u − kn )p+ dx =
−
k(u − kn )+ kpp,Bn
kp
|Bn |
(2.7)
p(1+pδ)
kn )+ kp,Bn .
,
b=2
and rewrite the previous recursive inequalities as
N pδ
p
γbpn
1+ N
pρ
1+pδ
Yn+1 ≤
.
+ γ∗ p Yn
Yn
(1 − τ )p
k
N +p
N
(2.8)
368
10 DEGIORGI CLASSES
Stipulate to take k so large that
k ≥ γ∗ ρN δ ,
k>
Z
Bρ
p1
up+ dx .
(2.9)
p
Then Yn ≤ 1 for all n and YnN ≤ Ynpδ . With these remarks and stipulations,
the previous recursive inequalities take the form
Yn+1 ≤
γbpn
Y 1+pδ
(1 − τ )p n
for all n = 1, 2, . . .
(2.10)
From the fast geometric convergence Lemma 15.1 of Chapter 9, it follows that
{Yn } → 0 as n → ∞, provided
Z
1
1
1
− 1
Y1 = p
up+ dx ≤ b pδ2 γ − pδ (1 − τ ) δ .
k Bρ
Therefore, taking also into account (2.9), choosing
1
1 Z
p1 b (pδ)2 γ p2 δ
p
u
dx
k = max γ∗ ρN δ ;
1
+
Bρ
(1 − τ ) pδ
one derives
Y∞
1
= p
k
Z
Bτ ρ
(u − k)p+ dx = 0
=⇒
ess sup u+ ≤ k.
Bτ ρ
If γ∗ = 0, then (2.8) are already in the form (2.10) with δ =
1
N.
2.2 Proof of Theorem 2.1 for p = N
The main difference occurs in the application of the embedding inequality
(3.2) of Theorem 3.1 of Chapter 9 to the functions (u − kn+1 )+ ζn over the
balls B̃n to derive inequalities analogous to (2.4). Let q > N to be chosen and
estimate
k(u − kn+1 )+ kpp,Bn+1 ≤ k(u − kn+1 )+ ζn kpp,B̃
n
≤ k(u −
p
q
1− p
q
kn+1 )+ ζn kq,B̃ |A+
kn+1 ,ρ̃n |
n
p
p
1− p
1− p
q
q
|A+
≤ γ(N, q) k∇[(u − kn+1 )+ ζn ]kp,B̃q k(u − kn+1 )+ ζkp,
kn+1 ,ρ̃n |
B̃n
n
2pn
1− p
pδ
q .
≤ γ(N, q)
|A+
k(u − kn+1 )+ kpp,Bn + γ∗ |A+
kn+1 ,ρn |
kn+1 ,ρn |
p
p
(1 − τ ) ρ
Choose q = 2/δ, estimate |A+
kn+1 ,ρn | as in (2.5)–(2.6), and arrive at the analogues of (2.7), which now take the form
3 Hölder Continuity of Functions in the DG Classes
369
p
k(u − kn+1 )+ kpp,Bn+1 ≤ γ
2np(2− 2 δ)
1
p(2− p δ)
k(u − kn )+ kp,Bn 2
p
p
p
p(2−
δ)
2
(1 − τ ) ρ k
p
+ γγ∗
Set
Yn =
1
kp
Z
Bn
2np(1+ 2 δ)
p(1+ p
2 δ)
k(u
−
k
)
k
.
p
n
+
p,Bn
k p(1+ 2 δ)
(u − kn )p+ dx
p
b = 22− 2 δ
and
and rewrite the previous recursive inequalities as
Np
p
2δ
γbpn
1+ p
1+ p
pρ
2 δ+(1− 2 δ)
2δ
Yn+1 ≤
Yn
+ γ∗ p Yn
.
(1 − τ )p
k
Stipulate to take k as in (2.9) with δ replaced by 12 δ, and recast these recursive
inequalities in the form (2.10) with δ replaced by 21 δ.
3 Hölder Continuity of Functions in the DG Classes
For a function u ∈ DG(E, p, γ, γ∗ , δ) and B2ρ (y) ⊂ E set
µ+ = ess sup u,
B2ρ (y)
µ− = ess inf u,
B2ρ (y)
ω(2ρ) = µ+ − µ− = ess osc u.
B2ρ (y)
(3.1)
These quantities are well defined since u ∈ L∞
loc (E).
Theorem 3.1 (DeGiorgi [47]). Let u ∈ DG(E, p, γ, γ∗ , δ). There exist constants C > 1 and α ∈ (0, 1) depending only upon the data and independent of
u, such that for every pair of balls Bρ (y) ⊂ BR (y) ⊂ E
n
ρ α
o
; γ∗ ρN δ .
ω(ρ) ≤ C max ω(R)
(3.2)
R
The Hölder continuity is local to E, with Hölder exponent αo = min{α; N δ}.
An upper bound for the Hölder constant is
{Hölder constant} ≤ C max{2M R−α ; γ∗ },
where
M = kuk∞ .
This implies that the local Hölder estimates deteriorate near ∂E. Indeed, fix
x, y ∈ E and let
R = min{dist{x; ∂E} ; dist{y; ∂E}}.
If |x − y| < R, then (3.2) implies
|u(x) − u(y)| ≤ C max{ω(R)R−αo ; γ∗ }|x − y|αo .
If |x − y| ≥ R, then
|u(x) − u(y)| ≤ 2M R−αo |x − y|αo .
370
10 DEGIORGI CLASSES
Corollary 3.1 Let u be a local weak solution of (1.1)–(1.4). Then for every
compact subset K ⊂ E, and for every pair x, y ∈ K
2MK
|u(x) − u(y)| ≤ C max
; γ∗ |x − y|αo
dist{K; ∂E}α
where MK = ess supK |u|.
3.1 On the Proof of Theorem 3.1
Although the parameters δ and p are fixed, in view of the value of δ in (1.5),
which naturally arises from quasi-linear equations, we will assume δ ≤ N1 . The
value δ = N1 would occur if ε → ∞ in the integrability requirements (1.3).
For homogeneous DG classes γ∗ = 0, while immaterial, we take δ = 1/N .
The proof will be carried on for 1 < p < N . The case p = N only differs in
the application of the embedding Theorem 3.1 of Chapter 9, and the minor
modifications needed to cover this case can be modeled after almost identical
arguments in Section 2.2 above. In what follows we assume that u ∈ DG is
given, the ball B2ρ (y) ⊂ E is fixed, µ± and ω(2ρ) are defined as in (3.1), and
denote by ω any number larger than ω(2ρ).
4 Estimating the Values of u by the Measure of the Set
Where u Is Either Near µ+ or Near µ−
Proposition 4.1 For every a ∈ (0, 1), there exists ν ∈ (0, 1) depending only
on the data and a, but independent of ω, such that if for some ε ∈ (0, 1)
[u > µ+ − εω] ∩ Bρ (y) ≤ ν|Bρ |
(4.1)+
then either εω ≤ γ∗ ρN δ or
u ≤ µ+ − aεω
a.e. in B 21 ρ (y).
(4.2)+
[u < µ− + εω] ∩ Bρ (y) ≤ ν|Bρ |
(4.1)−
Similarly, if
then either εω ≤ γ∗ ρN δ or
u ≥ µ− + aεω
a.e. in B 12 ρ (y).
(4.2)−
Proof. We prove only (4.1)+ –(4.2)+ , the arguments for (4.1)− –(4.2)− being
analogous. Set y = 0 and consider the sequence of balls {Bn } and {B̃n }
introduced in (2.2) for τ = 12 and the cutoff functions ζn introduced in (2.3).
For n ∈ N, introduce also the increasing levels {kn }, the nested sets {An },
and their relative measure {Yn } by
5 Reducing the Measure of the Set Where u is Either Near µ+ or Near µ−
kn = µ+ − aεω −
1−a
εω,
2n
An = [u > kn ] ∩ Bn ,
Yn =
371
|An |
.
|Bn |
Apply (1.6) to (u − kn )+ over the pair of concentric balls B̃n ⊂ Bn , for which
(1 − σ) = 2−n , to get
k∇(u − kn )+ kpp,B̃ ≤
n
p
γ2np
k(u − kn )+ kpp,Bn + γ∗p |An |1− N +pδ .
ρp
If 1 < p < N , by the embedding (3.1) of Theorem 3.1 of Chapter 9
p
(1 − a)εω
|An+1 | = (kn+1 − kn )p |An+1 | ≤ k(u − kn )+ ζn kpp,B̃
n
2n+1
p
p
≤ k(u − kn )+ ζn kpp∗ ,B̃ |An | N ≤ k∇[(u − kn )+ ζn ]kpp,B̃ |An | N
n
n
np
p
p
γ2
p
+pδ
p
1− N
|An | N
k(u − kn )+ kp,Bn + γγ∗ |An |
≤
ρp
p
γ2np εω p
|An |1+ N + γγ∗p |An |1+pδ .
≤ p
n
ρ
2
From this, in dimensionless form, in terms of Yn one derives
p
p
γ2np
γ2np
γ∗ ρN δ
1+ N
1+pδ
Yn+1 ≤
≤
Y
Y
Y 1+pδ
+
n
n
(1 − a)p
εω
(1 − a)p n
provided εω > γ∗ ρN δ . It follows from these recursive inequalities that {Yn } →
0 as n → ∞, provided (Lemma 15.1 of Chapter 9)
Y1 =
[u > µ+ − εω] ∩ Bρ
(1 − a)1/δ def
≤ 1/pδ 1/pδ2 = ν.
|Bρ |
γ
2
(4.3)
Remark 4.1 This formula provides a precise dependence of ν on a and the
data. In particular, ν is independent of ε.
5 Reducing the Measure of the Set Where u is Either
Near µ+ or Near µ−
Proposition 5.1 Assume that
[u ≤ µ+ − 12 ω] ∩ Bρ ≥ θ|Bρ |
(5.1)+
for some θ ∈ (0, 1). Then for every ν ∈ (0, 1) there exists ε ∈ (0, 1) that can
be determined a priori only in terms of the data and θ, and independent of ω,
such that either εω ≤ γ∗ ρN δ or
[u > µ+ − εω] ∩ Bρ ≤ ν Bρ .
(5.2)+
372
10 DEGIORGI CLASSES
Similarly, if
[u ≥ µ− + 21 ω] ∩ Bρ ≥ θ|Bρ |
(5.1)−
for some θ ∈ (0, 1), then for every ν ∈ (0, 1) there exists ε ∈ (0, 1) depending
only on the data and θ, and independent of ω, such that either εω ≤ γ∗ ρN δ or
[u < µ− + εω] ∩ Bρ ≤ ν Bρ .
(5.2)−
5.1 The Discrete Isoperimetric Inequality
Proposition 5.2 Let E be a bounded convex open set in RN , let u ∈ W 1,1 (E),
and assume that |[u = 0]| > 0. Then
kuk1 ≤ γ(N )
(diam E)N +1
k∇uk1 .
|[u = 0]|
(5.3)
Proof. For almost all x ∈ E and almost all y ∈ [u = 0]
Z
|u(x)| =
|y−x|
0
∂
u(x + nρ)dρ ≤
∂ρ
Z
|y−x|
0
|∇u(x + nρ)|dρ,
n=
x−y
.
|x − y|
Integrating in dx over E and in dy over [u = 0] gives
Z Z
Z |y−x|
|[u = 0]| kuk1 ≤
|∇u(x + nρ)|dρdy dx.
E
[u=0]
0
The integral over [u = 0] is computed by introducing polar coordinates with
center at x. Denoting by R(x, y) the distance from x to ∂E along n
Z
[u=0]
Z
0
≤
|y−x|
Z
|∇u(x + nρ)|dρdy
R(x,y)
sN −1 ds
0
Z
|n|=1
Z
R(x,y)
0
|∇u(x + nρ)|dρdn.
Z Z
|∇u(y)|
dydx.
|x − y|N −1
Combining these remarks, we arrive at
1
|[u = 0]| kuk1 ≤ (diam E)N
N
E
E
Inequality (5.3) follows from this, since
Z
dx
sup
≤ ωN diam E.
|x
−
y|N −1
y∈E E
For a real number ℓ and u ∈ W 1,1 (E), set
ℓ
if u > ℓ
uℓ =
u
if u ≤ ℓ.
5 Reducing the Measure of the Set Where u is Either Near µ+ or Near µ−
Apply (5.3) to the function (uℓ − k)+ for k < ℓ to obtain
Z
(diam E)N +1
(ℓ − k)|[u > ℓ]| ≤ γ(N )
|∇u|dx.
|[u < k]|
[k<u<ℓ]
373
(5.4)
This is referred to as a discrete version of the isoperimetric inequality ([47]).
A continuous version is in [77].
5.2 Proof of Proposition 5.1
We will establish (5.2)+ starting from (5.1)+ . Set
ks = µ+ −
1
ω,
2s
As = [u > ks ] ∩ Bρ ,
for s = 1, 2, . . . , s∗
(5.5)
where s∗ is a positive integer to be chosen. Apply (5.4) for the levels ks < ks+1
over the ball Bρ . By virtue of (5.1)+
for all s ∈ N.
[u < ks ] ∩ Bρ ≥ θ|Bρ |
Therefore
ω
2
|As+1 | ≤
s+1
γ(N )
ρ
θ
≤
γ(N )
ρ
θ
Z
As −As+1
Z
Bρ
|∇u|dx
p1
As − As+1
|∇(u − ks )+ |p dx
p−1
p
Take the p-power of both sides and estimate the term involving ∇(u − ks )+ by
making use of the DG classes (1.6) over the pair of balls Bρ ⊂ B2ρ for which
(1 − σ) = 12 . This gives
γ p ρp
ωp
p
|A
|
≤
s+1
2sp
θp
≤
γ p ρN
θp
p N −p+N pδ
As − As+1
+
γ
ρ
∗
ρp
s
p 2
ωp
p−1
Nδ
1
+
γ
ρ
As − As+1
.
∗
2sp
ω
k(u − ks )+ kpp,B2ρ
p−1
Let ε = 2−(s∗ +1) and stipulate that the term in [· · · ] is majorized by 2. Then,
1
power of both sides, this
after we divide through by (ω/2s )p and take the p−1
inequality yields
p
|As+1 | p−1 ≤
p
γ p−1
θ
N
ρ p−1 As − As+1 .
Add both sides over s = 1, . . . , s∗ and observe that the sum on the right-hand
side can be majorized by a telescopic series, which in turn is majorized by
|Bρ |. On the left-hand side the sum is carried over the constant minorizing
term |As∗ +1 |. Thus
374
10 DEGIORGI CLASSES
p
s∗ |As∗ +1 | p−1 ≤
≤
s∗
P
s=1
p
|As | p−1
p
γ p−1
θ
N
ρ p−1
∞
P
s=1
From this
As − As+1 ≤
p
γ p−1
θ
p
|Bρ | p−1 .
1 γ
def
|Bρ | = ν|Bρ |.
θ
|As∗ | ≤
p−1
p
s∗
6 Proof of Theorem 3.1
Consider the assumption (5.1)± with θ = 21 . Since ω ≥ ω(2ρ), by the definitions (3.1)
S
[u ≥ µ− + 12 ω] ∩ Bρ ⊃ Bρ .
[u ≤ µ+ − 21 ω] ∩ Bρ
Therefore not both of (5.1)± can be violated. Assuming the first is in force,
fix the number ν as the one claimed by Proposition 4.1 for the choice a = 12 ,
and then, such a number being fixed, determine s∗ and hence ε = 2−(s∗ +1) by
the procedure of Proposition 5.1. Then by Proposition 4.1, either εω ≤ γρN δ ,
or (4.2)+ holds. The latter implies
ess sup u ≤ ess sup − 21 ε ess osc u.
2
(6.1)
B2ρ
B2ρ
B1ρ
Now
− ess inf u ≤ − ess inf u.
B1ρ
2
B2ρ
Adding these inequalities gives
ω( 12 ρ) ≤ ηω(2ρ),
1
where η = 1 − ε.
2
(6.2)
Let BR (y) ⊂ E be fixed and set ρn = 4−n R. The previous remarks imply that
δ
ω(ρn+1 ) ≤ max{ηω(ρn ) ; ε−1 γ∗ ρN
n }
(6.3)
and by iteration
δ
ω(ρn+1 ) ≤ max{η n ω(R) ; ε−1 γ∗ ρN
n }.
Compute
ρn = 4−n R =⇒ −n = ln
ρ ln14
n
R
=⇒ η n =
ρ α
n
R
for α = −
ln η
.
ln 4
7 Boundary DeGiorgi Classes: Dirichlet Data
375
7 Boundary DeGiorgi Classes: Dirichlet Data
Let ∂E be the finite union of portions of (N − 1)-dimensional surfaces of class
C 1 , so that the trace of a function u ∈ W 1,p (E) can be defined except possibly
1
on an (N −2)-dimensional subset of ∂E. Given ϕ ∈ W 1− p ,p (∂E), the Dirichlet
problem for the quasi-linear equation (1.1) consists in finding u ∈ W 1,p (E)
such that tr(u) = ϕ and u satisfies the PDE in the weak form (1.4), with the
equality sign, for all v ∈ Wo1,p (E). Weak sub(super)-solutions of the Dirichlet
problem are functions u ∈ W 1,p (E) with tr(u) ≤ (≥)ϕ and satisfying (1.4)
for all non-negative v ∈ Wo1,p (E). If ϕ ∈ C(∂E), it is natural to ask whether
a solution of the Dirichlet problem, whenever it exists, is continuous up the
boundary ∂E. The issue builds on the Lebesgue counterexample of Section 7.1
of Chapter 2, and can be rephrased by asking what requirements are needed
on ∂E for the interior continuity of functions in the DG classes to extend up
to ∂E. Assume that ∂E satisfies the property of positive geometric density,
that is, there exist β ∈ (0, 1) and R > 0 such that for all y ∈ ∂E
Bρ (y) ∩ (RN − E) ≥ β Bρ
for all 0 < ρ ≤ R.
(7.1)
Fix y ∈ ∂E, assume up to a possible translation that it coincides with the
origin, and consider nested concentric balls Bσρ ⊂ Bρ for some ρ > 0 and
σ ∈ (0, 1). Let ϕ ∈ C(∂E) and set
ϕ+ (ρ) = sup ϕ,
ϕ− (ρ) = inf ϕ
∂E∩Bρ
∂E∩Bρ
(7.2)
+
−
ωϕ (ρ) = ϕ (ρ) − ϕ (ρ) = osc ϕ.
∂E∩Bρ
Let ζ be a non-negative, piecewise smooth cutoff function, that equals 1 on
Bσρ (y), vanishes outside Bρ (y), and such that |∇ζ| ≤ [(1 − σ)ρ]−1 , and let
u be a local sub(super)-solution of the Dirichlet problem associated to (1.1)
for the given ϕ. In the weak formulation (1.4), take as test functions v, the
localized truncations ±ζ p (u−k)± . While ζ vanishes on ∂Bρ , it does not vanish
of ∂E ∩ Bρ ; however
ζ p (u − k)+ is admissible if k ≥ ϕ+ (ρ)
ζ p (u − k)− is admissible if k ≤ ϕ− (ρ).
(7.3)
Putting these choices in (1.4), all the calculations and estimates of Section 1
can be reproduced verbatim, with the understanding that the various integrals
are now extended over Bρ ∩ E. However, since ζ p (u − k)± ∈ Wo1,p (Bρ ∩ E),
we may regard them as elements of Wo1,p (Bρ ) by defining them to be zero
outside E. Then the same calculations lead to the inequalities (1.6), with
the same stipulations that the various functions vanish outside E and the
various integrals are extended over the full ball Bρ . Given ϕ ∈ C(∂E), the
±
boundary DeGiorgi classes DG±
ϕ = DGϕ (∂E, p, γ, γ∗ , δ) are the collection of
376
10 DEGIORGI CLASSES
all u ∈ W 1,p (E) such that for all y ∈ ∂E and all pairs of balls Bσρ (y) ⊂ Bρ (y)
the localized truncations (u − k)± satisfy (1.6) for all levels k subject to the
−
restrictions (7.3). We further define DGϕ = DG+
ϕ ∩ DGϕ and refer to these
classes as homogeneous if γ∗ = 0.
7.1 Continuity up to ∂E of Functions in the Boundary DG Classes
(Dirichlet Data)
Let R be the parameter in the condition of positive geometric density (7.1).
For y ∈ ∂E consider concentric balls Bρ (y) ⊂ B2ρ (y) ⊂ BR (y) and set
µ+ = ess sup u,
µ− = ess inf u
B2ρ (y)∩E
B2ρ(y) ∩E
+
(7.4)
−
ω(2ρ) = µ − µ = ess osc u.
B2ρ (y)∩E
Let also ωϕ (2ρ) be defined as in (7.2).
Theorem 7.1. Let ∂E satisfy the condition of positive geometric density
(7.1), and let ϕ ∈ C(∂E). Then every u ∈ DGϕ is continuous up to ∂E,
and there exist constants C > 1 and α ∈ (0, 1), depending only on the data
defining the DGϕ classes and the parameter β in (7.1), and independent of ϕ
and u, such that for all y ∈ ∂E and all balls Bρ (y) ⊂ BR (y)
ρ α
n
o
ω(ρ) ≤ C max ω(R)
; ωϕ (2ρ) ; γ∗ ρN δ .
(7.5)
R
The proof of this theorem is almost identical to that of the interior Hölder
continuity, except for a few changes, which we outline next. First, Proposition 4.1 and its proof continue to hold, provided the levels εω satisfy (7.3).
Next, Proposition 5.1 and its proof continue to be in force, provided the levels
ks in (5.5) satisfy the restriction (7.3) for all s ≥ 1. Now either one of the
inequalities
µ+ − 41 ω ≥ ϕ+ ,
µ− + 14 ω ≤ ϕ−
must be satisfied. Indeed, if both are violated
µ+ − 14 ω ≤ ϕ+
and
− µ− − 41 ω ≤ −ϕ− .
Adding these inequalities gives
ω(2ρ) ≤ 2ωϕ (2ρ)
and there is nothing to prove. Assuming the first holds, then all levels ks as
defined in (5.5) satisfy the first of the restrictions (7.3) for s ≥ 2 and thus are
admissible. Moreover, (u − k2 )+ vanishes outside E, and therefore
[u ≤ µ+ − 41 ω] ∩ Bρ ≥ β|Bρ |
8 Boundary DeGiorgi Classes: Neumann Data
377
where β is the parameter in the positive geometric density condition (7.1).
From this, the procedure of Proposition 5.1 can be repeated with the understanding that (u − ks )+ are defined in the full ball Bρ and are zero outside
E. Proposition 5.1 now guarantees the existence of ε as in (5.2)+ and then
Proposition 4.1 ensures that (6.1) holds.
Remark 7.1 If ϕ is Hölder continuous, then u is Hölder continuous up to
∂E.
Remark 7.2 The arguments are local in nature and as such they require only
local assumptions. For example, the positive geometric density (7.1) could be
satisfied on only a portion of ∂E, open in the relative topology of ∂E, and ϕ
could be continuous only on that portion of ∂E. Then the boundary continuity
of Theorem 7.1 continues to hold only locally, on that portion of ∂E.
Corollary 7.1 Let ∂E satisfy (7.1). A solution u of the Dirichlet problem
for (1.1) for a datum ϕ ∈ C(∂E) is continuous in Ē. If ϕ is Hölder continuous in ∂E, then u is Hölder continuous in Ē. Analogous statements hold if
∂E satisfies (7.1) on an open portion of ∂E and if ϕ is continuous (Hölder
continuous) on that portion of ∂E.
8 Boundary DeGiorgi Classes: Neumann Data
Consider the quasi-linear Neumann problem
− div A(x, u, ∇u) + B(x, u, ∇u) = 0
A(x, u, ∇u) · n = ψ
in E
on ∂E
(8.1)
where n is the outward unit normal to ∂E. The functions A and B satisfy
the structure (1.2), and the Neumann datum ψ satisfies

p N −1

if 1 < p < N
q =
p−1 N
(8.2)
ψ ∈ Lq (∂E),
where


any q > 1
if p = N.
A weak sub(super)-solution to (8.1) is a function u ∈ W 1,p (E) such that
Z
Z
ψv dσ
(8.3)
[A(x, u, ∇u)∇v + B(x, u, ∇u)v] dx ≤ (≥)
E
∂E
for all non-negative v ∈ W 1,p (E), where dσ is the surface measure on ∂E.
All terms on the left-hand side are well defined by virtue of the structure
conditions (1.2), whereas the boundary integral on the right-hand side is well
defined by virtue of the trace inequalities of Proposition 8.2 of Chapter 9.
In defining boundary DG classes for the Neumann data ψ, fix y ∈ ∂E,
assume without loss of generality that y = 0, and introduce a local change
378
10 DEGIORGI CLASSES
of coordinates by which ∂E ∩ BR for some fixed R > 0 coincides with the
hyperplane xN = 0, and E lies locally in {xN > 0}. Setting
Bρ+ = Bρ ∩ [xN > 0]
for all 0 < ρ ≤ R
+
+
we require that all “concentric” 21 -balls Bσρ
⊂ Bρ+ ⊂ BR
be contained in
E. Denote by ζ a non-negative piecewise smooth cutoff function that equals
1 on Bσρ (y), vanishes outside Bρ (y), and such that |∇ζ| ≤ [(1 − σ)ρ]−1 .
Notice that ζ vanishes on ∂Bρ and not on ∂Bρ+ . Let u be a local sub(super)solution of (8.1) in the sense of (8.3), and in the latter take the test functions
v = ±ζ p (u−k)± ∈ W 1,p (E). Carrying on the same estimations as in Section 1,
we arrive at integral inequalities analogous to (1.6) with the only difference
+
that the various integrals are extended over Bσρ
and Bρ+ , and that the righthand side contains the boundary term arising from the right-hand side of
(8.3). Precisely
γ
k(u − k)± kpp,B +
k∇(u − k)± ζkpp,B + ≤
ρ
ρ
(1 − σ)p ρp
Z
(8.4)
p
p
1− N +pδ
+
ψ(u
−
k)
ζ
dx̄
+ γ∗p |A±
|
±
k,ρ
xN =0
where δ is given by (1.5), γ∗p is defined in (1.7), the sets A±
k,ρ are redefined
accordingly, and x̄ = (x1 , . . . , xN −1 ). The requirement (8.2) merely ensures
that (8.3) is well defined. The boundary DG classes for Neumann data ψ
require a higher order of integrability of ψ. We assume that

N −1 N +ε


if 1 < p < N
q =
p−1
N
(8.5)
ψ ∈ Lq (∂E),
where



any q > 1
if p = N
for some ε > 0. Using such a q, define p̄ > 1 by
1 N
N p̄
1
,
p̄∗ =
,
1− = ∗
q
p̄ N − 1
N − p̄
1
p̄ − 1 N
=
.
q
p̄ N − 1
One verifies that for these choices, 1 < p̄ < p ≤ N and the trace inequality
(8.3) of Chapter 9 can be applied. With this stipulation, estimate the last
integral as
Z
ψ(u − k)± ζ p dσ
xN =0
≤ kψkq;∂E k(u − k)± ζkp̄∗ N −1 ;∂E
N
≤ kψkq;∂E k∇[(u − k)± ζ]kp̄ + 2γk(u − k)± ζkp̄
1
1
p̄ − p
≤ kψkq;∂E k∇[(u − k)± ζ]kp + 2γk(u − k)± ζkp |A±
k,ρ |
1
≤ k∇(u − k)± ζkpp,B + + k(u − k)± (ζ + |∇ζ|)kpp,B +
ρ
ρ
2
p
1
1
p
p−1
( p̄ − p ) p−1
|A±
.
+ γ(N, p)kψkq;∂E
k,ρ |
8 Boundary DeGiorgi Classes: Neumann Data
379
Combining this with (8.4) and stipulating ρ ≤ 1 gives
k∇(u − k)± kpp,B + ≤
σρ
p
γ
p
1− N
+pδ
k(u − k)± kpp,B + + γ∗∗
|A±
k,ρ |
ρ
(1 − σ)p ρp
(8.6)
where δ is given by (1.5), and
p
p
p−1
p
γ∗∗
= γ(N, p) kf kpN +ε + kfo k p−1
N +ε + kψkq;∂E .
(8.7)
p
Given ψ ∈ Lq (∂E) as in (8.5), the boundary DeGiorgi classes DG±
ψ =
±
1,p
DGψ (∂E, p, γ, γ∗∗ , δ) are the collection of all u ∈ W (E) such that for all
+
y ∈ ∂E and all pairs of 12 -balls Bσρ
(y) ⊂ Bρ+ (y) for ρ < R, the localized
−
truncations (u − k)± satisfy (8.6). We further define DGψ = DG+
ψ ∩ DGψ and
refer to these classes as homogeneous if γ∗∗ = 0.
8.1 Continuity up to ∂E of Functions in the Boundary DG Classes
(Neumann Data)
Having fixed y ∈ ∂E, assume after a flattening of ∂E about y that ∂E coincides with the hyperplane xN = 0 within a ball BR (y). Consider the “concen+
+
tric” 12 -balls Bρ+ (y) ⊂ B2ρ
(y) ⊂ BR
(y) and set
µ+ = ess sup u,
+
B2ρ
(y)
µ− = ess
inf u,
+
B2ρ (y)
ω(2ρ) = µ+ − µ− = ess+ osc u.
(8.8)
B2ρ (y)
Theorem 8.1. Let ∂E be of class C 1 satisfying the segment property. Then
every u ∈ DGψ is continuous up to ∂E, and there exist constants C > 1 and
α ∈ (0, 1), depending only on the data defining the DGψ classes and the C 1
structure of ∂E, and independent of ψ and u, such that for all y ∈ ∂E and
+
(y)
all 21 -balls Bρ+ (y) ⊂ BR
n
ρ α
o
ω(ρ) ≤ C max ω(R)
(8.9)
; γ∗∗ ρN δ .
R
The proof of this theorem is almost identical to that of the interior Hölder
continuity, the only difference being that we are working with “concentric”
1
2 -balls instead of balls. Proposition 4.1 and its proof continue to hold. Since
(u − k)± ζ do not vanish on ∂Bρ+ , the embedding Theorem 2.1 of Chapter 9
is used instead of the multiplicative embedding. Next, Proposition 5.1 relies
on the discrete isoperimetric inequality of Proposition 5.2, which holds for
convex domains, and thus for 21 -balls. The rest of the proof is identical with
the indicated change in the use of the embedding inequalities.
Remark 8.1 The regularity of ψ enters only in the requirement (8.5) through
the constant γ∗∗ .
Remark 8.2 The arguments are local in nature, and as such they require
only local assumptions.
380
10 DEGIORGI CLASSES
Corollary 8.1 Let ∂E be of class C 1 satisfying the segment property. A weak
solution u of the Neumann problem for (8.1) for a datum ψ satisfying (8.5),
is Hölder continuous in Ē. Analogous local statements are in force, if the
assumptions on ∂E and ψ hold on portions of ∂E.
9 The Harnack Inequality
Theorem 9.1 ([53, 48]). Let u ∈ DG(E, p, γ, γ∗ , δ) be non-negative. There
exists a positive constant c∗ that can be quantitatively determined a priori in
terms of only the parameters N, p, γ and independent of u, γ∗ , and δ such that
Nδ
or
for every ball B4ρ (y) ⊂ E, either u(y) ≤ c−1
∗ γ∗ ρ
c∗ u(y) ≤ inf u.
Bρ (y)
(9.1)
This inequality was first proved for non-negative harmonic functions (Section 5.1 of Chapter 2). Then it was shown to hold for non-negative solutions
of quasi-linear elliptic equations of the type of (1.1) ([186, 235, 261]). It is
quite remarkable that they continue to hold for non-negative functions in the
DG classes, and it raises the still unsettled question of the structure of these
classes, versus Harnack estimates, and weak forms of the maximum principle.
The first proof of Theorem 9.1 is in [53]. A different proof that avoids
coverings is in [48]. This is the proof presented here, in view of its relative
flexibility.
9.1 Proof of Theorem 9.1. Preliminaries
Fix B4ρ (y) ⊂ E, assume u(y) > 0, and introduce the change of function and
variables
u
x−y
w=
,
x→
.
u(y)
ρ
Then w(0) = 1, and w belongs to the DG classes relative to the ball B4 , with
the same parameters as the original DG classes, except that γ∗ is now replaced
by
γ∗
Γ∗ = (2ρ)N δ
.
(9.2)
u(y)
In particular, the truncations (w − k)± satisfy
k∇(w−k)± kpp,Bσr (x∗ ) ≤
p
γ
+pδ
1− N
(9.3)
k(w−k)± kpp,Br (x∗ ) +Γ∗p |A−
k,r |
p
p
(1 − σ) r
for all Br (x∗ ) ⊂ B4 and for all k > 0. By these transformations, (9.1) reduces
to finding a positive constant c∗ that can be determined a priori in terms of
only the parameters of the original DG classes, such that
c∗ ≤ max{inf w ; Γ∗ }.
B1
(9.4)
9 The Harnack Inequality
381
9.2 Proof of Theorem 9.1. Expansion of Positivity
Proposition 9.1 Let M > 0 and B4r (x∗ ) ⊂ B4 . If
[w ≥ M ] ∩ Br (x∗ ) ≥ 21 |Br |
(9.5)
then for every ν ∈ (0, 1) there exists ε ∈ (0, 1) depending only on the data and
ν, and independent of Γ∗ , such that either εM ≤ Γ∗ rN δ or
[w < 2εM ] ∩ B4r (x∗ ) ≤ ν B4r .
(9.6)
As a consequence, either εM ≤ Γ∗ rN δ or
w ≥ εM
in B2r (x∗ ).
(9.7)
Proof. The assumption (9.5) implies that
[w ≥ M ] ∩ B4r (x∗ ) ≥ θ|B4r |,
where θ =
1 1
.
2 4N
Then Proposition 5.1 applied for such a θ and for ρ replaced by 4r implies
that (9.6) holds, for any prefixed ν ∈ (0, 1). This in turn implies (9.7), by
virtue of Proposition 4.1, applied with ρ replaced by 4r.
Remark 9.1 Proposition 4.1 is a “shrinking” proposition, in that information on a ball Bρ , yields information on a smaller ball B 12 ρ . Proposition 9.1
is an “expanding” proposition in the sense that information on a ball Br (x∗ )
yields information on a larger ball B2r (x∗ ). This “expansion of positivity” is
at the heart of the Harnack inequality (9.1).
9.3 Proof of Theorem 9.1
For s ∈ [0, 1) consider the balls Bs and the increasing families of numbers
Ms = sup u,
Bs
Ns = (1 − s)−β
where β > 0 is to be chosen. Since w ∈ L∞ (B2 ), the net {Ms } is bounded.
One verifies that
Mo = No = 1,
lim Ms < ∞,
s→1
and
lim Ns = ∞.
s→1
Therefore the equation Ms = Ns has roots, and we denote by s∗ the largest
of these roots. Since w is continuous in B2 , there exists x∗ ∈ Bs∗ such that
sup w = w(x∗ ) = (1 − s∗ )−β .
Bs∗ (x∗ )
Also, since s∗ is the largest root of Ms = Ns
382
10 DEGIORGI CLASSES
sup w ≤
BR (x∗ )
1 − s∗
2
−β
,
where
R=
1 − s∗
.
2
By virtue of the Hölder continuity of w, in the form (3.2), for all 0 < r < R
and for all x ∈ Br (x∗ )
r α
Nδ
+ Γ∗ r
w(x) − w(x∗ ) ≥ −C
sup w − inf w
R
BR (x∗ )
BR (x∗ )
(9.8)
r α
h
i
β
−β
Nδ
≥ −C 2 (1 − s∗ )
+ Γ∗ r
.
R
Next take r = ǫ∗ R, and then ǫ∗ so small that
(
N δ )
1
1
−
s
∗
Nδ
≤ (1 − s∗ )−β .
C 2β (1 − s∗ )−β ǫα
∗ + Γ∗ ǫ∗
2
2
The choice of ǫ∗ depends on C, α, Γ∗ , N, δ, which are quantitatively determined
parameters; it depends also on β, which is still to be chosen; however the choice
of ǫ∗ can be made independent of s∗ . For these choices
1
1
def
w(x) ≥ w(x∗ ) − (1 − s∗ )−β = (1 − s∗ )−β = M
2
2
for all x ∈ Br (x∗ ). Therefore
[w ≥ M ] ∩ Br (x∗ ) ≥
1
2
Br .
(9.9)
From this and Proposition 9.1, there exists ε ∈ (0, 1) that can be quantitatively
determined in terms of only the nonhomogeneous parameters in the DG classes
and is independent of β, r, Γ∗ , and w such that either
εM ≤ Γ∗ rN δ ,
where
r = 21 ǫ∗ (1 − s∗ )
or
w ≥ εM
on
B2r (x∗ ).
Iterating this process from the ball B2j r (x∗ ) to the ball B2j+1 r (x∗ ) gives the
recursive alternatives, either
εj M ≤ Γ∗ (2j r)N δ
or
w > εj M
on B2j+1 r (x∗ ).
(9.10)
After n iterations, the ball B2n+1 r (x∗ ) will cover B1 if n is so large that
2 ≤ 2n+1 r = 2n+1 12 ǫ∗ (1 − s∗ ) ≤ 4
from which
2εn M = εn (1 − s∗ )−β ≤ (2β ε)n ǫβ∗ ≤ 2β εn (1 − s∗ )−β = 2β+1 εn M.
(9.11)
10 Harnack Inequality and Hölder Continuity
383
In these inequalities, all constants except s∗ and β are quantitatively determined a priori in terms of only the nonhomogeneous parameters of the DG
classes. The parameter ǫ∗ depends on β but is independent of s∗ . The latter is
determined only qualitatively. The remainder of the proof consists in selecting
β so that the qualitative parameter s∗ is eliminated. Select β so large that
ε2β = 1. Such a choice determines ǫ∗ , and
def
εn M = εn 12 (1 − s∗ )−β ≥ 2−(β+1) ǫβ∗ = c∗ .
Returning to (9.10), if the first alternative is violated for all j = 1, 2, . . . , n,
then the second alternative holds recursively and gives
w ≥ ε n M ≥ c∗
in B1 .
If the first alternative holds for some j ∈ {1, . . . , n}, then a fortiori it holds
for j = n, which, taking into account the definition (9.2) of Γ∗ and (9.11),
implies
c∗ u(y) ≤ γ∗ (2ρ)N δ .
10 Harnack Inequality and Hölder Continuity
The Hölder continuity of a function u in the DG classes in the form (3.2) has
been used in an essential way in the proof of Theorem 9.1. For non-negative
solutions of elliptic equations, the Harnack estimate can be established independent of the Hölder continuity, and indeed, the former implies the latter
([186]).
Let µ± and ω(2ρ) be defined as in (3.1). Applying Theorem 9.1 to the two
non-negative functions w+ = µ+ − u and w− = u − µ− , gives either
Nδ
ess sup w+ = µ+ − ess inf u ≤ c−1
∗ γ∗ ρ
Bρ (y)
Bρ (y)
−
Nδ
ess sup w = ess sup u − µ− ≤ c−1
∗ γ∗ ρ
Bρ (y)
or
(10.1)
Bρ (y)
c∗ (µ+ − ess inf u) ≤ µ+ − ess sup u
Bρ (y)
Bρ (y)
−
c∗ (ess sup u − µ ) ≤ ess inf u − µ− .
Bρ (y)
(10.2)
Bρ (y)
If either one of (10.1) holds, then
Nδ
ω(ρ) ≤ ω(2ρ) ≤ c−1
.
∗ γ∗ ρ
Otherwise, both inequalities in (10.2) are in force. Adding them gives
c∗ ω(2ρ) + c∗ ω(ρ) ≤ ω(2ρ) − ω(ρ).
(10.3)
384
10 DEGIORGI CLASSES
From this
ω(ρ) ≤ ηω(2ρ),
where
η=
1 − c∗
.
1 + c∗
(10.4)
The alternatives (10.3)–(10.4) yield recursive inequalities of the same form
as (6.3), from which the Hölder continuity follows. These remarks raise the
question whether the Harnack estimate for non-negative functions in the DG
classes can be established independently of the Hölder continuity. The link
between these two facts rendering them essentially equivalent, is the next
lemma of real analysis.
11 Local Clustering of the Positivity Set of Functions in
W 1,1 (E)
For R > 0, denote by KR (y) ⊂ RN a cube of edge R centered at y and
with faces parallel to the coordinate planes. If y is the origin on RN , write
KR (0) = KR .
Lemma 11.1 ([54]) Let v ∈ W 1,1 (KR ) satisfy
kvkW 1,1 (KR ) ≤ γRN −1
and
|[v > 1]| ≥ ν|KR |
(11.1)
for some γ > 0 and ν ∈ (0, 1). Then for every ν∗ ∈ (0, 1) and 0 < λ < 1,
there exist x∗ ∈ KR and ǫ∗ = ǫ∗ (ν, ν∗ , λ, γ, N ) ∈ (0, 1) such that
|[v > λ] ∩ Kǫ∗ R (x∗ )| > (1 − ν∗ )|Kǫ∗ R |.
(11.2)
Remark 11.1 Roughly speaking, the lemma asserts that if the set where u
is bounded away from zero occupies a sizable portion of KR , then there exists
at least one point x∗ and a neighborhood Kǫ∗ R (x∗ ) where u remains large in
a large portion of Kǫ∗ R (x∗ ). Thus the set where u is positive clusters about
at least one point of KR .
Proof (of Lemma 11.1). It suffices to establish the lemma for u continuous and
R = 1. For n ∈ N partition K1 into nN cubes, with pairwise disjoint interior
and each of edge 1/n. Divide these cubes into two finite sub-collections Q+
and Q− by
Qj ∈ Q+ ⇐⇒ |[v > 1] ∩ Qj | > 12 ν|Qj |
Qi ∈ Q−
⇐⇒
|[v > 1] ∩ Qi | ≤ 21 ν|Qi |
and denote by #(Q+ ) the number of cubes in Q+ . By the assumption
P
P
|[v > 1] ∩ Qi | > ν|K1 | = νnN |Q|
|[v > 1] ∩ Qj | +
Qj ∈Q+
Qi ∈Q−
where |Q| is the common measure of the Qℓ . From the definitions of Q±
11 Local Clustering of the Positivity Set of Functions in W 1,1 (E)
νnN <
P
Qj ∈Q+
385
P |[v > 1] ∩ Qi |
|[v > 1] ∩ Qj |
+
|Qj |
|Qi |
Qi ∈Q−
< #(Q+ ) + 12 ν(nN − #(Q+ )).
Therefore
ν
nN .
2−ν
#(Q+ ) >
(11.3)
Fix ν∗ , λ ∈ (0, 1). The integer n can be chosen depending on ν, ν∗ , λ, γ, and
N , such that
for some Qj ∈ Q+ .
|[v > λ] ∩ Qj | ≥ (1 − ν∗ )|Qj |
(11.4)
This would establish the lemma for ǫ∗ = 1/n. We first show that if Q is a
cube in Q+ for which
|[v > λ] ∩ Q| < (1 − ν∗ )|Q|
(11.5)
then there exists a constant c = c(ν, ν∗ , λ, N ) such that
kvkW 1,1 (Q) ≥ c(ν, ν∗ , λ, N )
1
.
nN −1
(11.6)
From (11.5)
|[v ≤ λ] ∩ Q| ≥ ν∗ |Q|
h
1
1 + λi
∩ Q > ν|Q|.
v>
2
2
and
For fixed x ∈ [v ≤ λ] ∩ Q and y ∈ [v > (1 + λ)/2] ∩ Q
1−λ
≤ v(y) − v(x) =
2
Z
|y−x|
∇u(x + tn) · ndt,
0
n=
y−x
.
|x − y|
Let R(x, n) be the polar representation of ∂Q with pole at x for the solid angle
n. Integrate the previous relation in dy over [v > (1 + λ)/2] ∩ Q. Minorize the
resulting left-hand side, by using the lower bound on the measure of such a
set, and majorize the resulting integral on the right-hand side by extending
the integration over Q. Expressing such integration in polar coordinates with
pole at x ∈ [v ≤ λ] ∩ Q gives
ν(1 − λ)
|Q| ≤
4
Z
|n|=1
≤N
N/2
Z
0
|Q|
= N N/2 |Q|
R(x,n)
r
Z
|n|=1
Z
Q
N −1
Z
0
Z
|y−x|
0
|∇v(x + tn)| dt dr dn
R(x,n)
|∇v(x + tn)| dt dn
|∇v(z)|
dz.
|z − x|N −1
386
10 DEGIORGI CLASSES
Now integrate in dx over [u ≤ λ] ∩ Q. Minorize the resulting left-hand side
using the lower bound on the measure of such a set, and majorize the resulting
right-hand side, by extending the integration to Q. This gives
Z
νν∗ (1 − λ)
1
1,1
dx
|Q|
≤
kvk
sup
W
(Q)
N −1
4N N/2
z∈Q Q |z − x|
≤ C(N )|Q|1/N kvkW 1,1 (Q)
for a constant C(N ) depending only on N , thereby proving (11.6).
If (11.4) does not hold for any cube Qj ∈ Q+ , then (11.6) is verified for
all such Qj . Adding (11.6) over such cubes and taking into account (11.3)
ν
c(ν, ν∗ , λ, N )n ≤ kukW 1,1 (K1 ) ≤ γ.
2−ν
Remark 11.2 While the lemma has been proved for cubes, by reducing the
number ǫ∗ if needed, we may assume without loss of generality that it continues to hold for balls.
12 A Proof of the Harnack Inequality Independent of
Hölder Continuity
Introduce the same transformations of Section 9.1 and reduce the proof to
establishing (9.4). Following the same arguments and notation of Section 9.3,
for β > 0 to be chosen, let s∗ be the largest root of Ms = Ns and set
M∗ = (1 − s∗ )−β
1 − s∗
R=
4
M ∗ = 2β (1 − s∗ )−β
1 + s∗
Ro =
2
so that M ∗ = 2β M∗ and BR (x) ⊂ BRo for all x ∈ Bs∗ , and
ess sup w ≤ M ∗
BR (x)
for all x ∈ Bs∗ .
Proposition 12.1 There exists a ball BR (x) ⊂ BRo such that either
M ∗ ≤ Γ∗ RN δ
u(y) ≤ γ∗ ρN δ
(12.1)
|[w > M ∗ − εM ∗ ] ∩ BR (x)| > νa |BR |
(12.2)
=⇒
or
where
(1 − a)1/δ
νa = 1/δ 1/pδ2 ,
γ 2
a=ε=
r
1−
1
2β
(12.3)
and where γ is the quantitative constant appearing in (4.3) and dependent
only on the inhomogeneous data of the DG classes.
12 A Proof of the Harnack Inequality Independent of Hölder Continuity
387
Proof. If the first alternative (12.1) holds for some BR (x) ⊂ BRo , there is
nothing to prove. Thus assuming (12.1) fails for all such balls, if (12.2) holds
for some of these balls, there is nothing to prove. Thus we may assume that
(12.1) and (12.2) are both violated for all BR (x) ⊂ BRo . Apply Proposition 4.1
with ε = a and conclude, by the choice of a and νa , that
w < (1 − a2 )M ∗ = M∗
in all balls B 21 R (x) ⊂ BRo .
Thus w < M∗ in Bs∗ , contradicting the definition of M∗ .
A consequence is that there exists BR (x) ⊂ BRo such that
|[v > 1] ∩ BR (x)| > ν|BR |,
where
v=
w
.
(1 − a)M ∗
(12.4)
Write the inequalities (9.3) for w+ (k = 0) over the pair of balls BR (x) ⊂
B2R (x) ⊂ BRo , and then divide the resulting inequalities by [(1 − a)M ∗ ]p .
Taking into account the definition (9.2) of Γ∗ , and (12.2), this gives
k∇vkpp,BR (x) ≤ γ2pβ/2 RN −p .
From this
k∇vk1,BR (x) ≤ γ(β)RN −1 .
(12.5)
Thus the function v satisfies the assumptions of Lemma 11.1, with given and
fixed constants γ = γ(data, β) and ν = νa (data, β). The parameter β > 1
has to be chosen. Applying the lemma for λ = ν = 12 yields the existence of
x∗ ∈ BR (x) and ǫ∗ = ǫ∗ (data, β) such that
|[w > M ] ∩ Br (x∗ )| ≥ 12 |Br |,
where
r = ǫ∗ R
and where M = 41 M∗ . This is precisely (9.9), and the proof can now be
concluded as before.
Remark 12.1 The Hölder continuity was used to ensure, starting from (9.8)
that w is bounded below by M in a sizable portion of a small ball Bǫ∗ R (x∗ )
about x∗ . In that process, the parameter ǫ∗ had to be chosen in terms of the
data and the still to be determined parameter β. Thus ǫ∗ = ǫ∗ (data, β). The
alternative proof based on Lemma 11.1, is intended to achieve the same lower
bound on a sizable portion of Bǫ∗ R (x∗ ). The discussion has been conducted
in order to trace the dependence of the various parameters on the unknown
β. Indeed, also in this alternative argument, ǫ∗ = ǫ∗ (data, β), but this is the
only parameter dependent on β, whose choice can then be made by the very
same argument, which from (9.9) leads to the conclusion of the proof.
11
LINEAR PARABOLIC EQUATIONS IN
DIVERGENCE FORM WITH
MEASURABLE COEFFICIENTS
1 Parabolic Spaces and Embeddings
Let E be a bounded domain in RN with boundary ∂E of class C 1 , and for
0 < T < ∞ let ET denote the cylindrical domain E × (0, T ]. The space
Lr,q (ET ) for q, r ≥ 1 is the collection of functions f defined and measurable
in ET such that
Z T Z
rq 1r
kf kq,r;ET =
|f |q dx dτ
< ∞.
0
Lq,r
loc (ET ),
Also, f ∈
[t1 , t2 ] ⊂ (0, T ]
E
if for every compact subset K ⊂ E and every sub-interval
Z
t2
t1
Z
K
rq
|f |q dx dτ < ∞.
Whenever q = r we set Lq,q (ET ) = Lq (ET ). These definitions are extended in
the obvious way when either q or r is infinity.
We introduce spaces of functions, depending on (x, t) ∈ ET , that exhibit
different behavior in the space and time variables. These are spaces where
solutions of parabolic equations are typically found.
The set of all functions that are continuous in ET is denoted by C(ET ).
Given two points (x1 , t1 ), (x2 , t2 ) ∈ ET , we define
def
1
d((x1 , t1 ), (x2 , t2 )) = |x1 − x2 | + |t1 − t2 | 2 ,
α
and for α ∈ (0, 1), we let C α, 2 (ET ) be the subspace of C(ET ) consisting of
all functions f such that the norm
kf kC α, α2 (ET ) =
sup
(x,t)∈ET
|f (x, t)| +
sup
(xi ,ti )∈ET
(x1 ,t1 )6=(x2 ,t2 )
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_12
|f (x1 , t1 ) − f (x2 , t2 )|
[d((x1 , t1 ), (x2 , t2 ))]α
389
390
11 LINEAR PARABOLIC EQUATIONS
is finite. This is the space of the so-called Hölder continuous functions. Finally,
C 2,1 (ET ) is the set of all continuous functions in ET having continuous derivatives fxi , fxi xj with i, j = 1, . . . , N , ft in ET . The previous definitions can
be extended from ET to ET ; moreover, exactly as we did above for Lq,r (ET )
spaces, we say that
αα
2,1
(ET ),
f ∈ Cloc (ET ), Cloc2 (ET ), Cloc
whenever the previous definitions hold on any compact set K ⊂ ET .
Let m, p ≥ 1 and consider the Banach spaces
V m,p (ET ) = L∞ 0, T ; Lm(E) ∩ Lp 0, T ; W 1,p(E) ,
Vom,p (ET ) = L∞ 0, T ; Lm(E) ∩ Lp 0, T ; Wo1,p(E)
both equipped with the norm
kvkV m,p (ET ) = ess sup kv(·, t)km;E + kDvkp;ET .
0<t<T
When m = p, set Vop,p (ET ) = Vop (ET ) and V p,p (ET ) = V p (ET ). In the following we will be interested in V 2 (ET ) and Vo2 (ET ). Both spaces are embedded
in Lq (ET ) for some q > 2. In a precise way we have the following.
Proposition 1.1 There exists a constant γ depending only on N such that
for every v ∈ Vo2 (ET )
ZZ
ZZ
|v(x, t)|q dx dt ≤ γ q
|Dv(x, t)|2 dx dt
ET
ET
× ess sup
where
q=2
0<t<T
Z
E
N2
|v(x, t)|2 dx ,
(1.1)
N +2
.
N
Moreover,
kvkq;ET ≤ γkvkV 2 (ET ) .
(1.2)
Remark 1.1 The multiplicative inequality (1.1) and the embedding (1.2)
continue to hold for functions v ∈ V 2 (ET ) such that
Z
v(x, t) dx = 0 for a.e. t ∈ (0, T )
E
provided ∂E is piecewise smooth. In such a case the constant γ also depends
on the structure of ∂E, but not on its size.
1 Parabolic Spaces and Embeddings
391
Proposition 1.2 Assume that ∂E is piecewise smooth. There exists a constant γ depending only on N and the structure of E, such that for every
u ∈ V 2 (ET )
q1
T
kvkq;ET ≤ γ 1 +
kvkV 2 (ET ) ,
(1.3)
2
|E| N
where, again,
N +2
.
q=2
N
The next corollaries follow from Propositions 1.1–1.2 by applying Hölder’s
inequality.
Corollary 1.1 There exists a constant γ depending only on N , such that for
every v ∈ Vo2 (ET ),
kvk2;ET ≤ γ [|v| > 0]
1
N +2
kvkV 2 (ET ) .
Corollary 1.2 Assume that ∂E is piecewise smooth. There exists a constant
γ depending only on N and the structure of E, such that for every v ∈ V 2 (ET ),
NN+2
2
T
kvk22;ET ≤ γ 1 +
[|v| > 0] N +2 kvk2V 2 (ET ) .
2
|E| N
Working with Lq,r (ET ) spaces, Propositions 1.1–1.2 take the following form
Proposition 1.3 There exists a constant γ depending only on N , q, r, such
that for every v ∈ Vo2 (ET ),
kvkq,r;ET ≤ γkvkV 2 (ET ) ,
where the numbers q, r ≥ 1 are linked by
N
1 N
+
= ,
r
2q
4
(1.4)
and their admissible range is
if N = 1,
q ∈ (2, ∞],
if N = 2,
q ∈ [2, ∞),
if N > 2,
q ∈ 2, N2N
−2 ,
r ∈ [4, ∞);
r ∈ 2, ∞ ;
(1.5)
r ∈ [2, ∞].
Proposition 1.4 Assume that ∂E is piecewise smooth. There exists a constant γ depending only on N , q, r and the structure of E, such that for every
v ∈ V 2 (ET ),
1r
T
kvkq,r;ET ≤ γ 1 +
kvkV 2 (ET ) ,
(1.6)
2
|E| N
where q and r satisfy (1.4)–(1.5).
392
11 LINEAR PARABOLIC EQUATIONS
We conclude this section by stating a Lemma concerning the truncated functions (v − k)± .
Lemma 1.1 Let v ∈ V 2 (ET ). Then, (v − k)± ∈ V 2 (ET ) for all k ∈ R.
Assume in addition that ∂E is piecewise smooth and that the trace of v(·, t)
on ∂E is essentially bounded and
ess sup kv(·, t)k∞;∂E ≤ M
for some M > 0.
0<t<T
Then, (v − k)± ∈ Vo2 (ET ) for all k ≥ M .
Besides V 2 (ET ) and Vo2 (ET ), in the sequel we will also need the following
functional spaces:
V˜2 (ET ) = C([0, T ]; L2 (E)) ∩ L2 (0, T ; W 1,2 (E)),
V˜2 (ET ) = C([0, T ]; L2 (E)) ∩ L2 (0, T ; W 1,2 (E)),
o
o
W 2 (ET ) = W 1,2 ([0, T ]; L2(E)) ∩ L2 (0, T ; W 1,2 (E)),
Wo2 (ET ) = W 1,2 ([0, T ]; L2(E)) ∩ L2 (0, T ; Wo1,2 (E)).
Finally, Wqk,m (ET ) for integers k, m, and q > 1 is the Banach space consisting of the elements of Lq (ET ) having k generalized space derivatives and m
generalized time derivatives also in Lq (ET ) (see Section 1c of Chapter 9 for
higher order generalized derivatives).
Estimates analogous to (1.6) can be derived for the norm kukq,r;Σ , where
Σ = ∂E × (0, T ].
Proposition 1.5 Assume that ∂E is piecewise smooth. There exists a constant γ depending only on N , q, r and the structure of E, such that for every
v ∈ V 2 (ET ),


! 21 − NN−1
N
q
2
T
 kvkV 2 (ET ) ,
kvkq,r;Σ ≤ γ 1 +
|E|
where the numbers q, r ≥ 1 are linked by
1 N −1
N
+
= ,
r
2q
4
and their admissible range is
if N = 2,
q ∈ [1, ∞),
if N > 2,
q∈
2(N −1)
N
−1) , 2(N
N −2 ,
r ∈ 2, ∞ ;
r ∈ [2, ∞].
2 Weak Formulations
393
1.1 Steklov Averages
Let v ∈ L1 (ET ) and let 0 < h < T . The Steklov averages vh (·, t) and vh̄ (·, t)
are defined by
 Z t+h
1


v(·, τ )dτ for t ∈ (0, T − h],
vh = h t



0,
for t > T − h.
 Z t
1


v(·, τ )dτ

vh̄ = h t−h



0,
for t ∈ (h, T ],
for t < h.
Lemma 1.2 Let v ∈ Lq,r (ET ). Then, as h → 0, vh → v in Lq,r (ET −ε ) for
every ε ∈ (0, T ). If v ∈ C(0, T ; Lq (E)), then vh (·, t) → v(·, t) in Lq (E) for
every t ∈ (0, T − ε) for all ε ∈ (0, T ).
A similar statement holds for vh̄ . The proof of the lemma is straightforward
from the theory of Lp spaces.
2 Weak Formulations
Denote by (aij ) an N × N symmetric matrix with entries aij ∈ L∞ (ET ), and
satisfying the ellipticity condition
λ|ξ|2 ≤ aij (x, t)ξi ξj ≤ Λ|ξ|2
(2.1)
for all ξ ∈ RN and all (x, t) ∈ ET , for some 0 < λ ≤ Λ. The number Λ
is the least upper bound of the eigenvalues of (aij ) in ET , and λ is their
greatest lower bound. A vector-valued function f = (f1 , . . . , fN ) : ET →
RN is said to be in Lploc (ET ), for some p ≥ 1, if all the components fj ∈
Lploc (ET ). Given a scalar function f ∈ L1loc (ET ) and a vector-valued function
f ∈ L1loc (ET ), consider the formal partial differential equation in divergence
form (Section 3.1 of the Preliminaries)
ut − aij uxj xi = f − div f in ET .
(2.2)
Expanding formally the indicated derivatives gives a PDE of the type of (3.1)
of Chapter 1, which, in view of the ellipticity condition (2.1), admits one
family of real characteristic surfaces (Section 3 of Chapter 1). In this formal
sense, (2.2) is a second-order parabolic equation.
Now multiply (2.1) by an arbitrary smooth function ϕ = ϕ(x, t) which
vanishes on ∂E × (0, T ), integrate both sides over ET , and carry out an integration by parts in the terms containing the coefficients aij . As a result, we
obtain
394
11 LINEAR PARABOLIC EQUATIONS
ZZ
(ut ϕ + aij uxj ϕxi )dx dt =
ZZ
(2.3)
(f ϕ + fi ϕxi )dx dt.
ET
ET
As it is easy to find out, (2.2) and (2.3) are equivalent, if all the terms aij , f ,
fi are regular.
The above formulation contains the term ut , which we want to get rid of
(the motivation for doing so lies in the possibility of proving suitable existence
and uniqueness theorems, as we will show in the next paragraphs). If we
perform a further formal integration by parts with respect to the variable t,
we finally arrive at
ZZ
ZZ
Z
T
(f ϕ + fi ϕxi )dx dt. (2.4)
(−uϕt + aij uxj ϕxi )dx dt =
uϕ dx +
0
E
ET
ET
Equation (2.4) is well defined, provided we make the right integrability
asZ
sumptions on u and ∇u. Moreover, we also need to give meaning to
T
uϕ dx.
E
0
Leaving these issues aside for a moment, we say that (2.4) is the weak formulation of (2.3).
3 The Homogeneous Dirichlet Problem
Consider the homogeneous Cauchy–Dirichlet problem
ut − aij uxj xi = f − div f
in ET ,
on ∂E × (0, T ),
in E,
u=0
u(·, 0) = uo
(3.1)
where
with
f ∈ L2 (ET ), uo ∈ L2 (E), f ∈ Lq,r (ET ),

2N


, 2 , r ∈ [1, 2] for N ≥ 3,
q∈


N +2


1 N
N
q ∈ (1, 2], r ∈ [1, 2) for N = 2,
+
=1+
and

r 2q
4


4


q ∈ [1, 2], r ∈ 1,
for N = 1.
3
(3.2)
(3.3)
The PDE is meant in the weak sense by requiring that u ∈ Vo2 (ET ), and
such that for almost all τ ∈ [0, T ] and ∀ ϕ ∈ Wo2 (ET ) the following identity is
satisfied
Z τZ
Z
Z
uϕt dx dt
uo (x)ϕ(x, 0)dx −
u(x, τ )ϕ(x, τ )dx −
0
E
E
E
Z τZ
Z τZ
(3.4)
+
aij uxj ϕxi dx dt =
(fi ϕxi + f ϕ)dx dt.
0
E
0
E
3 The Homogeneous Dirichlet Problem
395
It is a matter of straightforward computations to show that if the integrability
conditions (3.2)–(3.3) are satisfied, then all the integrals in (3.4) are finite for
any functions u and ϕ in the indicated classes.
There is a different but equivalent way to define a weak solution of the
homogeneous Cauchy–Dirichlet problem (3.1): we say that u ∈ Vo2 (ET ) is a
weak solution of (3.1) if
Z TZ
Z TZ
aij uxj ϕxi dx dt
−
uϕt dx dt +
E
E
0
0
(3.5)
Z
Z TZ
=
uo (x)ϕ(x, 0)dx.
(fi ϕxi + f ϕ)dx dt +
0
E
E
for all ϕ ∈ Wo2 (ET ), which vanish for t = T .
Proposition 3.1 The two notions of solutions given in (3.4) and (3.5) are
equivalent.
Proof. That (3.4) implies (3.5) is trivial. Let us show that the opposite implication is also true. In the weak formulation (3.5), consider the test function
ϕǫ (x, t) = ζ(x, t)ηǫ (t),
where ζ ∈ Wo2 (ET ), and ηǫ is a piecewise linear function, which is equal to 1
τ −t
for t ≤ τ − ǫ, vanishes for t ≥ τ , and is equal to
for t ∈ (τ − ǫ, τ ). If we
ǫ
pass to the limit as ǫ → 0, it is apparent that
Z TZ
Z τZ
aij uxj ϕǫ,xi dx dt →
aij uxj ζxi dx dt,
Z
0
0
T
Z
E
E
0
→
(fi ϕǫ,xi + f ϕǫ )dx dt
Z
uo (x)ϕǫ (x, 0)dx
→
E
Z
Z0
E
τ
Z
(fi ζxi + f ζ)dx dt,
E
uo (x)ζ(x, 0)dx.
E
As for the first term in (3.5), we have
Z
Z
Z TZ
Z TZ
1 τ
uζdx dt.
uζt ηǫ dx dt +
−
uϕǫ,t dx dt = −
ǫ τ −ǫ E
0
E
0
E
Since the function
w(t) =
Z
u(x, t)ζ(x, t)dx
E
is summable on [0, T ], we conclude that, possibly up to a subsequence,
Z TZ
Z τZ
Z
−
uϕǫ,t dx dt → −
u(x, τ )ζ(x, τ )dx.
uζt dx dt +
0
E
0
E
E
396
11 LINEAR PARABOLIC EQUATIONS
Remark 3.1 Formulation (3.5) will be useful in the proof of existence of
solutions.
We can also give a formulation in terms of the Steklov averages. Namely, let
0 < h < T . In the weak formulation (3.4) take as test function the Steklov
average
Z
1 t
ϕh̄ (x, t) =
ϕ̂(x, τ ) dτ,
h t−h
where ϕ̂ is an arbitrary element of Wo2 (ET ), which vanishes for t ≥ T − h and
for t ≤ 0. If we take τ = T in (3.4), the first two terms on the left-hand side
vanish because of the assumption on ϕ̂, whereas the third one is transformed
in the following way
ZZ
ZZ
ZZ
uht ϕ dx dt.
uϕh̄t dx dt = −
uh ϕt dx dt =
−
ET
ET
ET
Indeed, for functions v, w ∈ L2 (−h, T ), one of which vanishes in (−h, 0) and
(T − h, T ), we have
Z T −h
Z T
vh w dt,
vwh̄ dt =
0
0
where we recall from Section 1.1 the definition of vh . Moreover, we have interchanged the order of integration with respect to t and τ . In all the other
terms, in a similar way, we transfer the averaging from ϕ to the corresponding
factor, taking into account that averaging and differentiation with respect to
x commute. We conclude
ZZ
ZZ
uht ϕ + (aij uxj )h ϕxi dx dt =
(fh ϕ + fi,h ϕxi ) dx dt.
ET −h
ET −h
By a density argument, it is not difficult to see that the previous equality
holds for any function ϕ that vanishes for t > t1 with t1 ≤ T − h and belongs
to V˜o2 (ET ). Therefore, we end up with the weak formulation
ZZ
ZZ
uht ϕ + (aij uxj )h ϕxi dx dt =
(fh ϕ + fi,h ϕxi ) dx dt
(3.6)
Et1
Et1
for any function ϕ ∈ V˜o2 (Et1 ), provided that t1 ≤ T − h.
4 The Energy Inequality
We have the following result, which is important in itself, and at the same
time plays a fundamental role in the existence proof for (3.1)–(3.3).
5 Existence of Solutions by Galerkin Approximations
397
Proposition 4.1 Let u ∈ V˜o2 (ET ) be a solution of the homogeneous Cauchy–
Dirichlet problem (3.1) with data satisfying (3.2)–(3.3). Then, the Energy
Inequality
h
i
kukV 2 (ET ) ≤ γ̃ kuo k2;E + kf k2;ET + kf kq,r;ET
(4.1)
holds true, where γ̃ > 0 depends only on N and λ.
Proof. In (3.6) choose ϕ = uh , take into account that
ZZ
Z
1
uht uh dx dt =
u2 (x, t)dx
2 E h
Et1
t=t1
t=0
and let h → 0. The regularity of all the functions guarantees that this can be
done, and yields
Z
Z t1 Z
Z t1 Z
t=t1
1
(f u + fi uxi )dx dt.
u2 (x, t)dx
+
aij uxj uxi dx dt =
2 E
t=0
0
E
0
E
Z
t=t1
The convergence of the term
u2h (x, t)
is ensured by the continuity of
t=0
E
u with respect to t in L2 (E), which we have assumed.
By the ellipticity condition (2.1) and the integrability conditions (3.2)–
(3.3), if q ′ and r′ denote the Hölder conjugate exponent of q and r, we obtain
Z t1 Z
Z
1
|∇u|2 dx dt
u2 (x, t1 )dx + λ
2 E
0
E
12
Z t1 Z
12 Z t1 Z
Z
1
2
2
2
≤
|∇u| dx dt
u (x, 0)dx +
|f | dx dt
2 E
0
E
0
E

 1′
r
r′′
qr ! r1 Z t1 Z
Z t1 Z
q
′

dt
dt
+
|u|q dx
|f |q dx
0
≤
1
2
Z
E
0
E
E
u2 (x, 0)dx + γ kf k2;Et1 + kf kq,r;Et1 kukV 2 (Et1 ) ,
′
′
since by Proposition 1.3 we have u ∈ Lq ,r (ET ) and kukq′ ,r′ ;Et1 ≤ γkukV 2 (Et1 ) .
Hence,
i
h
kuk2V 2 (Et ) ≤ γ̃ kuo k2;E + kf k2;Et1 + kf kq1 ,r1 ;Et1 kukV 2 (Et1 ) .
1
Now, choose t1 = T on the right-hand side, and by the arbitrariness of t1 take
the supremum on the left-hand side, to conclude.
5 Existence of Solutions of the Homogeneous
Cauchy–Dirichlet Problem (3.1) by Galerkin
Approximations
We finally come to the solvability of the homogeneous Dirichlet problem (3.1).
398
11 LINEAR PARABOLIC EQUATIONS
Theorem 5.1. If conditions (3.2)–(3.3) are satisfied, then Problem (3.1) has
a solution in Vo2 (ET ).
Proof. Let {wk } be a countable, complete, orthonormal system for Wo1,2 (E).
The existence of a such a system follows by the same argument as in Section 7
of Chapter 9. For the sake of simplicity, we assume it to be normalized with
respect to the inner product in L2 (E), that is (wi , wj ) = δij . We look for an
approximate solution uM of (3.1) of the form
uM (x, t) =
M
X
cM
k (t)wk (x),
k=1
M
where the coefficients cM
k (t) = (u (x, t), wk (x)) are determined from the
Cauchy problem
Z
Z
Z
d
dx
=
(fi wk,xi + f wk ) dx,
(5.1)
uM wk dx +
aij uM
w
xj k,xi
dt E
E
E
Z
uo wk dx,
(5.2)
cM
(0)
=
k
E
with k = 1, . . . , M . Conditions (5.1) are a system of linear ordinary differential
equations, which we can rewrite as
d M
c (t) + Akl cM
l (t) = Fk (t),
dt k
where
k = 1, . . . , M,
(5.3)
Z
aij wk,xi wl,xj dx,
Akl =
ZE
(fi wk,xi + f wk ) dx,
Fk =
E
2
and both belong to L (0, T ), as can be easily seen from (2.1) and (3.2)–(3.3).
In Section 5c we discuss a general result, which guarantees that under these
conditions, (5.3)–(5.2) have a unique solution; we can then conclude that the
approximate solutions uM are uniquely determined for any M .
Let us first show that the coefficients cM
k are equibounded. If we multiply
each of (5.1) by cM
,
sum
what
we
have
obtained
over k from 1 to M , and
k
integrate with respect to t from 0 to t1 for any t1 ∈ (0, T ), we obtain
Z t1 Z
Z
Z t1 Z
t1
1
M
M M
M 2
(fi uM
aij uxj uxi dx dt =
|u | (x, t)dx +
xi + f u )dx dt.
2 E
0
0
E
0
E
We can then work as in the proof of the Energy Inequality of Section 4 and
conclude that
i
h
(5.4)
kuM kV 2 (ET ) ≤ γ kf k2;ET + kf kq,r;ET + kuo k2;E ,
5 Existence of Solutions by Galerkin Approximations
399
where γ depends only on N , λ, q, r, and we have taken into account that
kuM (x, 0)k22;E =
M
X
2
.
|ckM (0)|2 ≤ kuo k2;E
k=1
M
From (5.4) it follows that the coefficients cM
k = ck (t) are all bounded by the
same quantity for any k and for any M .
Let us now prove that for fixed k and arbitrary M ≥ k the coefficients ckM
are equicontinuous. It suffices to show that
M
|cM
k (t + h) − ck (t)| ≤ ǫ(h, k),
where ǫ(h, k) tends to zero as h → 0, and depends on h and k, but does not
depend on M . From (5.1)
M
|cM
k (t + h) − ck (t)| ≤
Z
t+h
Z h
i
|aij uM
+
|
|f
w
+
w
|
|f
w
|
i k,xi
k dx dt.
xj k,xi
E
t
By (2.1) and (3.2)–(3.3) we have
Z
t+h
t
E
≤Λ
Z
"Z
t+h
t
Z
t
Z
t+h
t
Z
E
t+h
Z
|aij uM
xj wk,xi |dx dt
E
Z
E
M 2
|∇u | dx dt
|fi wk,xi |dx dt ≤
|f wk |dx dt ≤ γ
# 21 Z
E
"Z
t+h
t
"Z
t
t+h
12
1
|∇wk | dx h 2 ,
2
Z X
N
E i=1
Z
E
2
|fi | dx dt
rq
|f | dx
dt
q
# 21 Z
E
# r1 Z
E
21
1
|∇wk |2 dx h 2 ,
12
1
|∇wk | dx h r′ .
2
Since all the integrals on the right-hand sides tend to zero as h tends to zero,
we conclude that
M
|cM
k (t + h) − ck (t)| ≤ ǫ(h)k∇wk k2;E .
We can now apply the Ascoli–Arzelà Theorem, and for any fixed k, select
i
a subsequence {cM
k } that converges uniformly on [0, T ] to some continuous
function ck , k ≥ 1 as Mi tends to ∞. Notice that, in general, the sequence
{Mi } depends on k. However, such a dependence can be dispensed with, owing
to a diagonal selection process.
With these coefficients ck we build a function
u(x, t) =
∞
X
k=1
ck (t)wk (x).
400
11 LINEAR PARABOLIC EQUATIONS
We claim that u ∈ Vo2 (ET ) and is a weak solution of (3.1).
If we let
Mi
X
Mi
u =
ckMi wk ,
k=1
it is matter of straightforward computations to check that, as Mi → ∞,
uMi → u
uMi → u
∇uMi → ∇u
weakly in L2 (E) uniformly with respect to t,
weakly in L2 (ET ),
weakly in L2 (ET ).
Moreover, by the properties of the weak convergence, u will satisfy
i
h
kukV 2 (ET ) ≤ γ kf k2;ET + kf kq,r;ET + kuo k2;E ,
just as uM , and we conclude that u ∈ Vo2 (ET ).
In order to show that u is a weak solution, multiply each (5.1) by a smooth
function dk (t), which vanishes for t = T , sum over k from 1 to M ′ ≤ M , and
integrate with respect to t from 0 to T . If we let
M′
Φ
′
(x, t) =
M
X
dk (t)wk (x),
k=1
after an integration by parts we have
Z
Z TZ h
i
′
′
=
dt
dx
− uM ΦtM + aij uxMj ΦM
xi
0
0
E
+
T
Z
Z ′
M′
dxdt
fi ΦM
xi + f Φ
E
′
uM (x, 0)ΦM (x, 0)dx.
E
By the weak convergence of the subsequence {uMi } studied above, we can
pass to the limit with respect to M and obtain
Z TZ Z TZ h
i
′
′
M′
M′
dxdt
dx
dt
f
=
+
f
Φ
Φ
− uΦtM + aij uxj ΦM
i
xi
xi
0
0
E
Z E
′
+
uo (x)ΦM (x, 0)dx,
E
where we have taken into account that
Z
Z
′
′
uo (x)ΦM (x, 0)dx.
uM (x, 0)ΦM (x, 0)dx →
E
E
Now, by Lemma 4.12 of Chapter II of Ladyzhenskaya et al. [151], the functions
′
ΦM are dense in the space of all functions ϕ ∈ Wo2 (ET ), which vanish at t = T .
Therefore, by density we can conclude that
5 Existence of Solutions by Galerkin Approximations
Z
T
Z h
E
0
Z
i
− uϕt + aij uxj ϕxi dx dt =
0
+
T
Z
Z
401
(fi ϕxi + f ϕ) dxdt
E
(5.5)
uo (x)ϕ(x, 0)dx,
E
for all ϕ ∈ Wo2 (ET ), which vanish at t = T . By Proposition 3.1 we conclude
that u is a weak solution.
Proposition 5.1 Under the same assumptions as in Theorem 5.1, any solution u ∈ Vo2 (ET ) of Problem (3.1) is strongly continuous in t in the L2 (E)
norm.
Proof. By the previous construction, u ∈ L∞ (0, T ; L2(E)); now we show that
it is actually more regular. We let
Fi = fi − aij uxj ,
so that we can rewrite (5.5) as
−
Z
0
T
Z
E
uϕt dx dt −
Z
uo (x)ϕ(x, 0)dx =
E
Z
T
0
Z
(Fi ϕxi + f ϕ) dx dt.
E
We now extend u, Fi , f onto the infinite cylinder E∞ = E × (−∞, +∞) by
setting


u(x, t) t ∈ [0, T ]
∗
u = u(x, −t) t ∈ [−T, 0) ,


0 |t| > T


Fi (x, t) t ∈ [0, T ]
∗
Fi = − Fi (x, −t) t ∈ [−T, 0) ,


0 |t| > T


f (x, t) t ∈ [0, T ]
∗
f = − f (x, −t) t ∈ [−T, 0) .


0 |t| > T
By the previous definitions we have
ZZ
ZZ
∗
−
u ϕt dx dt =
E∞
E∞
(Fi∗ ϕxi + f ∗ ϕ) dx dt
for any ϕ ∈ Wo2 (ET ), which vanishes for |t| ≥ T . Choosing ω = ω(t), a smooth
function that equals 1 for t ∈ [−(T − δ), T − δ] and 0 for |t| ≥ T , and setting
ϕ(x, t) = ω(t)Φ(x, t), where Φ ∈ Wo2 (E∞ ), yields
ZZ
ZZ
(Fi∗ ωΦxi + f ∗ ωΦ + u∗ ωt Φ) dx dt.
u∗ ωΦt dx dt =
−
E∞
E∞
402
11 LINEAR PARABOLIC EQUATIONS
If we let v = u∗ ω, then
ZZ
ZZ
−
vΦt dx dt =
E∞
E∞
(Fi∗ ωΦxi + (f ∗ ω + u∗ ωt )Φ) dx dt.
Consider η = η(x, t) a function in L2 (0, T ; Wo1,2(E)) and in the above identity
choose as Φ the Steklov average ηh̄ . Shifting the average from the second factor
to the first one, as we have already done in the proof of the Energy Inequality
of Section 4, yields
ZZ
ZZ
vh ηt dx dt =
[(Fi∗ ω)h ηxi + (f ∗ ω + u∗ ωt )h η] dx dt.
−
E∞
E∞
Let η(x, t) = χ(t)ζ(x), where χ is a smooth function of t, and ζ ∈ Wo1,2 (E).
We can then rewrite
Z ∞
Z
χ
−
vh (x, t)ζ(x)dx
t dt
−∞
E
Z ∞
Z
χ
=
dt
[(Fi∗ ω)h ζxi + (f ∗ ω + u∗ ωt )h ζ] dx,
−
Z
∞
−∞
χt (vh , ζ)dt =
Z
−∞
∞
−∞
E
χ [((Fi∗ ω)h , ζxi ) + ((f ∗ ω + u∗ ωt )h , ζ)] dt,
where (·, ·) stands for the inner product in L2 (E). Taking into account the
notion of distributional derivative,
d
(vh , ζ) = ((Fi∗ ω)h , ζxi ) + ((f ∗ ω + u∗ ωt )h , ζ) .
dt
By its definition, vh is strongly continuous in the L2 (E)-norm with respect to
time, and therefore, a fortiori, the inner product (vh , ζ) is continuous in time.
Moreover, for almost all t ∈ (−∞, ∞),
d
(vh , ζ) = (vh,t , ζ)
dt
and
(vh,t , ζ) = ((Fi∗ ω)h , ζxi ) + ((f ∗ ω + u∗ ωt )h , ζ) .
Choosing two different values h1 , h2 gives
(vh1 ,t − vh2 ,t , ζ)
= ((Fi∗ ω)h1 − (Fi∗ ω)h2 , ζxi ) + ((f ∗ ω + u∗ ωt )h1 − (f ∗ ω + u∗ ωt )h2 , ζ) .
Taking ζ = vh1 − vh2 , which is an admissible choice owing to the space regularity of vh , and integrating with respect to time in an arbitrary interval
[t1 , t2 ], gives
1
kvh1 − vh2 k2;E
2
t2
t1
=
Z
def
7 Traces of Functions on Σ = ∂E × (0, T ]
t2
t1
+
Z
403
((Fi∗ ω)h1 − (Fi∗ ω)h2 , ζxi ) dt
t2
t1
((f ∗ ω + u∗ ωt )h1 − (f ∗ ω + u∗ ωt )h2 , ζ) dt.
The right-hand side tends to zero as both h1 and h2 tend to zero. Therefore,
choosing t1 = −∞ and t2 arbitrary, it follows that we have strong convergence
in L2 (E), as h → 0, uniformly in t ∈ (−∞, ∞). Consequently, the limit
function v is equivalent to a strongly continuous function in the L2 (E)-norm
with respect to t, and by the definition of v, the same must hold for u.
Notice that, strictly speaking, we have proved the continuity up to T − δ,
with δ > 0, but this is immaterial, since we could first extend u to T + δ, and
then apply the previous argument to this extension.
6 Uniqueness of Solutions of the Homogeneous
Cauchy–Dirichlet Problem (3.1)
Theorem 6.1. The homogeneous Cauchy–Dirichlet problem (3.1) provided
with conditions (3.2)–(3.3) admits at most one weak solution u ∈ Vo2 .
Proof. It is a direct consequence of Theorem 5.1. Assume by contradiction
that two solutions u1 and u2 exist, with u1 6≡ u2 . By the linearity of the
homogeneous Cauchy–Dirichlet problem, their difference u ∈ V˜o2 (ET ) would
satisfy Problem (3.1) with f = f = uo = 0. By (4.1), it follows that u ≡ 0.
def
7 Traces of Functions on Σ = ∂E × (0, T ]
In Chapter 9 we characterized the traces on ∂E of functions in W 1,p (E), and
at the same time we studied the extension from ∂E to E of functions defined
in proper Sobolev spaces. Such results were instrumental in solving the elliptic
inhomogeneous Dirichlet Problem.
In order to deal with the same problem in the parabolic context, we need
to consider similar trace and extension theorems in space-time cylinders and
on their lateral boundary. Given a bounded set E with boundary ∂E, we
denote the lateral boundary ∂E × (0, T ) of the cylinder ET with Σ.
We are not going to present the theory in full generality. Here, we concentrate on an approach limited to Hilbert spaces, as developed, for example, in
Chapter 4 of Lions and Magenes [176]. For related comments, see Section 7c
in the Complements.
In order to properly define the spaces we are interested in, we need some
introductory material. For θ ∈ (0, 1), consider the so-called Slobodeckij seminorm, which we have already briefly considered in Section 8.3 of Chapter 9.
We let
404
11 LINEAR PARABOLIC EQUATIONS
def
[f ]θ;E =
Z Z
E
E
|f (x) − f (y)|2
dx dy
|x − y|N +2θ
21
.
We have already introduced Sobolev spaces of higher, integer order in Section 1c of Chapter 9. We now extend the definition. Indeed, for r > 0, r 6∈ N,
let ⌊r⌋ be its integer part, and set θ = r − ⌊r⌋. The space W r,2 (E) is defined
as
(
)
def
W r,2 =
f ∈ W ⌊r⌋,2 (E) :
sup [Dα f ]θ;E < ∞ .
|α|=⌊r⌋
It is a Hilbert space endowed with the norm
!2  21
sup [Dα f ]θ;E  .

kf kr,2;E = kf k2⌊r⌋,2;E +
|α|=⌊r⌋
For a given Hilbert space H, we now consider W s,2 (0, T ; H): if s ∈ N, then
W s,2 (0, T ; H) = {v ∈ L2 (0, T ; H) :
∂v
∂ sv
, . . . , s ∈ L2 (0, T ; H)},
∂t
∂t
def
and if s > 0 but s 6∈ N, as we have just done for W r,s (E), we let k = ⌊s⌋,
and define
n
W s,2 (0, T ; H) = v ∈ W k,2 (0, T ; H) :
Z
T
0
k
Z
T
0
∂ k v(·,t)
∂σk
|t −
2
∂ k v(·,τ )
∂σk
2;H
τ |1+2(s−k)
−
o
dt dτ < ∞ ,
k
v(·,τ )
v(·,t)
and ∂ ∂σ
stand for time derivatives of order k of v, evaluated
where ∂ ∂σ
k
k
respectively at time t and at time τ . Finally, let r and s be two non-negative
real numbers. For a bounded open set E, whose boundary ∂E is of class C 1,1
and satisfies the segment property (see Section 8.1 of Chapter 9) we define
H r,s (ET ) = L2 (0, T ; W r,2 (E)) ∩ W s,2 (0, T ; L2(E)),
which is a Hilbert space with the norm
kukH r,s (ET ) =
Z
0
T
ku(t)k2W r,2 (E) dt
+
kuk2W s,2 (0,T ;L2 (E))
! 12
.
The previous definition extends in a straightforward way to the case where
ET is replaced by its lateral boundary Σ: it suffices to use the spaces L2 (∂E)
and W r,2 (∂E), instead of L2 (E) and W r,2 (E). Therefore,
def
H r,s (Σ) = L2 (0, T ; W r,2 (∂E)) ∩ W s,2 (0, T ; L2(∂E)).
405
8 The Inhomogeneous Dirichlet Problem
As in the elliptic case, it turns out that the differential properties of the
boundary values of functions from spaces H r,s (ET ) and of certain derivatives
∂
of theirs can be exactly described in terms of the spaces H µj ,νj (Σ). Let ∂n
denote the normal derivative on Σ oriented toward the interior of ET . We
have
Theorem 7.1. For u ∈ H r,s (ET ) with r > 12 , s ≥ 0, we may define
Σ if j < r − 21 , j ≥ 0 is an integer; moreover,
∂j u
∈ H µj ,νj (Σ), where
∂nj
r − j − 21
µj
νj
=
=
, and νj = 0 if s = 0.
r
s
r
Finally, the mapping u →
H µj ,νj (Σ).
(7.1)
∂j u
is linear and continuous from H r,s (ET ) to
∂nj
Similarly, for s > 21 , r ≥ 0 we may define
is an integer, and
∂j u
on
∂nj
∂ku
on E if k < s − 21 , k ≥ 0
∂tk
∂ku
(x, 0) ∈ W pk ,2 (E), where
∂tk
1
r
s−k−
.
pk =
s
2
The mapping u →
∂ku
(x, 0) is linear and continuous from H r,s (ET ) to
∂tk
H pk (E).
Conversely, given r >
1
2
and s ≥ 0, µj > 0 and νj ≥ 0, which satisfy (7.1),
∂j u
and v ∈ H µj ,νj (Σ), there exists u ∈ H r,s (ET ) such that
= v.
∂nj
Proof. See Section 7c of the Complements.
Remark 7.1 We omit any extension with regard to the value of u on E, since
in the applications to come we are only interested in the boundary values.
8 The Inhomogeneous Dirichlet Problem
Assume that ∂E is of class C 1,1 and satisfies the segment property. Given f
1 1
and f satisfying (3.2)–(3.3) and ϕ ∈ H 2 , 2 (Σ), consider the Cauchy–Dirichlet
problem
ut − aij uxj x = f − div f
in ET
i
u(x, t) = ϕ
u(·, 0) = uo
on Σ
in E.
(8.1)
406
11 LINEAR PARABOLIC EQUATIONS
By Theorem 7.1, if we take r = s = 1, j = k = 0, and µj = νj =
exists v ∈ H 1,1 (ET ) such that
v ∈ L2 (ET ),
vt ∈ L2 (ET ),
1
v(·, 0) ∈ W 2 ,2 (E),
∇v ∈ L2 (ET ),
1
2,
v
Σ
there
= ϕ.
def
Let vo = v(·, 0). A solution of (8.1) is sought of the form u = w+v, where w ∈
Vo2 (ET ) is the unique weak solution of the auxiliary, homogeneous Cauchy–
Dirichlet problem
wt − aij wxj xi = f˜ − div f̃ in ET
on Σ
in E,
w(x, t) = 0
w(·, 0) = wo
where
2
f˜ = f − vt ,
f˜j = fj − aij vxi ,
wo = uo − vo .
Since vt ∈ L (ET ), if we choose q, r as in (3.3), we have
r
1
q
kvt kq,r;ET ≤ T 1− 2 |E| r − 2r kvt k2;E ,
and therefore, f˜ ∈ Lq,r (ET ); in a similar way, we have f̃ ∈ L2 (ET ), and
wo ∈ L2 (E). Hence, we conclude
Theorem 8.1. Assume that ∂E is of class C 1,1 and satisfies the segment
1 1
property. For every f and f satisfying (3.2)–(3.3) and ϕ ∈ H 2 , 2 (Σ), the
inhomogeneous Cauchy–Dirichlet problem (8.1) has a unique weak solution
u ∈ V 2 (ET ).
1 1
Remark 8.1 The class H 2 , 2 (Σ) where the boundary datum ϕ is assumed, is
not the most general one. In this context we are not interested in the minimal
regularity assumptions on the data, which ensure the existence of a unique
solution.
9 The Neumann Problem
Assume that ∂E is of class C 1 and satisfies the segment property. Given f
and f satisfying (3.2)–(3.3), consider the formal Cauchy–Neumann problem
ut − aij uxj x = f − div f
in ET
i
(aij uxi − fj ) nj = ψ
u(·, 0) = uo
on Σ
(9.1)
in E,
where n = (n1 , . . . , nN ) is the outward unit normal to ∂E and ψ ∈ Lq2 ,r2 (Σ)
for some proper (q2 , r2 ). If aij = δij , f = f = 0, and ψ is sufficiently regular,
9 The Neumann Problem
407
this is precisely the Neumann problem (1.3) of Chapter 5. Since aij ∈ L∞ (ET )
and f ∈ Lq,r (ET ), f ∈ L2 (ET ), neither the PDE nor the boundary condition
in (9.1) are well defined in the classical sense, and they have to be interpreted
in some weak form, as we have just done for the Cauchy–Dirichlet problem.
We first make proper hypotheses on the integrability of ψ, namely we
assume that
1
N −1
where
ψ ∈ Lq2 ,r2 (Σ),
+
=
r
2q2

i 2
h
−1)

, r2 ∈ [1, 2],
q2 ∈ 2(NN−1) , 2(N

N −2




q2 > 1, r2 ∈ [1, 2)






q2 = 43 , r2 = 34 ,
N
1
+
4
2
and
if N > 2
(9.2)
if N = 2
if N = 1.
By a weak solution of the Cauchy–Neumann problem (9.1), under assumptions
(3.2)–(3.3) and (9.2), we mean a function u ∈ V 2 (ET ) satisfying the identity
ZZ
ZZ
ZZ
(f ϕ + fi ϕxi )dx dt
−
uϕt dx dt +
aij uxj ϕxi dx dt =
ET
ET
E
ZZ T
Z
(9.3)
+
ψϕ dσ dt +
uo (x)ϕ(x, 0)dx
Σ
E
for any function ϕ ∈ W 2 (ET ), which vanishes for t = T . We formally obtain the previous relation, by multiplying the first of (9.1) by ϕ, performing
integration by parts with respect to x and t, and taking into account the second
Z Z and third of (9.1). It is easy to check that by (9.2), the surface integral
ψϕ dσ dt is well defined, whereas for the remaining terms, we are under
Σ
the same conditions that we considered for the Cauchy–Dirichlet problem.
As we did before, we can give an equivalent definition of a weak solution,
requiring that u ∈ V 2 (ET ) satisfies
Z
Z τZ
Z τZ
u(x, τ )ϕ(x, τ )dx −
uϕt dx dt +
aij uxj ϕxi dx dt
E
0
E
0
E
(9.4)
Z τZ
Z τZ
Z
ψϕdσ dt +
uo (x)ϕ(x, 0)dx
=
(f ϕ + fi ϕxi )dx dt +
0
E
0
∂E
E
for almost any τ ∈ (0, T ) and for any function ϕ ∈ W 2 (ET ).
In order to prove the existence of a weak solution, we apply Galerkin’s
method. In this instance, we require {wk } to be a countable, complete system for W 1,2 (E) (not for Wo1,2 (E) as for the homogeneous Cauchy–Dirichlet
problem), and we assume that (wi , wj ) = δij , where (·, ·) is the inner product
in L2 (E).
We look for an approximate solution uM = uM (x, t), given by
408
11 LINEAR PARABOLIC EQUATIONS
uM (x, t) =
M
X
cM
k (t)wk (x),
k=1
where the coefficients ckM (t) = (uM , wk ) are determined from the Cauchy
problem
Z
Z
Z
Z
d
aij uM
w
dx
=
(f
w
+
f
w
)dx
+
ψwk dσ,
uM wk dx +
i
k,x
k
k,x
i
i
xj
dt E
∂E
E
ZE
cM
uo wk dx.
k (0) =
E
Then, we proceed as with the homogeneous Cauchy–Dirichlet problem, and
we conclude that there exists at least one weak solution u ∈ V 2 (ET ). As
before, we also have that u ∈ V˜2 (ET ). Therefore, we have proved
Theorem 9.1. Let ∂E be of class C 1 and satisfy the segment property. Let
f and f satisfy (3.2)–(3.3), and ψ satisfy (9.2). Then, the Cauchy–Neumann
problem (9.1) admits a weak solution u ∈ V˜2 (ET ).
9.1 The Energy Inequality for the Neumann Problem
Proposition 9.1 Let ∂E be of class C 1 and satisfy the segment property. Let
u ∈ V˜2 (ET ) be a solution of the Cauchy–Neumann problem (9.1) with f, f
satisfying (3.2)–(3.3), and ψ satisfying (9.2). Then, the Energy Inequality
"
1′
r
T
kukV 2 (ET ) ≤γ kuo k2;E + kf k2;ET + 1 +
kf kq,r;ET
2
|E| N
(9.5)
#
1′ − 21 !
r
T
2
+ 1+
kψkq2 ,r2 ;Σ
2
|E| N
holds true, where γ > 0 depends only on N , λ, q, r, q2 , r2 .
Proof. In (9.4) take as test function the Steklov average
Z
1 h
ϕ̂(x, τ )dτ,
ϕh̄ (x, t) =
h t−h
where ϕ̂ is an arbitrary element of W 2 (ET ), which vanishes for t ≥ T − h and
for t ≤ 0. The regularity of all the involved quantities allows us to work as in
Sections 3–4, so that for any t1 ∈ (0, T ] we obtain
Z
ZZ
t=t1
1
u2 (x, t)dx
aij uxj uxi dx dt
+
2 E
t=0
Et1
ZZ
Z t1 Z
ψu dσ dt.
=
(f u + fi uxi )dx dt +
Et1
0
∂E
9 The Neumann Problem
The ellipticity condition (2.1) and Hölder’s inequality yield
ZZ
Z
1
2
u (x, t1 ) dx + λ
|∇u|2 dx dt
2 E
Et1
! 21 Z Z
! 12
Z
ZZ
1
u2 dx +
≤
|f |2 dx dt
|∇u|2 dx dt
2 E o
Et1
Et1

 1′
r
rq ! r1 Z t1 Z
r′′
Z t1 Z
q
′
q
q


+
|f | dx
dt
dt
|u| dx
0
+
Z
E
t1
0
Z
0
q2
∂E
|ψ| dσ
qr2
=I + II + III + IV.
2
409
E
! r1 Z
2

dt
t1
0
Z
∂E
q2′
|u| dσ
r′2′
q2
 r1′
2
dt
We now proceed with the estimates of I, II, III, IV ; relying on Propositions 1.4 and 1.5, and on the first of (9.2) gives
1
kuo k2;E kukV 2 (ET ) ;
2
II ≤ kf k2;ET kuk2;ET ;
I≤
III ≤ kf kq,r;ET kukq′ ,r′ ;ET
≤γ 1+

IV ≤ kψkq2 ,r2 ;Σ kukq2′ ,r2′ ;Σ ≤ γ 1 +
=γ
1+
T
2
|E| N
1
′
r2
− 12
!
T
|E|
1′
r
2
N
N
T2
|E|
kf kq,r;ET kukV 2 (ET ) ;
! 21 − N −1′ 
N q2
 kψkq2 ,r2 ;Σ kukV 2 (ET )
kψkq2 ,r2 ;Σ kukV 2 (ET ) .
Collecting all these estimates and taking the supremum over t1 on the lefthand side, we conclude.
As a direct consequence of Proposition 9.1 we have the following.
Theorem 9.2. The Cauchy–Neumann problem (9.1) provided with conditions
(9.2) admits at most one weak solution u ∈ V 2 .
Proof. Assume by contradiction that two solutions u1 and u2 exist, with u1 6≡
u2 . By the linearity of the Cauchy–Neumann problem, their difference u ∈
V˜2 (ET ) would satisfy Problem (9.1) with f = f = uo = ψ = 0. By (9.5), it
follows that u ≡ 0.
410
11 LINEAR PARABOLIC EQUATIONS
9.2 A Variant of Problems (3.1) and (9.1)
We can also consider the following mixed problem
ut − aij uxj xi = f − div f
in ET
on Σ1
on Σ2
(aij uxi − fj ) nj = ψ
u=0
u(·, 0) = uo
(9.6)
in E,
where ∂E is of class C 1 and satisfies the segment property, Σ1 = (∂E)1 ×
(0, T ), Σ2 = (∂E)2 ×(0, T ), with (∂E)1 ∪(∂E)2 = ∂E, and (∂E)1 ∩(∂E)2 = ∅,
f and f satisfy (3.2)–(3.3), and ψ satisfies (9.2).
We let
Wγ1,2
(E) = {v ∈ W 1,2 (E) : γo v = 0 on (∂E)2 },
o (∂E)2
where γo v is the trace of v on ∂E.
We say that
(E))
u ∈ C([0, T ]; L2 (E)) ∩ L2 (0, T ; Wγ1,2
o (∂E)2
is a weak solution of (9.6) if
ZZ
ZZ
ZZ
(f ϕ + fi ϕxi ) dx dt
aij uxj ϕxi dx dt =
uϕt dx dt +
−
ET
ET
Z
Z Z ET
uo (x)ϕ(x, 0)dx
ψϕ dσ dt +
+
Σ1
E
for any function
ϕ ∈ W 1,2 ([0, T ]; L2(E)) ∩ L2 (0, T ; Wγ1,2
(E)),
o (∂E)2
which vanishes for t = T . This can be seen as a combination of (3.5) and
(9.3). An equivalent formulation can be similarly obtained combining (3.4)
and (9.4). The existence of a weak solution is proved by Galerkin’s method,
the only difference being in the choice of the countable, complete, system for
the subspace of W 1,2 (E) of the functions that vanish on (∂E)2 . The existence
of such a system can be ensured by the arguments discussed in Sections 19–20
of Chapter 7 of DiBenedetto [50].
10 A Priori L∞ (ET ) Estimates for Solutions of the
Cauchy–Dirichlet Problem (8.1)
A weak sub(super)-solution of the Cauchy–Dirichlet problem (8.1) is a function u ∈ V˜2 (ET ), such that u(·, t) ≤ (≥)ϕ(·, t) on ∂E in the sense of the traces
of functions in W 1,2 (E) for a.e. t ∈ (0, T ), and such that
10 A Priori L∞ (ET ) Estimates for the Cauchy–Dirichlet Problem (8.1)
Z
Z
Z
τ Z
u(x, τ )ζ(x, τ )dx −
uζt dx dt
uo (x)ζ(x, 0)dx −
E
0
E
Z τZ
Z τZ
(fi ζxi + f ζ)dx dt,
aij uxj ζxi dx dt ≤ (≥)
+
E
0
0
E
411
(10.1)
E
for almost all τ ∈ [0, T ] and for all non-negative ζ ∈ Wo2 (ET ). It is apparent that a function u ∈ V˜2 (ET ) is a weak solution of the Cauchy–Dirichlet
problem (8.1), if and only if it is both a weak sub- and super-solution of that
problem.
Proposition 10.1 Let u ∈ V˜2 (ET ) be a weak sub-solution of (8.1) for N ≥ 2.
For 1 < p < NN+2 let
r = r(p) = 2
and assume
p
,
p−1
s = s(p) =
uo,+ ∈ L∞ (E),
f ∈ Lr(p) (ET ),
2p(N + 2)
,
(N + 4)p − (N + 2)
ϕ+ ∈ L∞ (Σ),
f+ ∈ Ls(p) (ET ).
(10.2)
(10.2)+
Then, u+ ∈ L∞ (ET ) and there exists a constant Cp that can be determined a
priori only in terms of λ, N , p, and the constant γ in the parabolic embedding
(1.2)–(1.3), such that
ess sup u+ ≤Cp max ess sup uo,+ ; ess sup ϕ+ ; kf kr(p);ET
ET
E
Σ
(10.3)+
δ
+ kf+ ks(p);ET |ET | 2 ,
where
δ=
1
2
+
− 1.
p N +2
A similar statement holds for super-solutions. Precisely:
Proposition 10.2 Let u ∈ V˜2 (ET ) be a weak super-solution of (8.1) for
N ≥ 2. For 1 < p < NN+2 let r = r(p) and s = s(p) as in (10.2), and assume
uo,− ∈ L∞ (E),
f ∈ Lr(p) (ET ),
ϕ− ∈ L∞ (Σ),
f− ∈ Ls(p) (ET ).
(10.2)−
Then, u− ∈ L∞ (ET ) and
ess sup u− ≤Cp max ess sup uo,− ; ess sup ϕ− ; kf kr(p);ET
ET
E
δ
+ kf− ks(p);ET |ET | 2
Σ
for the same constants Cp and δ as in Proposition 10.1.
(10.3)−
412
11 LINEAR PARABOLIC EQUATIONS
Remark 10.1 The integrability exponents r(p) and s(p) are both decreasing
with respect to p. In particular,
lim r(p) = N + 2,
+2
p→ NN
lim s(p) =
+2
p→ NN
N +2
.
2
Remark 10.2 The constant Cp tends to infinity and δ tends to zero, as
p → NN+2 . Indeed, the propositions are false for p = NN+2 .
11 Proof of Propositions 10.1–10.2
It suffices to establish Proposition 10.1. Let u ∈ V˜2 (ET ) be a weak subsolution of the Cauchy–Dirichlet problem (8.1), in the sense of (10.1) for all
non-negative ζ ∈ Wo2 (ET ). Let
k ≥ max{kϕ+ k∞,Σ , kuo,+ k∞,E }
(11.1)
be chosen and consider the Steklov average uh of u; according to Lemma 1.1 it
is apparent that (uh (·, t) − k)+ ∈ Wo1,2 (E) for all t ∈ (0, T − h), and therefore,
the test function
ζ(x, t) = (uh (x, t) − k)+
is admissible in the weak formulation (10.1).
If we rewrite (10.1) in terms of the Steklov average uh , and take ζ as above,
we have
ZZ
ZZ
uh,t (uh (x, t) − k)+ dx dt +
(aij uxj )h ((uh (x, t) − k)+ )xi dx dt
Et1
≤
Et1
ZZ
Et1
[fi,h ((uh (x, t) − k)+ )xi + f+,h (uh (x, t) − k)+ ] dx dt,
that is
ZZ
ZZ
1
∂t ((uh (x, t) − k)+ )2 dx dt +
(aij uxj )h ((uh (x, t) − k)+ )xi dx dt
2
Et1
Et1
ZZ
≤
[fi,h ((uh (x, t) − k)+ )xi + f+,h (uh (x, t) − k)+ ] dx dt,
Et1
where t1 is any value in (0, T −h). Integrating with respect to time, and taking
(11.1) into account gives
ZZ
Z
1
2
(uh − k)+ (x, t1 )dx +
(aij uxj )h ((uh (x, t) − k)+ )xi dx dt
2 E
Et1
ZZ
≤
[fi,h ((uh (x, t) − k)+ )xi + f+,h (uh (x, t) − k)+ ] dx dt.
Et1
11 Proof of Propositions 10.1–10.2
413
Passing to the limit as h → 0, taking the supremum with respect to t1 over
(0, T ], and using the ellipticity condition (2.1) yields
Z
ZZ
1
sup
(u − k)2+ (x, t)dx + λ
|∇(u − k)+ |2 dx dt
2 0<t<T E
ET
ZZ
[fi ((u − k)+ )xi + f+ (u − k)+ ] dx dt.
≤
ET
Since
ZZ
ET
fi ((u − k)+ )xi dx dt
λ
≤
2
we obtain
sup
0<t<T
Z
E
ZZ
ET
(u −
≤ γ1
1
|∇(u − k)+ | dx dt +
2λ
2
k)2+ (x, t)dx
ZZ
ET
+
ZZ
ET
ZZ
N
X
|fi |2 χ[u>k] dx dt,
ET i=1
|∇(u − k)+ |2 dx dt
f+ (u − k)+ dx dt + γ1
ZZ
N
X
ET i=1
|fi |2 χ[u>k] dx dt.
Moreover, by Proposition 1.1
ZZ
f+ (u − k)+ dx dt ≤ kf+ χ[u>k] k2 N +2 ;ET k(u − k)+ k2 N +2 ;ET
N +4
ET
N
≤ γk(u − k)+ kV 2 (ET ) kf+ χ[u>k] k2 N +2 ;ET
N +4
1
≤
k(u − k)+ k2V 2 (ET ) + C(γ, γ1 )kf+ χ[u>k] k22 N +2 ;ET .
N +4
4γ1
Hence,
sup
0<t<T
≤ γ2
Z
E
(u − k)2+ (x, t)dx +
ZZ
N
X
ET i=1
For any p ∈ 1,
ZZ
ZZ
ET
|∇(u − k)+ |2 dx dt
|fi |2 χ[u>k] dx dt + γ2
N +2
N
N
X
ET i=1
Z Z
2 N +2
ET
f+N +4 χ[u>k] dx dt
+4
N
N +2
(11.2)
.
we have
|fi |
2χ
[u>k] dx dt
def
≤
Z Z
ET
2p′
|f |
1′
p
dx dt
Ak
1
p
,
N +4
where Ak = {(x, t) ∈ ET : u(x, t) > k}; by Hölder’s inequality for q = p N
+2
and q ′ =
p(N +4)
p(N +4)−(N +2)
we obtain
414
11 LINEAR PARABOLIC EQUATIONS
ZZ
2 N +2
ET
f+N +4 χ[u>k] dx dt
≤
Z Z
N +4
N
+2
2(N +2)p
ET
f+p(N +4)−(N +2) dx dt
+2)
p(N +4)−(N
(N +2)p
Ak
1
p
.
Therefore,
sup
0<t<T
Z
E
(u − k)2+ (x, t)dx +
≤γ2 Ak
+
1
p
"Z Z
|f |2p dx dt
2(N +2)p
p(N +4)−(N +2)
ET
ET
′
ET
Z Z
ZZ
f+
1′
dx dt
where we have set
Z Z
1′ Z Z
p
2p′
+
Cf =
|f | dx dt
ET
ET
Let
kn = k 2 −
|∇(u − k)+ |2 dx dt
p
+2)
p(N +4)−(N
(N +2)p
2(N +2)p
p(N +4)−(N +2)
f+

 = γ2 Cf Ak
dx dt
1
p
,
+2)
p(N +4)−(N
(N +2)p
.
1 ,
Akn = [u > kn ],
n = 1, 2, . . . ,
2n−1
and rewrite the previous estimate in terms of kn ; this yields
Z
ZZ
1
sup
(u − kn )2+ (x, t)dx +
|∇(u − kn )+ |2 dx dt ≤ γCf |An | p , (11.3)
0<t<T
E
ET
where An = Akn . Moreover,
ZZ
ZZ
2
(u − kn )+ dx dt ≥
ET
ET
(u − kn )2+ χ[u>kn+1 ] dx dt
(11.4)
k2
≥ (kn − kn+1 ) |An+1 | = n |An+1 |,
4
2
and, by the parabolic embedding of Proposition 1.1
ZZ
2 N +2
(u − kn )+ N dx dt
ET
≤γ
ZZ
ET
2
|∇(u − kn )+ | dx dt
sup
0<t<T
Z
E
(u −
N2
kn )2+ dx
Therefore, taking into account (11.3) and the previous two estimates
ZZ
k2
(u − kn )2+ dx dt
|A
|
≤
n+1
4n
ET
.
11 Proof of Propositions 10.1–10.2
≤
≤
Z Z
ET
(u −
sup
0<t<T
·
Z Z
Z
E
2 N +2
kn )+ N dx dt
(u −
NN+2
N2+2
|∇(u − kn )+ | dx dt
1
2
|An | N +2
kn )2+ (x, t)dx
2
ET
415
2
NN+2
2
|An | N +2
2
1
≤γ3 Cf |An | p |An | N +2 = γ3 Cf |An | p + N +2 .
1
p
By our assumption on p, it is apparent that
1 + δ, with δ = δ(p, N ), we obtain
|An+1 | ≤ γ3
+ N2+2 > 1. If we set
4n
Cf |An |1+δ ,
k2
1
p
+ N2+2 =
∀ n ∈ N.
If |An | → 0 as n → ∞, then u ≤ 2k a.e. in ET . By Lemma 15.1 of Chapter 9
this occurs if
−1/δ −1/δ 2 −1/δ 2/δ
|A1 | ≤ |ET | ≤ γ3
4
Cf
k ,
and this is in turn satisfied if k is chosen from
1/2
k = 21/δ γ3
1/2
Cf
|ET |δ/2 ,
which finally implies
u ≤ γ4 (λ, N, p, γ)|ET |
δ
2
"Z Z
ET
|f |
r(p)
dx dt
1
r(p)
+
Z Z
ET
s(p)
f+ dx dt
1 #
s(p)
.
Taking into account the original assumptions on k, we conclude.
Remark 11.1 In (11.2), the last term on the right-hand side can be estimated
in a slightly different way, namely
ZZ
ET
2 N +2
f+N +4 χ[u>k] dx dt
≤
≤
+4
!N
N +2
Z Z
ET
Z Z
ET
2 N +2 q′
f+N +4 dx dt
2 N +2 q′
f+N +4 dx dt
N +4
(N +2)q′
N +4
(N +2)q′
Ak
Ak
N +4
(N +2)q
1
q
ET
2
(N +2)q
,
+4
where now q > 1, whereas in (11.2) q > p N
N +2 with p > 1.
We can relabel q with p, and then proceed as before. This is particularly
useful when considering with this different estimate the limiting case of p → 1,
which entails that both f and f+ end up belonging to L∞ (ET ). In such a case
we have
416
11 LINEAR PARABOLIC EQUATIONS
h
ess sup u+ ≤C1 max ess sup uo,+ ; ess sup ϕ+ ; kf k∞;ET
ET
E
∂E
i
o
1
1
+|ET | N +2 kf+ k∞;ET |E| N +2
where the constant C1 is the stable limit of Cp as p → 1. Moreover, apart
from change from N to N + 2, we have a perfect correspondence with the
analogous estimates of Section 15 of Chapter 9.
Remark 11.2 The fact that most elliptic estimates can be repeated almost
verbatim in the parabolic framework up to a change from N to N + 2 is not
a casual fact, but it is a direct consequence of the so-called parabolic scaling,
as we shall more clearly see in Chapter 12.
12 A Priori L∞ (ET ) Estimates for Solutions of the
Neumann Problem (9.1)
A weak sub(super)-solution of the Neumann problem (9.1) is a function u ∈
V˜2 (ET ) satisfying
Z
Z
u(x, τ )ζ(x, τ )dx −
uo (x)ζ(x, 0)dx
E
E
Z τZ
Z τZ
−
uζt dx dt +
aij uxi ζxj dx dt
(12.1)
0
E
0
E
Z τZ
Z τZ
≤ (≥)
(fj ζxj + f ζ)dx dt +
ψz dσ dt
0
E
0
∂E
for all non-negative test functions ζ ∈ W 2 (ET ). A function u ∈ V˜2 (ET ) is a
weak solution of the Cauchy–Neumann problem (9.1), if and only if it is both
a weak sub- and super-solution of that problem.
Proposition 12.1 Let ∂E be of class C 1 and satisfy the segment property.
+2
Let u ∈ V˜2 (ET ) be a weak sub-solution of (9.1) for N ≥ 2. For 1 < p < N
N +1
let
p
N −1 p
,
r(p) =
,
q = q(p) =
N p−1
p−1
(12.2)
2p(N + 2)
s = s(p) =
,
(N + 4)p − (2 − p)(N + 2)
and assume
uo,+ ∈ L∞ (E),
f ∈ Lr(p) (ET ),
ψ+ ∈ Lq(p),r(p) (Σ),
f+ ∈ Ls(p) (ET ).
(12.3)+
Then, u+ ∈ L∞ (ET ), and there exists a constant Cp that can be determined
a priori only in terms of λ, N , p, T , the constant γ in Proposition 1.2, the
12 A Priori L∞ (ET ) Estimates for Solutions of the Neumann Problem (9.1)
417
constant γ in the embeddings of Theorem 2.1 of Chapter 9, the constant γ of
the trace inequality of Proposition 8.2 of Chapter 9, and the structure of ∂E
through the parameters h and ω of its cone condition such that
ess sup u+
ET
(
≤Cp max ess sup uo,+ ; 1 +
+kf kr(p);ET
E
+ 1+
T
2
|E| N
T
2
2(NN+2)
|E| N
2(NN+2)
where
δ=
|ET |
kf+ ks(p);ET
δ
2
"
kψ+ kq(p),r(p);Σ
#)
(12.4)+
,
2
2
+
− 2.
p N +2
(12.5)
Proposition 12.2 Let ∂E be of class C 1 and satisfy the segment property.
Let u ∈ V˜2 (ET ) be a weak sub-solution of (9.1) for N ≥ 2. Assume that
uo,− ∈ L∞ (E),
ψ− ∈ Lq(p),r(p) (Σ),
f ∈ Lr(p) (ET ),
f− ∈ Ls(p) (ET ),
(12.3)−
for the same q(p), r(p), s(p) as in (12.2). Then, u− ∈ L∞ (ET ) and
ess sup u+
ET
(
≤Cp max ess sup uo,− ; 1 +
E
+kf kr(p);ET + 1 +
T
2
|E| N
T
2
2(NN+2)
|E| N
2(NN+2)
|ET |
kf− ks(p);ET
δ
2
"
kψ− kq(p),r(p);Σ
#)
(12.4)−
where the parameters Cp , δ are the same as in (12.4)+ and (12.5).
Remark 12.1 The constant Cp depends on the embedding constants of Theorem 2.1 of Chapter 9. As such, they depend on the structure of ∂E through
the parameters h and ω of its cone condition. Because of this dependence, Cp
tends to ∞ as either h → 0 or ω → 0.
Remark 12.2 The integrability exponents q(p), r(p) and s(p) are all decreasing with respect to p. In particular,
lim q(p) =
(N − 1)(N + 2)
,
N
lim s(p) =
N +2
,
2
+2
p→ N
N +1
+2
p→ N
N +1
lim r(p) = N + 2,
N +2
p→ N
+1
and as far as f and f are concerned, we find the same limiting conditions as
in Propositions 10.1–10.2.
418
11 LINEAR PARABOLIC EQUATIONS
Remark 12.3 The constant Cp tends to infinity and δ tends to zero, as
+2
N +2
p→ N
N +1 . Indeed, the propositions are false for p = N +1 .
13 Proof of Propositions 12.1–12.2
It suffices to establish Proposition 12.1. Let u ∈ V˜2 (ET ) be a sub-solution
of the Cauchy–Neumann problem (9.1), in the sense of (12.1). Let k ≥
ess supE uo,+ to be chosen.
Working with Steklov averages, and then passing to the limit with respect
to the parameter h as in the previous section, we have
Z
ZZ
1
2
sup
(u − k)+ (x, t)dx + λ
|∇(u − k)+ |2 dx dt
0<t<T 2 E
ET
ZZ
ZZ
(fi ((u − k)+ )xi + f+ (u − k)+ )dx dt +
ψ+ (u − k)+ dσ dt.
≤
Σ
ET
Now we have to estimate the right-hand side, taking into account that (u(·, t)−
k)+ ∈ W 1,2 (E) for almost every t ∈ (0, T ], but in general it does not have
zero trace. Since
ZZ
fi ((u − k)+ )xi dx dt
ET
λ
≤
2
ZZ
ET
we have
sup
0<t<T
≤ γ1
+ γ1
ZZ
1
|∇(u − k)+ | dx dt +
2λ
2
Z
E
ZZ
ZZ
(u −
ET
Σ
k)2+ (x, t)dx
+
ZZ
|f |2 χ[u>k] dx dt + γ1
ET
N
X
ET i=1
ZZ
|fi |2 χ[u>k] dx dt,
|∇(u − k)+ |2 dx dt
ET
f+ (u − k)+ dx dt
ψ+ (u − k)+ dσ dt.
By Proposition 1.2
ZZ
f+ (u − k)+ dx dt ≤ kf+ χ[u>k] k2 N +2 ;ET k(u − k)+ k2 N +2 ;ET
N +4
ET
≤γ 1 +
≤
T
2
|E| N
2(NN+2)
N
k(u − k)+ kV 2 (ET ) kf+ χ[u>k] k2 N +2 ;ET
1
k(u − k)+ k2V 2 (ET )
4γ1
(NN+2)
T
+ C(γ, γ1 ) 1 +
kf+ χ[u>k] k22 N +2 ;ET ,
2
N +4
|E| N
N +4
13 Proof of Propositions 12.1–12.2
419
which gives
sup
0<t<T
Z
E
ZZ
≤ γ2
(u − k)2+ (x, t)dx +
ET
+ γ2 1 +
|f |
2χ
T
2
|E| N
[u>k] dx dt
ZZ
ET
+ γ1
ZZ
|∇(u − k)+ |2 dx dt
Σ
NN+2 Z Z
ET
ψ+ (u − k)+ dσ dt
2 N +2
f+N +4 χ[u>k] dx dt
By Remark 12.2, q > N − 1 and by assumption 1 < p <
can determine p and p∗ from the following conditions
1
1
= ∗,
q
p
1−
p∗ =
N +4
N
+2
N +2
N +1 ;
.
therefore, we
1
p−1 N
=
.
q
p N −1
Np
,
N −p
We can the apply the trace inequality (8.3) of Chapter 9, to obtain
Z TZ
Z T
ψ+ (u − k)+ dσ dt ≤
kψ+ (·, t)kq;∂E k(u(·, t) − k)+ kp∗ N −1 ;∂E dt
0
∂E
T
≤
≤
≤
Z
kψ+ (·, t)kq;∂E [k∇(u − k)+ kp;E + 2γk(u − k)+ kp;E ] dt
0
Z
T
0
1
4
N
0
1
1
kψ+ (·, t)kq;∂E [k∇(u − k)+ k2;E + 2γk(u − k)+ k2;E ] |Ak (t)| p − 2 dt
ZZ
ET
|∇(u − k)+ |2 dx dt + γ
Z
0
T
2
kψ+ (·, t)k2q;∂E |Ak (t)| p −1 dt
ZZ
Z T
2
1
2
(u − k)+ dx dt + γ
kψ+ (·, t)k2q;∂E |Ak (t)| p −1 dt
+
4T
ET
0
ZZ
Z
1
2
2
≤
|∇(u − k)+ | dx dt
(u − k)+ (x, t)dx +
sup
4 0<t<T E
ET
! 2(p−1)
Z
p
T
p
p−1
dt
kψ+ (·, t)kq;∂E
+γ
0
|Ak |
2−p
p
,
where Ak (t) = {x ∈ E : u(x, t) > k} and Ak = {(x, t) ∈ ET : u(x, t) > k}.
Hence, collecting all the terms yields
ZZ
Z
2
|∇(u − k)+ |2 dx dt
(u − k)+ (x, t)dx +
sup
0<t<T
E
≤γ2 1 +
+ γ2
T
2
ZZ
|E| N
ET
NN+2 Z Z
ET
ET
2 N +2
f+N +4 χ[u>k] dx dt
|f |2 χ[u>k] dx dt + γ3
Z
T
0
+4
N
N +2
p
p−1
kψ+ (·, t)kq;∂E dt
! 2(p−1)
p
|Ak |
2−p
p
.
420
11 LINEAR PARABOLIC EQUATIONS
Moreover,
ZZ
|f |
ET
2χ
≤ |Ak |
[u>k] dx dt
2−p
p
Z Z
ET
|f |
p
p−1
dx dt
2(p−1)
p
,
and
Z Z
ET
≤ |Ak |
2 N +2
f+N +4 χ[u>k] dx dt
Z Z
2−p
p
+4
N
N +2
2p(N +2)
p(N +4)−(2−p)(N +2)
ET
f+
dx dt
+2)
p(N +4)−(2−p)(N
p(N +2)
(13.1)
.
Hence,
sup
0<t<T
Z
E
(u − k)2+ (x, t)dx +
ZZ
ET
|∇(u − k)+ |2 dx dt ≤ γCf,ψ |Ak |
2−p
p
,
where
def
Cf,ψ =
Z Z
p
ET
|f | p−1 dx dt
+ 1+
+
Z
NN+2 Z Z
T
2
|E| N
2p(N +2)
p(N +4)−(2−p)(N +2)
ET
T
p
p−1
kψ+ (·, t)kq;∂E dt
0
Let
2(p−1)
p
kn = k 2 −
f+
! 2(p−1)
p
dx dt
+2)
p(N +4)−(2−p)(N
p(N +2)
.
1 ,
Akn = [u > kn ],
n = 1, 2, . . . ,
2n−1
and rewrite the previous estimate in terms of kn . By Proposition 1.2
ZZ
(u − kn )2+ dx dt
ET
≤
Z Z
ET
≤γ 1 +
+
ZZ
(u −
T
2
|E| N
2 N +2
kn )+ N dx dt
NN+2 sup
0<t<T
2
ET
NN+2
Z
E
2
|An | N +2
(u − kn )2+ (x, t)dx
2
|∇(u − kn )+ | dx dt |An | N +2 ,
where An = Akn , and therefore,
13 Proof of Propositions 12.1–12.2
ZZ
ET
(u −
≤ γCf,ψ 1 +
kn )2+ dx dt
2−p
p
By the assumption on p, we have
+
NN+2
T
|E|
2
N +2
2
N
|An |
2−p
2
p + N +2
421
.
> 1. We set
2
2
2−p
def 2
+
=1+δ ⇒ δ = +
− 2.
p
N +2
p N +2
As we have already seen in (11.4)
ZZ
k2
(u − kn )2+ dx dt ≥ n |An+1 |;
4
ET
and therefore,
|An+1 | ≤ γCf,ψ 1 +
T
2
|E| N
NN+2
4n
|An |1+δ .
k2
We can now apply Lemma 15.1 of Chapter 9, and conclude that u ≤ 2k a.e.
in ET , provided we have
2
−1/δ
|A1 | ≤ |ET | ≤ γ −1/δ 4−1/δ Cf,ψ k 2/δ 1 +
T
2
|E| N
− δ(NN+2)
,
and this is in turn satisfied if k is chosen from
2(NN+2)
1
1
1
δ
T
2
k = 2 δ γ 2 |ET | 2 Cf,ψ 1 +
,
2
|E| N
which implies
ess sup u ≤γ 1 +
ET
T
+ 1+

+
Z
0
2
|E| N
T
2(NN+2)
T
2
|ET | N
Z
∂E
|ET |
δ
2
( Z Z
ET
2(NN+2) Z Z
ET
kψ+ (·, t)kq(p) dσ
|f |
r(p)
s(p)
f+ dx dt
r(p)
q(p)
dx dt
1
s(p)
1
 s(p)
)
dt
1
r(p)
.
If we take into account the original stipulation on k, we conclude.
Remark 13.1 In (13.1), the last term on the right-hand side can be estimated
in a slightly different way, namely
ZZ
ET
2 N +2
f+N +4 χ[u>k] dx dt
+4
!N
N +2
422
11 LINEAR PARABOLIC EQUATIONS
≤
≤
Z Z
ET
Z Z
ET
2 N +2 q′
f+N +4 χ[u>k] dx dt
2 N +2 q′
f+N +4 χ[u>k] dx dt
N +4
(N +2)q′
N +4
(N +2)q′
Ak
Ak
N +4
(N +2)q
2−p
p
N +4
− 2−p
p
(N +2)q
ET
,
where q > 1.
We can relabel q with p, and then proceed as before. This is particularly
useful when considering with this different estimate the limiting case of p → 1,
which entails that f , f+ , ψ+ all end up belonging to L∞ (ET ). In such a case
we have
ess sup u+
ET
(
≤C1 max ess sup uo,+ ; 1 +
+kf k∞;ET
E
+ 1+
T
2
|E| N
T
2
|E| N
2(NN+2)
2(NN+2)
|ET |
1
N +2
|ET |
1
N +2
"
kψ+ k∞;Σ
kf+ k∞;ET
#)
,
where the constant C1 is the stable limit of Cp as p → 1. Moreover, apart
from change from N to N + 2, we have a perfect correspondence with the
analogous estimates of Section 16 of Chapter 9.
14 Miscellaneous Remarks on Further Regularity
1,2
A function u ∈ Cloc (0, T ; L2loc (E))∩L2loc (0, T ; Wloc
(E)) is a local weak solution
to (2.2), irrespective of possible boundary data, if for every compact set K ⊂ E
and for every sub-interval [t1 , t2 ] ⊂ (0, T ]
Z
Z t2 Z
Z t2 Z
t2
uϕdx +
aij uxj ϕxi dx dt =
[f ϕ + fi ϕxi ] dx dt
K
t1
t1
t1
K
K
for all ϕ ∈ Cloc (0, T ; L2 (K))∩L2loc (0, T ; Wo1,2 (K)). On the data f and f assume
f ∈ Lr (ET ),
f± ∈ Ls (ET ),
with r > N + 2, s >
N +2
.
2
(14.1)
The set of parameters {N, λ, Λ, r, s, kf kr;ET , kf ks;ET } are the data, and we
say that a constant C, γ, . . . depends on the data if it can be quantitatively
determined a priori in terms of these quantities only. Continue to assume
that the boundary ∂E is of class C 1 and with the segment property, and let
def
∂p ET = Σ ∪ (E × {0}). For a compact set K ⊂ ET define
1
|x − y| + |t − s| 2 .
dist(K, ∂p ET ) =
inf
(x,t)∈K
(y,s)∈∂p ET
15 Gaussian Bounds on the Fundamental Solution
423
1,2
Theorem 14.1. Let u ∈ Cloc (0, T ; L2loc (E)) ∩ L2loc (0, T ; Wloc
(E)) be a local
weak solution of (2.2) and let (14.1) hold. Then, u is locally bounded and
locally Hölder continuous in ET . Moreover, for every compact set K ⊂ ET ,
there exist positive constants γK > 1 and α ∈ (0, 1), depending only on the
data, such that
!α
1
|x1 − x2 | + |t1 − t2 | 2
|u(x1 , t1 ) − u(x2 , t2 )| ≤ γK
dist(K, ∂p ET )
for every pair of points (x1 , t1 ), (x2 , t2 ) ∈ K.
Theorem 14.2. Let u ∈ V˜2 (ET ) be a solution of the Cauchy–Dirichlet problem (8.1), with f and f satisfying (14.1), uo ∈ C αo (Ē) for some exponent
1 1
αo ∈ (0, 1), ϕ ∈ H 2 , 2 (Σ) and Hölder continuous in Σ for some exponent
ǫ ∈ (0, 1). Then, u is Hölder continuous in ET and there exist constants
γ > 1 and α ∈ (0, 1), depending only upon the data, the C 1 structure of ∂E,
the Hölder norm k|ϕ|kǫ;Σ , and the Hölder norm k|uo |kαo ;Ē such that
α
1
|u(x1 , t1 ) − u(x2 , t2 )| ≤ γ |x1 − x2 | + |t1 − t2 | 2
for every pair of points (x1 , t1 ), (x2 , t2 ) ∈ ET .
Theorem 14.3. Let u ∈ V˜2 (ET ) be a solution of the Cauchy–Neumann
problem (9.1), with f and f satisfying (12.1), ψ ∈ LN +1+σ (∂E) for some
σ ∈ (0, 1), uo ∈ C αo (Ē) for some exponent αo ∈ (0, 1). Then, u is Hölder
continuous in ET , and there exist constants γ and α ∈ (0, 1), depending on the
data, the C 1 structure of ∂E, kψkN +1+σ;∂E , and the Hölder norm k|uo |kαo ;Ē
such that
α
1
|u(x1 , t1 ) − u(x2 , t2 )| ≤ γ |x1 − x2 | + |t1 − t2 | 2
for every pair of points (x1 , t1 ), (x2 , t2 ) ∈ ET .
The precise structure of these estimates in terms of the initial condition uo ,
Dirichlet data ϕ, or Neumann data ψ, as well as the dependence on the structure of ∂E, is specified in more general theorems for functions in the parabolic
DeGiorgi classes (Theorem 8.1 and Theorem 9.1 of the next Chapter). These
are the key, seminal facts in the theory of regularity of solutions of parabolic
equations in divergence form. They can be used, by boot-strap arguments,
to establish further regularity on the solutions, whenever further regularity is
assumed on the coefficients.
15 Gaussian Bounds on the Fundamental Solution
In Section 2 of Chapter 5 we studied the fundamental solution of the heat
equation, and in particular, in Theorem 2.1 we discussed the role it plays in
424
11 LINEAR PARABOLIC EQUATIONS
solving the Cauchy problem in RN . The linearity of the heat equation is a
central feature in these estimates.
When considering (2.2) with f = f = 0 in RN × (0, T ), since the linearity
is preserved, it is quite natural to define a fundamental solution in this more
general framework as well, and inquire about its structure.
In other words, we seek a smooth function (t, x, y) 7→ Γ (x, t; y) defined on
RN × (0, +∞) × RN such that, for every y ∈ RN , Γ (x, t; y) is a solution of
(∂t − div aij (x, t)∇) u = 0,
satisfies
Z
(15.1)
Γ (x, t; y) dx = 1
RN
∀ y ∈ RN , ∀ t > 0, and for any ϕ ∈ C0∞ RN
Z
Γ (x, t; y)ϕ(y) dy
u(x, t) =
RN
tends to ϕ(x) as t → 0. In other words, Γ allows us to solve the Cauchy
problem for (15.1) in RN . When the initial condition is assigned not at time
0 but at a general time s, the corresponding fundamental solution depends
on s too, and we write Γ (x, t; y, s). As a straightforward consequence of its
definition, we have
Z
Γ (x, t; y, s) =
Γ (x, t; z, τ )Γ (z, τ ; y, s) dz.
RN
This is usually referred to as reproducing property. We assume that
aij ∈ C ∞ (RN × (0, T )) ∩ L∞ (RN × (0, T ))
and the N × N matrix aij satisfies the structure (2.1). The C ∞ regularity of
the coefficients aij allows us to easily justify all the computations to follow,
but otherwise plays no role.
Whenever we need to highlight the correspondence with the matrix aij ,
we will write Γa .
Quite surprisingly, it turns out that the estimates above and below for Γ
very strongly resemble the fundamental solution of the heat equation. Indeed,
there exists a positive constant C, which depends only on λ, Λ, N , such that
for s < t we have
2
|x−y|2
exp − C|x−y|
C
exp
−
t−s
C(t−s)
≤ Γ (x, t; y, s) ≤
.
N
N
C(t − s) 2
(t − s) 2
Upper and lower bounds on Γ were first shown by Aronson ([11]), and then
by Davies ([44]). In the first case, the proof relies on the parabolic Harnack
15 Gaussian Bounds on the Fundamental Solution
425
inequality (see Chapter 12), whereas in the second instance such an inequality
is not used. Anyway, in both examples, the connection with Nash’s ideas,
developed in his foundational paper [190], is not clear.
Here, we derive the bound above on Γ , following Fabes’ interpretation of
the original ideas by Nash [68, 190] (see also Fabes and Stroock [69]). For
the bound below, again we rely on Nash’s paper, combining the presentation
given in Fabes [68] and Semenov [229].
As shown, for example, in Fabes and Stroock [69], once the estimates on the
fundamental solution Γ are available, the local Hölder continuity of its local
solutions, and the Harnack inequality can be proven. We refrain from studying
such properties in this chapter; we will deal with them in Chapter 12, in the
more general context of parabolic DeGiorgi classes. Here, we limit ourselves
to the bounds on Γ , which represent a very interesting issue in themselves.
In both coming sections, whenever we refer to the data, we mean the set
{λ, Λ, N }. Moreover, we denote with ∂t and ∂xi the partial derivatives with
respect to t and xi respectively.
15.1 The Gaussian Upper Bound
We state and prove a number of introductory results, before coming to the
main statement.
Proposition 15.1 There exists a positive constant γ, depending only on the
data, such that for s < t and all x ∈ RN we have
Z
1
|x − y|Γ (x, t; y, s) dy ≤ γ(t − s) 2 .
RN
In order to prove Proposition 15.1, we need some introductory Lemmata.
Lemma 15.1 There exists a positive constant γ, depending only on the data,
such that for s < t and all x, y ∈ RN we have
Γ (x, t; y, s) ≤
γ
N
(t − s) 2
.
Proof. Fix t > 0, let p(x, t) = Γ (x, t; 0, 0), and set
Z
p2 (x, t) dx.
u(t) =
RN
Differentiating, we have
Z
Z
p div (aij ∇p) dx
(∂t p) p dx = 2
u′ (t) = 2
RN
RN
Z
Z
|∇p|2 dx.
= −2
aij ∂xj p ∂xi p dx ≤ −2λ
RN
RN
426
11 LINEAR PARABOLIC EQUATIONS
If we rely on the so-called Nash inequality (see Section 15c of the Complements for the proof)
Z
1+ N2
Z
p2 dx
≤ γ(N )
RN
Z
and take into account that
RN
Z
|∇p|2 dx
RN
N4
,
p dx
(15.2)
p dx = 1, we have
RN
Z u(t)
Z t
λ
du
2
u1+ N ,
≤
−γ
ds,
1+2/N
γ(N )
u(t/2) u
t/2
N −2/N
N
t
N
N
u
(t/2) − u−2/N (t) ≤ −γ ,
γt + u−2/N (t/2) ≤ u−2/N (t),
2
2
2
2
2
1
1
2/N
−2/N
u
≤
,
γt + u
(t/2) ≤ 2/N ,
u
(t)
γt + u−2/N (t/2)
u′ (t) ≤ −2
and therefore,
u ≤ γt−N/2
⇔
Z
RN
Γ 2 (x, t; 0, 0) dx ≤ γt−N/2 .
By the translation invariance and the reproducing property, we easily obtain
Z
Γ 2 (x, t; y, s) dy ≤ γ(t − s)−N/2
RN
whenever s < t, and also
Z
RN
Γ 2 (x, t; y, s) dx ≤ γ(t − s)−N/2 .
Since for s < t
Γ (x, t; y, s) =
Z
RN
t + s t + s
Γ x, t; z,
; y, s dz,
Γ z,
2
2
by the Hölder inequality
Γ (x, t; y, s) ≤
Z
RN
1/2 Z
t+s
1/2
t + s
Γ 2 x, t; z,
dz
; y, s dz
Γ 2 z,
2
2
RN
≤ γ(t − s)−N/2 .
Lemma 15.2 There exists a constant γ > 0, depending only on the data,
such that for all y ∈ RN ,
Z
|x − y|Γ (x, t; y, s) dx ≤ γ(t − s)1/2 .
RN
15 Gaussian Bounds on the Fundamental Solution
427
Proof. As in the proof of Lemma 15.1, we can take y = 0, s = 0. Moreover,
as before, we let p(x, t) = Γ (x, t; 0, 0), and define
Z
M1 (t) =
|x| p(x, t) dx.
RN
We have
M1′ (t) =
Z
|x|∂t p(x, t) dx =
Z
|x| div (aij ∂xi p) dx
Z
xj
1 √ xj
aij ∂xi p
=−
aij ∂xi p √ p
dx = −
dx
|x|
p
|x|
RN
RN
1/2 Z
1/2
Z
∂xi p ∂xj p
xi xj
dx
p dx
aij
≤
aij
p
|x| |x|
RN
RN
Z
1/2
|∇p|2
=C
dx
,
p
RN
RN
Z
RN
where the constant C > 0 depends only on the data. Now, we introduce the
entropy
Z
Q(t) = −
p(x, t) ln p(x, t) dx.
RN
If we differentiate, we have
Z
′
(∂t p ln p + ∂t p) dx
Q (t) = −
N
ZR
=−
div aij ∂xj p ln p + div aij ∂xj p dx
N
Z R
∂x p ∂xi p
dx.
=
aij j
p
RN
Hence, by the ellipticity condition
λ
Z
RN
|∇p|2
dx ≤ Q′ (t) ≤ Λ
p
Z
RN
2
|∇p|
dx,
p
and we conclude that
M1′ (t)
′
≤ C [Q (t)]
1/2
1
= C Q (t)t
t
′
1/2
≤ Ct
1/2
1
Q (t) +
.
t
′
If we integrate from ǫ > 0 to t, we have
Z t
Z t
1
M1 (t) − M1 (ǫ) ≤C
τ 1/2 Q′ (τ ) dτ + C
dτ
1/2
τ
ǫ
ǫ
i
h
≤2C t1/2 − ǫ1/2 + C Q(t)t1/2 − Q(ǫ)ǫ1/2
(15.3)
428
11 LINEAR PARABOLIC EQUATIONS
−
C
2
Z
t
Q(τ )
√ dτ,
τ
ǫ
def
with C > 0. If we let as usual Q− = max{0; −Q}, we obtain
Z
C t Q− (τ )
√
dτ.
M1 (t) − M1 (ǫ) ≤ 2Ct1/2 + CQ(t)t1/2 +
2 ǫ
τ
(15.4)
We need an estimate for Q− . Since from Lemma 15.1 we have
Γ (x, t; 0, 0) = p(x, t) ≤ γt−N/2 ,
we get
− ln p ≥ −C +
N
ln t
2
and also
because
Z
Z
Q(t) = −
p(x, t) ln p(x, t) dx
N
Z R
min (− ln p(x, t)) p(x, t) dx
≥
N
RN x∈R
Z N
N
−C +
≥
ln t p(x, t) dx = −C +
ln t,
2
2
RN
p(x, t) dx = 1. Therefore, we have proved that
RN
Q(t) ≥ −C +
N
ln t,
2
(15.5)
which implies that
N
| ln t|.
2
If we insert (15.6) in (15.4) and let t → 0+ , we conclude that
Z
C t | ln τ |
√ dτ.
M1 (t) ≤ 2Ct1/2 + CQ(t)t1/2 +
2 0
τ
Q− (t) ≤ C +
Now we want to estimate M1 from below. If we consider
g : (0, +∞) → R,
defined by
g(p) = p ln p + σp,
since g ′ (p) = ln p + σ + 1, we immediately obtain
min
p∈(0,+∞)
g(p) = −e−σ−1 .
Hence, if we choose σ = a|x| + b, we have
(15.6)
15 Gaussian Bounds on the Fundamental Solution
429
p(x, t) ln p(x, t) + (a|x| + b)p(x, t) ≥ −e−b−1 e−a|x|
and also, if we integrate over RN ,
Z
Z
−b−1
[p(x, t) ln p(x, t) + (a|x| + b)p(x, t)] dx ≥ −e
e−a|x| dx,
RN
RN
which implies
−Q(t) + aM1 (t) + b ≥ −e
=−
−b−1
e−b−1
aN
Z
dω
|ω|=1
Z
∞
e−aρ ρN −1 dρ
0
Z
dω
|ω|=1
Z
∞
e−aρ (aρ)N −1 d(aρ),
0
that is,
−Q(t) + aM1 (t) + b ≥ −CN e−b a−N .
If we choose a =
1
, e−b = aN , we obtain
M1 (t)
−Q(t) + 1 − N ln
1
≥ −CN ,
M1 (t)
that is,
Q(t) ≤ CN + 1 + N ln M1 (t),
which yields
1
CM1 ≥ e N Q(t) .
Hence, we have the following estimates
Q(t) ≥ −C +
N
ln t;
2
(15.7)
M1 (t) ≤ 2Ct1/2 + CQ(t)t1/2 +
1
CM1 (t) ≥ e N Q(t) .
C
2
Z
t
0
| ln τ |
√ dτ ;
τ
(15.8)
(15.9)
From (15.3) and (15.9), we have
1
γ1 e N Q(t) ≤ M1 (t) ≤ γ2
Z
t
[Q′ (τ )]1/2 dτ.
0
N
Now define N R(t) = Q(t) + C −
ln t. By what we have just seen, N R ≥ 0.
2
N
Moreover, Q′ (t) = N R′ (t) + . Hence,
2t
1/2
Z t
C
1
′
R(t)− N
+ 12 ln t
1/2
R (τ ) +
γ1 e
≤ M1 (t) ≤ N γ2
dτ,
2τ
0
430
11 LINEAR PARABOLIC EQUATIONS
"Z #
Z tr
1/2
t
√ R(t)
τ ′
1
γ3 t e
≤ M1 (t) ≤ γ2
dτ +
R (τ ) dτ ,
2τ
2
0
0
#
"
r
Z t
√
√ R(t)
1 R(τ )
t
√ √ dτ ,
R(t) −
γ3 t e
≤ M1 (t) ≤ γ2
2t +
2
τ
0 2 2
1
1
where we have used that when a and a + b are positive, (a + b) 2 ≤ a 2 +
this yields
b
1
2a 2
;
√ R(t)
√ √
1
γ3 t e
≤ M1 (t) ≤ γ2 t
2 + √ R(t) .
2
Since the right-hand side grows slower than the left-hand one, we conclude
that R(t) has to be bounded above and
√
√
γ4 t ≤ M1 (t) ≤ γ5 t.
Remark 15.1 Since
1
γ1 e N Q(t) ≤ M1 (t),
we have
1
γ1 e N Q(t) ≤ γ5 t1/2
⇒
Q(t) ≤ γ8 +
N
ln t.
2
Hence, we can say that
−C +
N
N
ln t ≤ Q(t) ≤ C +
ln t.
2
2
Proof of Proposition 15.1 – The bounds above and below on M1 (t) proved in
Lemma 15.2 amount to the bound required by Proposition 15.1, again because
of the properties of the fundamental solution.
Proposition 15.2 For each j ∈ N, there exists Cj > 0, depending only on
the data and j, such that
Z
def
Mj (t) =
|x|j Γ (x, t; 0, 0) dx ≤ Cj tj/2 .
RN
Proof. It obviously suffices to take j ≥ 2. As before, we let p(x, t) =
Γ (x, t; 0, 0). If we differentiate, we obtain
Z
Z
Mj′ (t) =
|x|j ∂t p dx =
|x|j div (aik ∂xk p) dx
RN
RN
Z
xi
dx
= −j
aik ∂xk p |x|j−1
|x|
N
ZR
Z
j
|∇p| √
p dx
≤ Cj
|x|j−1 |∇p| dx = Cj
|x| 2 |x|j/2−1 √
p
N
N
R
R
15 Gaussian Bounds on the Fundamental Solution
Z
≤ Cj
RN
1/2
= Cj Mj
1/2 Z
|x| p(x, t) dx
|x|
|∇p|2
dx
p
.
j
Z
RN
|x|j−2
RN
1/2
j−2 |∇p|
2
p
431
1/2
dx
Hence,
1/2
Z t Z
q
2
j−2 |∇p|
Mj (t) ≤ Cj
|x|
dx
ds.
p
0
RN
Z
If we now define Qj (t) = −
|x|j p ln p dx, we easily obtain
RN
Z
Q′j = −
RN
=−
Z
=j
(15.10)
Z
|x|j [∂t p ln p + ∂t p] dx
|x|j [div (aik ∂xk p)] (1 + ln p) dx
Z
|x|j−2 aik ∂xk p xi (1 + ln p) dx +
RN
RN
|x|j aik
RN
∂xk p ∂xi p
dx.
p
Therefore,
Z
Z
|∇p|2
|x|j
dx ≤ ΛQ′j + Λ2 j
|x|j−1 |∇p|(1 + | ln p|) dx
p
N
N
R
R
Z
j−1 √
j−1 |∇p|
′
2
=ΛQj + Λ j
|x| 2 p(1 + | ln p|) |x| 2 √ dx
p
RN
≤ΛQ′j
+ Λ2 j
Z
RN
≤ΛQ′j
+ Cj Λ
Z
RN
|x|j−1
|x|
1/2 Z
1/2
|∇p|2
dx
|x|j−1 p(1 + | ln p|)2 dx
p
RN
j−1 |∇p|
p
2
1/2 Z
dx
Mj−1 +
RN
|x|
j−1
1/2
p| ln p| dx
.
2
We conclude that
Z
|∇p|2
dx ≤ ΛQ′j
|x|j
p
N
R
Z
Z
|∇p|2
|x|j−1 p| ln p|2 dx .
dx + Mj−1 +
+ Cj
|x|j−1
p
RN
RN
If we iterate, we have
Z
X
X
|∇p|2
|x|j
dx ≤
Ckj Q′k (t) +
C̃kj Mk (t)
p
RN
k≤j
k<j
Z
X
+
|x|k p | ln p|2 dx
C̃kj
k<j
RN
(15.11)
432
11 LINEAR PARABOLIC EQUATIONS
where Ckj > 0, C̃kj ≥ 0 depend only on k, j, and the data. Inserting (15.11)
in (15.10) yields
1/2
Mj
(t) ≤
Z th X
0
Ckj Q′k (s) +
k≤j−2
+
X
C̃kj
k<j−2
Z
RN
X
C̃kj Mk (s)
k<j−2
i1/2
|x|k p | ln p|2 dx
ds.
(15.12)
Now, we need to control the right-hand side of (15.12). This is given by the
following result.
Lemma 15.3 Set
Pk (t) =
Z
RN
|x|k p(x, t)| ln p(x, t)|2 dx.
Then there exists a positive constant Ck , depending only on k, and the data,
such that
1
Pk (t) ≤ Ck 1 + ln2 t Mk (t) + √ Mk+1 (t) + tk/2 .
t
Proof. Since
√ x
x
−N/2
√
t √ , t · 1; 0, 0 = t
Γa( t· ,t· ) √ , 1; 0, 0 ,
Γa (x, t; 0, 0) = Γa
t
t
it suffices to assume t = 1. We let p(x) = Γ (x, 1; 0, 0), and we want to show
that
Z
|x|k p(x)| ln p(x)|2 dx ≤ Ck [Mk (1) + Mk+1 (1) + 1] .
RN
Since p is bounded by Lemma 15.1, we have
Z
Z
k
2
|x|k p2 dx
|x| p| ln p| dx ≤ C
RN
RN ∩{p>1/e}
Z
|x|k p dx = C̃Mk (1).
≤ C̃
RN
Moreover, for any r ∈ R+
√
√
max p ln2 p − r = e− 1+r−1 [2 1 + r + 2].
0≤p≤1/e
Hence,
Z
RN ∩{0<p<1/e}
|x|k p | ln p|2 dx
15 Gaussian Bounds on the Fundamental Solution
=
Z
|x|k p | ln p|2 − |x| + |x| dx
RN ∩{0<p<1/e}
=
Z
|x|k p | ln p|2 − |x| dx + Mk+1 (1)
RN ∩{0<p<1/e}
≤C
433
Z
RN ∩{0<p< e1 }
|x|k e−
√
1+|x|
dx + Mk+1 (1)
≤ Ck + Mk+1 (1).
If we now go back to (15.12), we have
Z th X
X
1/2
Mj (t) ≤
Ckj Q′k (s) +
0
k≤j−2
C̃kj (Mk (s) + Pk (s))
0≤k<j−2
i1/2
ds.
Now we work by induction, recalling that Mo (t) ≡ 1, M1 (t) ≤ C1 t1/2 . Take
j ≥ 2, and assume that Mk (t) ≤ Ck tk/2 for all 0 ≤ k < j. Relying on all the
previous estimates, we have
Z th X
X
i1/2
1/2
Ckj Q′k (s) + Cj
Mj (t) ≤
sk/2 1 + ln2 s
ds.
0
k≤j−2
0≤k<j−2
If we take t = 1, we have
Z 1h X
i1/2
1/2
Ckj Q′k (s) + Cj s−1
ds
Mj (1) ≤
0
k≤j−2
1/2
≤Cj
Z
1
s−1/2 ds +
0
1/2
≤2Cj
1
X
1
1/2
2Cj
Z
1
X
0 k≤j−2
Ckj Q′k (s)s1/2 ds
Ckj Qk (1)
1/2
2Cj k≤j−2
Z 1 X
1
1
−
Ckj Qk (s) 1/2 ds.
1/2
s
0 k≤j−2
4C
+
j
1/2
1/2
ej > 0 such
Since |Qk (t)| ≤ Pk (t)Mk (t), by induction we conclude that ∃C
that Mj (1) ≤ C̃j . Scaling back, we conclude.
As
Γa (x, t; y, s) = Γa(x+·,t−·) (y − x, t − s; 0, 0),
we also have the following.
Corollary 15.1 For each j ∈ N, there exists Cj > 0 such that ∀s < t
Z
|x − y|j Γ (x, t; y, s) dy ≤ Cj (t − s)j/2 .
RN
434
11 LINEAR PARABOLIC EQUATIONS
We now have all the elements, to state and prove the Gaussian upper bound
on the fundamental solution
Theorem 15.1. There exists a constant C > 0, depending only on the data,
such that
2
Γ (x, t; y, s) ≤ C(t − s)−N/2 e−|x−y| /C(t−s) .
As with Propositions 15.1–15.2, we need a couple of auxiliary results.
Lemma 15.4 Let
ϕj (t) =
Z
RN
|x|2j Γ 2 (x, t; 0, 0) dx.
Then, there exists a positive constant C, depending only on the data, such that
N
ϕj (t) ≤ C j j! t− 2 +j .
Proof. From Lemma 15.1 and Proposition 15.2
Z
ϕj (t) ≤ Ct−N/2
|x|2j Γ (x, t; 0, 0) dx ≤ CCj t−N/2+j .
RN
Hence, for 0 ≤ j ≤
that
Now, take j >
N 2
N + 1, we can find B, depending only on the data, such
N
ϕj (t) ≤ B j j! t− 2 +j .
+ 1. Setting as usual p(x, t) = Γ (x, t; 0, 0), we have
Z
2j
′
|x| p ∂t p dx = 2
|x|2j p div (aik ∂xi p) dx
ϕj (t) = 2
RN
RN
Z
Z
xk
p|x|2j−1 aik ∂xi p
= −2
|x|2j aik ∂xi p ∂xk p dx − 4j
dx
|x|
N
N
R
R
Z
Z
|x|2j−1 p |∇p| dx − 2λ
|x|2j |∇p|2 dx
≤ C̃j
RN
RN
Z
Z
j−1
j
= C̃j
|x|
p|x| |∇p| dx − 2λ
|x|2j |∇p|2 dx
RN
RN
Z
≤ C̃j 2
|x|2(j−1) p2 dx,
2
Z
RN
that is
Since j >
N 2
If we assume
ϕ′j (t) ≤ C̃j 2 ϕj−1 (t).
+ 1, we have ϕj (0) = 0. Hence,
ϕj (t) ≤ C̃j 2
Z
0
t
ϕj−1 (s) ds.
15 Gaussian Bounds on the Fundamental Solution
435
N
ϕj−1 (s) ≤ C j−1 (j − 1)! s− 2 +(j−1) ,
we obtain
N
ϕj (t) ≤ C̃j 2 C j−1 (j − 1)!
For j >
N 2
j
t− 2 +j
= C̃j! C j−1
N
−2 +j
j−
N
N
2
t− 2 +j .
⌊ N2 ⌋+1
= DN . Hence, choosing
⌊ N2 ⌋+1− N2
o
n
C ≥ max B, C̃DN ,
j
j− N
2
+ 1, we have
≤
we can conclude by induction.
Corollary 15.2 There exists a constant C > 0, depending only on the data,
such that, for all j ∈ N
|x|j Γ (x, t; 0, 0) ≤ C j (j!)1/2 t−N/2+j/2 .
Proof. By the reproducing property of the fundamental solution, we have
Z
t t
Γ (x, t; 0, 0) =
Γ x, t; y,
Γ y, ; 0, 0 dy,
2
2
RN
and also
Z
t t
Γ y, ; 0, 0 dy
|x − y|j Γ x, t; y,
2
2
RN
Z
t
t
Γ y, ; 0, 0 dy.
+ 2j
|y|j Γ x, t; y,
2
2
RN
|x|j Γ (x, t; 0, 0) ≤2j
If we now apply the Schwarz inequality, and Lemma 15.4, we conclude.
Eventually, we finish the proof of Theorem 15.1.
Proof. As always, we can take y = 0, s = 0. By Corollary 15.2,
e
δ|x|2
t
Γ (x, t; 0, 0) =
≤
∞
X
δj
|x|2j Γ (x, t; 0, 0)
j
j!
t
j=0
∞
X
δj
j=0
j!
whence
tN/2 e
δ|x|2
t
Γ (x, t; 0, 0) ≤
C 2j t−N/2 [(2j)!]1/2 ,
∞
X
δ j C j ((2j)!)1/2
j=0
j!
where we have relabeled C 2 as C. Now, choosing δ =
1
4C
,
yields
436
11 LINEAR PARABOLIC EQUATIONS
|x|2
tN/2 e 4Ct Γ (x, t; 0, 0) ≤
∞
X
((2j)!)1/2
4j j!
j=0
.
n n
√
2πn
. Hence, for n sufficiently large
e
√
2j 1/2
∞
∞
4πj 2j
X
X
e
1
.
=
=
√
j
1/4
j
π j 1/4 2j
4j 2πj
j=n
j=n
We recall the Stirling formula n! ∼
∞
X
((2j)!)1/2
4j j!
j=n
e
Since the series on the right-hand side has finite sum, we conclude that
Γ (x, t; 0, 0) ≤
C
|x|2
tN/2
e− 4Ct .
15.2 The Gaussian Lower Bound
We now come to the lower bound. Assume for the moment that we have the
following.
Proposition 15.3 There exists a positive constant C, depending only on the
data, such that ∀ xo ∈ RN , ∀ x, y ∈ B√t−s (xo ), ∀s < t, we have
Γ (x, t; y, s) ≥
1
.
C(t − s)N/2
Assuming the proposition for the moment, we prove this lower bound, by an
argument due to Fabes and Stroock [69].
Theorem 15.2. There exists a constant C > 0, depending only on the data,
such that
C|x − y|2
1
.
exp
−
Γ (x, t; y, s) ≥
t−s
C(t − s)N/2
Proof. We assume y = 0, s = 0. Moreover, by the usual properties of the
fundamental solution, we can also take t = 1. Hence, it suffices to prove
Γ (x, 1; 0, 0) ≥
1
exp −C|x|2 .
C
In view of Proposition 15.3, we may also take |x| > 1. Given x ∈ RN with
|x| > 1, let k ∈ N be the smallest integer such that k > 4|x|2 , and let
S=
k−1
Y
l=1
B
1
√
2 k
l
x .
k
For (ξ1 , ξ2 , . . . , ξk−1 ) ∈ S, by simple computation we have
15 Gaussian Bounds on the Fundamental Solution
1
|ξ1 | < √ ,
k
1
max |ξl − ξl−1 | < √ ,
k
1<l<k
437
1
|x − ξk−1 | < √ .
k
Hence, by Proposition 15.3
Z
Z
k−1
·
Γ (x, 1; 0, 0) =
···
Γ x, 1; ξk−1 ,
k
RN
RN
k−1
1
k−2
· Γ ξk−1 ,
· · · Γ ξ1 , ; 0, 0 dξ1 . . . dξk−1
; ξk−2 ,
k
k
k
N/2 k
k
|S|
≥
C
N/2 k "
N #k−1
k
1
=
ωN
C
21/2 k 1/2
k 2N/2 k N/2
ω
N
=
N/2
ωN
C2
and from here we immediately conclude.
Proof of Proposition 15.3 – Without loss of generality, we may assume s = 0,
and work with t > 0. For simplicity, we let p(x, t, y) = Γ (x, t; y, 0). We want
to show that there exists k > 0 such that ∀ xo ∈ RN , ∀ x, y ∈ B√t (xo ), ∀ t > 0
p(x, t, y) ≥ kt−N/2 ,
where k depends only on the data.
Again, without loss of generality, we can assume xo = 0. In the following
we let
1
|x − y|2
pH (x, t, y) =
exp −
4t
(4πt)N/2
and
po (x, t) = pH (x, δt; 0)
for any δ > Λ. We have
Γ (x, 2t; y, 0) =
Z
Γ (x, 2t; z, t) Γ (z, t; y, 0) dz
RN
=
=
Z
ZR
N
RN
Γa(·,·+t) (x, t; z, 0) Γ (z, t; y, 0) dz
Γa(·,·+t) (x, t; z, 0) Γa(·,t−·)(y, t; z, 0) dz.
If we let
p+ (x, t, z) = Γa(·,·+t) (x, t; z, 0),
p− (y, t, z) = Γa(·,t−·) (y, t; z, 0),
438
11 LINEAR PARABOLIC EQUATIONS
this yields
p(x, 2t, y) =
Z
RN
Z
p+ (x, t, z)p− (y, t, z) dz
po (z, t)
p+ (x, t, z) p− (y, t, z) dz
kpo k∞
Z
= (4πδt)N/2
pH (z, δt, 0) p+ (x, t, z) p− (y, t, z) dz.
≥
RN
RN
By Jensen’s inequality
Z
ln
pH (z, δt, 0) p+ (x, t, z) p− (y, t, z) dz
RN
Z
≥
ln[p+ (x, t, z) p− (y, t, z)] pH (z, δt, 0) dz
N
ZR
Z
≥
ln p+ (x, t, z) pH (z, δt, 0) dz +
ln p− (y, t, z) pH (z, δt, 0) dz.
RN
RN
Hence,
N
N
ln(4πδ) +
ln t + Θx+ (t) + Θy− (t),
2
2
Z
where we have set Θz± (t) =
ln p± (z, t, s) pH (s, δt, 0) ds. Therefore, if we
ln p(x, 2t, y) ≥
RN
can show that
N
ln t + γ
(15.13)
2
for some positive γ, depending only on the data, we have finished. In the
following we proceed with a general p(x, t, z), irrespective of any time-shift.
In order to prove (15.13), we introduce Nash’s function
Z
G(τ ) =
pH (0, δt, s) ln p(z, τ, s) ds
∀z ∈ B√t
Θz± (t) ≥ −
RN
for
t
2
≤ τ ≤ t. It is apparent that
Z
pH (0, δt, s) ln p(z, t, s) ds = Θz (t).
G(t) =
RN
If we differentiate, we obtain
15 Gaussian Bounds on the Fundamental Solution
Z
439
1
∂τ p(z, τ, s) ds
p(z, τ, s)
Z
1
div aij ∂sj p ds
pH (0, δt, s)
=
p(z, τ, s)
RN
Z
pH
=−
aij ∂si p ∂sj
ds
p
N
R
Z
Z
∂s p ∂sj pH
∂s p ∂sj p
=−
aij j
pH ds +
aij i
pH ds
p
pH
p
p
RN
RN
Z
Z
∂s p ∂sj p
1
∂s p ∂sj p
≥
aij i
pH ds −
aij i
pH ds
p
p
2 RN
p p
RN
Z
1
1
∂s pH ∂sj pH
1
pH ds = − Jo + J,
aij i
−
2 RN
pH
pH
2
2
G′ (τ ) =
pH (0, δt, s)
RN
where we have set
Z
∂s pH ∂sj pH
pH ds,
Jo =
aij i
pH
pH
RN
J=
Z
aij
RN
∂si p ∂sj p
pH ds.
p
p
If we now let
def p
ϕo (0, t, s) =
pH (0, t, s),
def
ϕ̂o (s) = ϕo (0, δt, s),
it is a matter of straightforward computation to check that
Z
N
N
|s|2
2
ϕ̂
−
|∇ϕ̂o | ds ≤
ϕ̂
=
0,
and
.
−∆s ϕ̂o +
o
o
(4δt)2
4δt
4δt
N
R
Hence, we can estimate Jo from above in the following way
Z
Z
|∇pH |2
2
Jo ≤ Λ
ds = 4Λ
|∇ϕ̂o | ds
pH
RN
RN
ΛN
Λ d N
=
=2
ln t .
δ t
δ dt 2
In order to estimate J from below, we rely on the following Poincaré-type
inequality
Z
Z 2
|∇f |2 ϕo ds
f − ϕ2o f ϕo ds ≤ 2t
N
N
R
R
Z
2
ϕo f ds. Hence, we have
where ϕ2o f =
RN
J=
Z
aij
RN
Z
∂si p ∂sj p
pH ds
p
p
aij ∂si ln p ∂sj ln p pH ds
Z
λ
2
|∇ ln p| pH ds =
≥λ
(ln p − G(τ ))2 pH ds.
2δt RN
RN
=
RN
Z
440
11 LINEAR PARABOLIC EQUATIONS
We can then conclude that
Z
Λ d N
λ
′
G (τ ) ≥ −
ln t +
pH (ln p − G(τ ))2 ds.
δ dt 2
4δt RN
Setting Q̃(t) =
N
2
ln t, it is immediate to check that for τ ≤ t, we have
Q̃′ (τ ) ≥ Q̃′ (t).
Taking into account that δ ≥ Λ, we obtain
Z
λ
′
(G(τ ) + Q̃(τ )) ≥
pH (ln p − G(τ ))2 ds.
4δt RN
(15.14)
By the upper bound of Theorem 15.1 and thanks to the triangle inequality
t
∀ z ∈ B√t , τ ∈
, t , δ > Λ, pH (0, δt, s) ≥ k p(z, τ s),
(15.15)
2
where k > 0 depends only on the data. Inequality (15.15) and the Cauchy
inequality yield
Z
2 Z
Z
k
p| ln p − G(τ )| ds ≤
p| ln p − G(τ )|2 ds
p ds k
RN
RN
RN
Z
≤
pH (ln p − G(τ ))2 ds,
RN
so that
d
dτ
Z
2
N
λk
G(τ ) +
ln τ ≥
p| ln p − G(τ )| ds
2
4δt RN
and also
Z
Z
p| ln p − G(τ )| ds ≥
p(ln p − G(τ )) ds
N
RN
ZR
Z
p ln p ds − G(τ )
=
RN
RN
p ds = −Q(τ ) − G(τ ).
Taking into account (15.5), we have
Z
N
ln τ − C.
p |ln p − G(τ )| ds = −G(τ ) −
2
N
R
We distinguish two alternatives:
a) If ∀ τ ∈ 2t , t − G(τ ) − N2 ln τ − 2C ≥ 0, then
′
2
N
N
λk
G(τ ) +
−G(τ ) −
ln τ ≥
ln τ − C
2
4δt
2
2
N
λk
−G(τ ) −
ln τ ,
≥
16δt
2
3c The Homogeneous Dirichlet Problem
whence, after integration over τ in the interval
G(t) +
b) If for some τ ∈
441
t 2, t ,
32δ
N
ln t ≥
.
2
λk
t N
2 , t we have −G(τ ) − 2 ln τ − 2C ≤ 0, then by (15.14)
G(t) +
N
N
ln t ≥ G(τ ) +
ln τ ≥ −2C
2
2
and in any case we have finished.
Problems and Complements
3c The Homogeneous Dirichlet Problem
In Section 5 of Chapter 9 the homogeneous Cauchy–Dirichlet problem (4.1)
was solved using the Riesz Representation Theorem, which is usually referred
to as the Lax–Milgram Theorem in this context. It turns out that in the
parabolic case there exists a corresponding result, which plays a comparable
role.
Let H be a Hilbert space, endowed with its inner product (·, ·) and the
norm |·|. We identify H with its dual space H ′ . Moreover, we consider a second
Hilbert space V , with norm k · k. We assume that V ⊂ H, with continuous
and dense imbedding, such that
V ֒→ H ֒→ V ′ .
Choose T > 0 and suppose that for a.e. t ∈ [0, T ] we have a bilinear form
a(t; ·, ·) : V × V → R, which satisfies the following properties:
1. ∀ u, v ∈ V the application t 7→ a(t; ·, ·) is measurable;
2. there exists M > 0 such that for a.e. t ∈ [0, T ], ∀ u, v ∈ V we have
|a(t; u, v)| ≤ M kuk kvk;
3. there exist α, C > 0 such that for a.e. t ∈ [0, T ], ∀ v ∈ V we have a(t; v, v)+
C|v|2 ≥ αkvk2 .
We have the following abstract result.
Theorem 3.1c. [174] Let f ∈ L2 (0, T ; V ′ ) and uo ∈ H. Then there exists a
unique function u ∈ C([0, T ]; H) ∩ L2 (0, T ; V ) satisfying
442
11 LINEAR PARABOLIC EQUATIONS
−
Z
T
(u(t), ϕ′ (t)) dt +
0
=
Z
Z
T
a(t; u(t), ϕ(t)) dt
0
T
(f (t), ϕ(t)) dt + (uo , ϕ(0))
0
for every ϕ ∈ Φ, where
Φ = {ϕ : ϕ ∈ C([0, T ]; V ), ϕ′ ∈ C([0, T ]; H), ϕ(0) = 0} .
For the proof, see, for example, page 46 of Lions [174], § 5 of Fujie and Tanabe
[88], or Chapter 3, § 1–4 of Lions and Magenes [175].
If we let
Z
aij (x, t)uxj vxi dx,
H = L2 (E), V = Wo1,2 (E), a(t; u, v) =
E
2
with aij as in Section 9.1, and f, f ∈ L (ET ), Theorem 3.1c ensures the
existence of a unique weak solution of the homogeneous Cauchy–Dirichlet
problem (3.1).
5c Existence of Solutions of the Homogeneous Dirichlet
Problem (3.1) by Galerkin Approximations
It is well-known that given an open set D ⊂ RN +1 and (to , xo ) ∈ D, for a
continuous vector field f : D → RN , the Cauchy problem
(
x′ (t) = f (t, x(t))
(5.1c)
x(to ) = xo
is equivalent to the integral equation
Z t
f (s, x(s)) ds.
x(t) = xo +
to
Moreover, for f continuous, any solution x = x(t) of the Cauchy problem is
of class C 1 . It is apparent that (5.1c) makes sense for a more general class of
functions f , if x is not required to have a continuous first-order derivative, but,
for example, it is assumed to be only absolutely continuous. This remark helps
us in extending the notion of solution of an ordinary differential equation.
Take an open set D ⊂ RN +1 and consider f : D → RN . We look for conditions on f , which ensure the existence of an absolutely continuous function
x : I ⊂ R → RN , such that (t, x(t)) ∈ D for t ∈ I and
x′ (t) = f (t, x(t))
(5.2c)
5c Existence of Solutions by Galerkin Approximations
443
for all t ∈ I, except on a set of zero Lebesgue measure. If such a function x
and interval I exist, we say that x is a solution of (5.2c). Moreover, a solution
x through the point (to , xo ) is said to be a solution of the Cauchy problem
(5.1c).
In the sequel, we will always assume that our conditions are satisfied except
on a set of zero Lebesgue measure, and therefore, we will not repeat it. We
have the following.
Theorem 5.1c (Carathéodory [29]). Let D ⊂ RN +1 be an open set. Suppose that f : D → RN , f = f (t, x) is measurable in t for each fixed x,
continuous in x for each t, and such that for each compact set K ⊂ D, there
exists an integrable function mK = mK (t) such that
|f (t, x)| ≤ mK (t)
(t, x) ∈ K.
Then, for any (to , xo ) ∈ D there exists a solution of (5.1c) in the sense given
before.
Moreover, if ϕ = ϕ(t) is a solution of (5.1c) on some interval J, ϕ can be
extended to a maximal interval of existence (a, b), and ϕ = ϕ(t) tends to the
boundary of D as t → a and t → b.
Finally, if under the previous conditions, for each compact set K ⊂ D,
there exists an integrable function h = hK (t) such that
|f (t, x) − f (t, y)| ≤ hK (t)|x − y|
(x, t) ∈ K, (t, y) ∈ K,
then for any (to , xo ) ∈ K there exists a unique solution of (5.1c) in the sense
given before.
Remark 5.1c It is apparent that for any linear system
x′ (t) = A(t)x(t) + b(t),
where A is a N × N matrix and b is an N vector, whose elements are integrable on any finite interval, all the previous conditions are satisfied, and
consequently, the corresponding Cauchy problem has a unique solution.
Proof of Theorem 5.1c – We follow the approach given in Hale [113]. Choose
α, β > 0 such that
def
R = {(t, x) : |t − to | ≤ α, |x − xo | ≤ β} ⊂ D,
def
define Iα = {t : |t − to | ≤ α}, let m(t) = mR (t), and M (t) =
Z
t
m(s) ds.
to
Pick ᾱ ∈ (0, α], β̄ ∈ (0, β] such that ∀ t ∈ Iᾱ , we have |M (t)| ≤ β̄, and let
A = ϕ ∈ C(Iᾱ ; RN ) : ϕ(to ) = xo , and ∀ t ∈ Iᾱ |ϕ(t) − xo | ≤ β̄ .
It is straightforward to check that A is a closed, bounded, convex subset of
C(Iᾱ ; RN ). For ϕ ∈ A and t ∈ Iᾱ , define
444
11 LINEAR PARABOLIC EQUATIONS
T ϕ(t) = xo +
Z
t
f (s, ϕ(s)) ds.
to
Since the fixed points of T in A coincide with the solutions of (5.1c), if we can
prove that such a fixed point exists, we conclude about existence. In order to
do so, we rely on the Schauder–Leray Fixed Point Theorem (see Theorem 3.1
of Chapter 14).
First of all, ∀ ϕ ∈ A, T is well-defined, since f (s, ϕ(s)) is integrable whenever ϕ ∈ A.
It is clear that T ϕ(to ) = xo , and ∀ t ∈ Iᾱ , T ϕ(t) is continuous at t.
Moreover, for all t ∈ Iᾱ
Z t
Z t
f (s, ϕ(s)) ds ≤
|T ϕ(t) − xo | =
|f (s, ϕ(s))| ds
to
t
≤
to
Z
mR (s) ds = |M (t)| ≤ β̄.
to
Hence, T : A → A.
Let us consider the continuity of T on A. Consider {ϕn } ⊂ A such that
ϕn → ϕ ∈ A as n → ∞. By the continuity of f (t, x) in x for each fixed t, we
have that ∀ t ∈ Iᾱ
f (t, ϕn (t)) → f (t, ϕ(t)),
as n → ∞. Since |f (t, ϕn (t))| ≤ m(t), by the Dominated Convergence Theorem, we conclude that for any t ∈ Iᾱ
Z t
Z t
f (s, ϕ(s)) ds
f (s, ϕn (s)) ds →
to
to
as n → ∞.
Finally, ∀ ϕ ∈ A and for all t, τ ∈ Iᾱ
Z
Z t
|T ϕ(t) − T ϕ(τ )| =
f (s, ϕ(s)) ds −
=
to
Z t
τ
≤
Z
τ
t
τ
f (s, ϕ(s)) ds
to
f (s, ϕ(s)) ds ≤
Z
τ
t
|f (s, ϕ(s))| ds
m(s) ds = |M (t) − M (τ )|.
By the continuity of M on Iᾱ , we have that M is also uniformly continuous on Iᾱ ; hence, the set T A is an equi-bounded and equi-continuous set of
C([0, T ]; RN ) and by the Ascoli–Arzelà Theorem is relatively compact. We
conclude that T is completely continuous. In this way, all the assumptions of
the Schauder–Leray Fixed Point Theorem are satisfied, and we finish.
We omit the proof of the second and third parts of the theorem, since they
can be obtained by straightforward adaptations of the corresponding classical
.
arguments for a Lipschitz continuous function f .
8c The Inhomogeneous Dirichlet Problem
445
def
7c Traces of Functions on Σ = ∂E × (0, T ]
The statement of Theorem 7.1 is taken from Chapter 4, § 2.2 of Lions and Magenes [176], where a full proof is given, which, however, requires the knowledge
of tools from interpolation theory. The same result is also given in Lemma 3.4
of Chapter II of Ladyzhenskaya et al. [151], without any proof, and in Theorem
4.2 of Grisvard [109], whose proof again is based on interpolation theory.
The sharpness of these embedding results is briefly discussed in Weidemaier [273]; the author provides an alternative proof in Weidemaier [272],
where he requires the boundary ∂E to be of class C 2 .
8c The Inhomogeneous Dirichlet Problem
Besides considering linear equations, one could deal with more general quasilinear equations
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u),
(8.1c)
where A : ET × R × RN → RN , B : ET × R × RN → R are measurable with
respect to (x, t) for all (u, ξ) ∈ RN +1 , continuous with respect to (u, ξ) for a.e.
(x, t) ∈ ET , and they satisfy the structure conditions

2

A(x, t, u, ξ) · ξ ≥ Co |ξ| − go (x, t),
|A(x, t, u, ξ)| ≤ C1 |ξ| + g1 (x, t),
(8.2c)


|B(x, t, u, ξ)| ≤ C|ξ| + g2 (x, t),
with 0 < Co ≤ C1 , C ≥ 0, and go , g12 , g2 ∈ Lq,r (ET ) as in (3.2)–(3.3).
Assume that ∂E is of class C 1,1 and satisfies the segment property. Given A
and B as above, we are interested in the Cauchy–Dirichlet problem
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u)
u(x, t) = ψ
u(·, 0) = uo
in ET ,
on Σ,
(8.3c)
in E,
where ψ ∈ L2 (0, T ; W 1,2 (E)).
A weak solution of the Cauchy–Dirichlet problem
(8.3c) is a measurable
function u ∈ C 0, T ; L2(E) ∩ L2 0, T ; W 1,2 (E) satisfying for all τ ∈ [0, T ]
ZZ
Z
u(x, τ )ϕ(x, τ )dx +
− uϕt + A(x, t, u, ∇u) · ∇ϕ dxdt
Eτ
E
Z
ZZ
uo ϕ(x, 0)dx
B(x, t, u, ∇u)ϕ dxdt +
=
Eτ
for all test functions ϕ ∈ Wo2 (ET ).
E
446
11 LINEAR PARABOLIC EQUATIONS
In addition, we take the boundary condition u = ψ to mean that (u −
ψ)(·, t) ∈ Wo1,2 (E) for a.e. t ∈ (0, T ].
Here, the boundary condition on Σ is given in a slightly different sense
with respect to what we did in Section 8. The solvability of (8.3c) is studied,
for example, in Chapter V of Ladyzhenskaya et al. [151].
One can also define local weak solutions, as we will do in Chapter 12, and,
precisely as in the elliptic context, it turns out that local solutions of (8.3c),
whenever they exist, show the same kind of local behavior as local solutions
of (8.1).
The situation is much more complicated, in terms of existence, uniqueness,
and above all regularity of solutions, if one deals with
ut − div A(x, t, u, ∇u) = 0,
where we dropped the lower order terms just for the sake of simplicity, and
A : ET × R × RN → RN is measurable with respect to (x, t) for all (u, ξ) ∈
RN +1 , continuous with respect to (u, ξ) for a.e. (x, t) ∈ ET , and it satisfies
the structure conditions
(
A(x, t, u, ξ) · ξ ≥ Co |ξ|p ,
|A(x, t, u, ξ)| ≤ C1 |ξ|p−1 ,
with 0 < Co ≤ C1 , and p > 1, p 6= 2. We refrain from going into further detail
here, and refer the interested reader to, for example, DiBenedetto [49] and the
references therein.
8.1c Parabolic Quasi-Minima
As we have seen in Section 8.2c of Chapter 9, the notion of Q-minimum can
be introduced, in order to provide a unifying framework to some regularity
results for elliptic equations, and also for systems.
The same approach can be followed in the parabolic context; it was started
by Wieser [280], and it has seen tremendous development in recent years.
Given an open set E ⊆ RN with N ≥ 2 and T > 0, consider a function
F : ET × R × RN → R, which is measurable in (x, t) for all (u, ξ) ∈ R × RN ,
and continuous in (u, ξ) for a.e. (x, t) ∈ ET , and assume that it satisfies the
growth conditions
Co |ξ|2 − b|u|γ − g(x, t) ≤ F (x, t, u, ξ) ≤ C1 |ξ|2 + b|u|γ + g(x, t),
where g ∈ L1 (ET ), g ≥ 0, b, γ ∈ R+ , 0 < Co ≤ C1 .
1,2
2
(0, T ; Wloc
(E)) ∩ Lγ (ET ) is
For Q ≥ 1 a function u : ET → R, u ∈ Lloc
called a parabolic Q-minimum, of the functional
ZZ
F (x, t, u, ∇u) dxdt,
J[u] =
ET
9c The Neumann Problem
447
if for every ϕ ∈ Co∞ (ET ) we have
ZZ
ZZ
−
uϕt dxdt +
F (x, t, u, ∇u) dxdt
K
ZZ K
F (x, t, u − ϕ, ∇u − ∇ϕ) dxdt,
≤Q
K
where K = supp ϕ.
Consider the quasi-linear parabolic equation (8.1c) with the structure conditions (8.2c). Then, along the same lines as the elliptic proof, it is not difficult
to show that a local weak solution of (8.1c) (see Chapter 12 for more details
on local solutions), is a parabolic Q-minimum for the functional
ZZ
|∇u|2 + h dxdt,
J[u] = λ
ET
with λ ∈ (0, 1) and h = h(x, t) = (1 + go (x, t) + g12 (x, t) + g22 (x, t)).
9c The Neumann Problem
It is quite natural to wonder whether the abstract approach discussed in Section 3c of the Complements, in the context of the homogeneous Cauchy–
Dirichlet problem, can also be used for the Cauchy–Neumann or mixed problem. In § 4.7.2–4.7.3 of Chapter 3 of Lions and Magenes [175], problems (9.1)
and (9.2) respectively are discussed with ψ = 0, and it is shown how the framework of Section 3c allows the unique existence of a solution u to be proved.
However, the abstract approach requires some care in dealing with the term f
on the right-hand side; indeed, in this case V ′ = (W 1,2 (E))′ , whose characterization is not straightforward (in particular, it is not a space of distributions).
We refrain from going into further detail here.
As we did in Section 8c of the Complements, for the Cauchy–Neumann
Problem, we can also consider a general quasi-linear operator. Namely, assume
that ∂E is of class C 1 and satisfies the segment property. Given A and B as
in (8.2c), we are interested in the problem
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u)
A(x, t, u, ∇u) · n = ψ
u(·, 0) = uo
in ET ,
on Σ,
in E,
(9.1c)
where n is the outward unit normal to ∂E, and ψ enjoys the same regularity
as in (9.2).
By a weak solution of the Cauchy–Neumann Problem (9.1c), under all the
previous assumptions, we mean a function u ∈ V 2 (ET ) satisfying the identity
448
−
11 LINEAR PARABOLIC EQUATIONS
ZZ
ZZ
ZZ
B(x, t, u, ∇u)ϕ dxdt
A(x, t, u, ∇u)∇ϕ dxdt =
ET
Z ZET
Z
uo (x)ϕ(x, 0)dx
+
ψϕ dσ dt +
uϕt dxdt +
ET
Σ
E
for any function ϕ ∈ W 2 (ET ), which vanishes for t = T .
10c A Priori L∞ (ET ) Estimates for Solutions of the
Dirichlet Problem (8.1)
For N ≥ 1, ǫ > 0, the equation
ut − ∆u = f,
where
f =−
(2N + 1)(T − t) + (2N − 3)|x|2
,
[(T − t) + |x|2 ]2
is solved in a cylindrical neighborhood ET = Bǫ × (T − ǫ, T ) of the point (0, T )
by
u(x, t) = ln[(T − t) + |x|2 ].
One verifies that f ∈ Ls (ET ) for any s <
unbounded in ET .
On the other hand, for N ≥ 3, if
N +2
2 ,
and it is apparent that u is
α−1 (2N + 1)(T − t) + (2N − 3)|x|2
f =α − ln((T − t) + |x|2 )
[(T − t) + |x|2 ]2
α−2
4|x|2
,
− α(α − 1) − ln((T − t) + |x|2 )
[(T − t) + |x|2 ]2
with α =
N −2
N +2 ,
the solution is
α
u(x, t) = − ln((T − t) + |x|2 ) .
In such a case one verifies that f ∈ L
ET .
N +2
2
(ET ), and again u is unbounded in
10.1. Extend Proposition 10.1 assuming
f ∈ Lr1 (p),r2 (p) (ET ),
f ∈ Ls1 (p),s2 (p) (ET ),
for proper (r1 , r2 ) and (s1 , s2 ), where the spaces Lp,q (ET ) have been defined in Section 1 (see also Section 12.1 of Chapter 14).
15c Gaussian Bounds on the Fundamental Solution
449
12c A Priori L∞ (ET ) Estimates for Solutions of the
Neumann Problem (9.1)
12.1. Extend Proposition 12.1 assuming
f ∈ Lr1 (p),r2 (p) (ET ),
f ∈ Ls1 (p),s2 (p) (ET ),
for proper (r1 , r2 ) and (s1 , s2 ).
15c Gaussian Bounds on the Fundamental Solution
We prove (15.2); it suffices to show that the following hold.
Proposition 15.1c [190] There exists a positive constant CN such that for
any function f ∈ C0∞ RN we have
Proof. For f ∈ C0∞
2
4
2(1+ N
)
2
kf k2
≤ CN
k∇f k22 kf k1N .
RN , we denote by fˆ its Fourier Transform
Z
ˆ
f (x)e−iξ·x dx.
f (ξ) =
RN
It is well-known that
sup |fˆ(ξ)| ≤
RN
Z
|f | dx,
(15.1c)
kf k2 .
(15.2c)
RN
N/2
kfˆk2 = (2π)
d| = |ξ||fˆ|. Starting from (15.2c) and relying on (15.1c), we have
Moreover, |∇f
1
kfˆk22
(2π)N
"Z
#
Z
1
|fˆ|2 dξ +
|fˆ|2 dξ
=
(2π)N
|ξ|≤R
|ξ|>R
Z
ωN R N ˆ 2
1
≤
kf k∞ +
|ξ|2 |fˆ|2 dξ
(2π)N
(2π)N R2 RN
Z
ωN R N
1
2
|∇f |2 dx.
≤
kf k1 +
(2π)N
(2π)N/2 R2 RN
kf k22 =
If we optimize over R, we obtain
4
with C =
h
2N
kf k22 ≤ C kf k1N +2 k∇f k2N +2
2
N
NN+2
N2+2 i ωNN +2
2
+
N
2
(2π)N
and we conclude.
12
PARABOLIC DEGIORGI CLASSES
1 Quasi-Linear Equations and DeGiorgi Classes
A quasi-linear parabolic equation in a set ET = E × (0, T ] ⊂ RN +1 is an
expression of the form
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u)
(1.1)
1,2
where for u ∈ Cloc (0, T ; L2loc(E)) ∩ L2loc (0, T ; Wloc
(E)), the functions
A x, t, u(x, t), ∇u(x, t) ∈ RN
ET ∋ (x, t) →
B(x, t, u(x, t), ∇u(x, t) ∈ R
are measurable and satisfy the structure conditions
A x, t, u, ξ · ξ ≥ Co |ξ|2 − [f (x, t)]2
|A x, t, u, ξ | ≤ C1 |ξ| + f (x, t)
|B x, t, u, ξ | ≤ C2 |ξ| + fo (x, t)
(1.2)
for given constants 0 < Co ≤ C1 and C2 > 0, and given non-negative functions
f ∈ LN +2+ε (ET ),
fo ∈ L
N +2+ε
2
(ET ),
for some ε > 0.
(1.3)
The Dirichlet and Neumann problems for these equations were introduced in
the Complements of Chapter 11, their solvability was established for a class
of functions A and B, and L∞ (ET ) bounds were derived for suitable data.
Here we are interested in the local behavior of these solutions, both at the
interior, and at the boundary, either with Dirichlet or with Neumann data.
1,2
A function u ∈ Cloc (0, T ; L2loc(E)) ∩ L2loc (0, T ; Wloc
(E)) is a local weak
sub(super)-solution of (1.1), if for every compact set K ⊂ E and every subinterval [t1 , t2 ] ⊂ (0, T ] we have
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_13
451
452
12 PARABOLIC DEGIORGI CLASSES
Z
t2
uϕ dx
K
+
Z
t2
t1
t1
Z
K
− uϕt + A(x, t, u, ∇u) · Dϕ dx dt
≤ (≥)
Z
t2
t1
Z
(1.4)
B(x, t, u, Du)ϕ dx dt
K
for all non-negative testing functions
1,2
ϕ ∈ Wloc
0, T ; L2(K) ∩ L2loc 0, T ; Wo1,2(K) .
(1.5)
This guarantees that all the integrals in (1.4) are convergent.
A local weak solution of (1.1) is a function u ∈ Cloc (0, T ; L2loc(E)) ∩
1,2
2
Lloc (0, T ; Wloc
(E)) satisfying (1.4) with the equality sign, for all
1,2
ϕ ∈ Wloc
0, T ; L2(K) ∩ L2loc 0, T ; Wo1,2(K) .
No further requirements are placed on A and B other than the structure
conditions (1.2). Specific examples of these PDEs are those introduced in
the previous chapter. In particular, they include the class of linear parabolic
equations with bounded and measurable coefficients (2.2) of Chapter 11. Even
though the coefficients are only measurable, nevertheless, local weak solutions
of (1.1) are locally Hölder continuous in ET . This follows from their membership in more general classes of functions called parabolic DeGiorgi classes,
which are introduced next.
Let Bρ (y) ⊂ E denote a ball of center y and radius ρ; if y is the origin,
write Bρ (0) = Bρ . For θ > 0 let (y, s) + Qρ± (θ) ⊂ ET denote the cylinders
(y, s) + Qρ− (θ) = Bρ (y) × (s − θρ2 , s],
(y, s) + Qρ+ (θ) = Bρ (y) × (s, s + θρ2 ];
If θ = 1, write (y, s) + Qρ± (1) = (y, s) + Qρ± .
Consider a piecewise smooth, cutoff function ζ vanishing on ∂Bρ (y), and
such that 0 ≤ ζ ≤ 1, and let u be a local sub(super)-solution of (1.1). For
k ∈ R, the localized truncations ±ζ 2 (u − k)± can be taken as test functions ϕ,
modulo a standard Steklov average, as considered in Section 11 of the previous
Chapter.
After a translation we may assume (y, s) = (0, 0). In (1.4) integrate over
Bρ × (−θρ2 , t], with t ∈ (−θρ2 , 0].
ZZ
uτ (u − k)± ζ 2 dx dτ
±
Bρ ×(−θρ2 ,t]
±
ZZ
≤±
Bρ ×(−θρ2 ,t]
ZZ
A(x, τ, u, ∇u)∇[(u − k)± ζ 2 ]dx dτ
Bρ ×(−θρ2 ,t]
B(x, τ, u, ∇u)(u − k)± ζ 2 dx dτ.
1 Quasi-Linear Equations and DeGiorgi Classes
453
As for the first term
ZZ
ZZ
1
±
uτ (u − k)± ζ 2 dx dτ =
[(u − k)2± ]τ ζ 2 dx dτ
2
2
2
Bρ ×(−θρ ,t]
Bρ ×(−θρ ,t]
ZZ
Z
Z
1 t
d
=
(u − k)2± ζζτ dx dτ
(u − k)2− ζ 2 dx dτ −
2 −θρ2 dτ Bρ
Bρ ×(−θρ2 ,t]
Z
Z
1
1
(u − k)2± ζ 2 (x, t)dx −
(u − k)2± ζ 2 (x, −θρ2 )dx
≥
2 Bρ
2 Bρ
ZZ
−
(u − k)2± ζ|ζτ |dx dτ.
Q−
ρ (θ)
The second integral is transformed and estimated as
ZZ
±
A(x, τ, u, ∇u) · ∇[(u − k)± ζ 2 ]dx dτ
B ×(−θρ2 ,t]
=
Z ρZ
Bρ
±2
≥ Co
−
×(−θρ2 ,t]
ZZ
ZZ
ZZ
−2
Bρ ×(−θρ2 ,t]
Bρ ×(−θρ2 ,t]
Q−
ρ (θ)
− 2C1
±A(x, τ, u, ∇u) · ∇(u − k)± ζ 2 dx dτ
ZZ
ZZ
f 2 χ[(u−k)± >0] ζ 2 dx dτ
Q−
ρ (θ)
≤
Co
4
ZZ
Bρ
+ γ(Co )
ZZ
Q−
ρ (θ)
≤
ZZ
|∇(u − k)± |(u − k)± ζ|∇ζ|dx dτ
f (u − k)± ζ|∇ζ|dx dτ.
Bρ ×(−θρ2 ,t]
2
|∇(u − k)± |2 ζ 2 dx dτ
Bρ ×(−θρ2 ,t]
By Young’s inequality
ZZ
2C1
and
(u − k)± ζA(x, τ, u, ∇u) · ∇ζ dx dτ
|∇(u − k)± |(u − k)± ζ|∇ζ|dx dτ
×(−θρ2 ,t]
ZZ
Q−
ρ (θ)
(u − k)2± |∇ζ|2 dx dτ
f (u − k)± ζ|∇ζ|dx dτ
Q−
ρ (θ)
|∇(u − k)± |2 ζ 2 dx dτ
(u − k)2± |∇ζ|2 dx dτ +
ZZ
Q−
ρ (θ)
f 2 χ[(u−k)± >0] ζ 2 dx dτ.
454
12 PARABOLIC DEGIORGI CLASSES
Combining these terms
ZZ
±
A(x, τ, u, ∇u) · ∇[(u − k)± ζ 2 ]dx dτ
Bρ ×(−θρ2 ,t]
ZZ
3
≥ Co
4
ZZ
Bρ ×(−θρ2 ,t]
Q−
ρ (θ)
Finally,
ZZ
±
Bρ ×(−θρ2 ,t]
≤ C2
+
≤
ZZ
ZZ
Co
4
|∇(u − k)± |2 ζ 2 dx dτ
k)2± |∇ζ|2 dx dτ
(u −
−
ZZ
Q−
ρ (θ)
f 2 χ[(u−k)± >0] ζ 2 dx dτ.
B(x, τ, u, ∇u)(u − k)± ζ 2 dx dτ
Bρ ×(−θρ2 ,t]
Bρ ×(−θρ2 ,t]
ZZ
|∇(u − k)± |(u − k)± ζ 2 dx dτ
fo (u − k)± ζ 2 dx dτ
Bρ ×(−θρ2 ,t]
+ γ(Co )C22
ZZ
|∇(u − k)± |2 ζ 2 dx dτ
Q−
ρ (θ)
(u − k)2± ζ 2 dx dτ +
ZZ
Q−
ρ (θ)
fo (u − k)± ζ 2 dx dτ.
Combining the previous estimates and recalling that t ∈ (−θρ2 , 0] is arbitrary
yields
Z
Z
1
1
ess sup
(u − k)2± ζ 2 (x, −θρ2 )dx
(u − k)2± ζ 2 (x, t)dx −
2 −θρ2 <t<0 Bρ
2 Bρ
ZZ
Co
+
|∇[(u − k)± ζ]|2 dx dτ
−
2
Qρ (θ)
ZZ
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
≤γ
Q−
ρ (θ)
+
ZZ
Q−
ρ (θ)
f 2 χ[(u−k)± >0] + fo (u − k)±
+γC22 (u − k)2± ζ 2 dx dτ.
We point out that the proof traces the dependence of the constant γ on the
parameters {N, Co , C1 , C2 }. Next, estimate
ZZ
1− N2+2 +2δ
f 2 χ[(u−k)± >0] dx dτ ≤ kf k2N +2+ε;Q− (θ) |A±
,
k,ρ |
Q−
ρ (θ)
where
ρ
455
1 Quasi-Linear Equations and DeGiorgi Classes
−
A±
k,ρ = [(u − k)± > 0] ∩ Qρ (θ)
and
δ=
ε
.
(N + 2)(N + 2 + ε)
(1.6)
The term involving fo is estimated by Hölder’s inequality and using Proposition 1.1 of Chapter 11. We have
ZZ
fo (u − k)± ζ 2 dx dτ
Q−
ρ (θ)
≤kfo χ[(u−k)± >0] k2 N +2 ;Q−
k(u − k)± ζ]k2 N +2 ;Q−
ρ (θ)
ρ (θ)
N +4
N
≤γ(N )k(u − k)± ζkV 2 (Q−
kfo χ[(u−k)± >0] k2 N +2 ;Q−
ρ (θ))
ρ (θ)
N +4
Co 1
≤ min
k(u − k)± ζk2V 2 (Q− (θ))
;
ρ
4 4
+ γ(N, Co )kfo χ[(u−k)± >0] k22 N +2 ;Q− (θ)
ρ
N +4
Z
1
≤ ess sup
(u − k)2± ζ 2 (x, t)dx
4 −θρ2 <t<0 Bρ
ZZ
Co
+
|∇[(u − k)± ζ]|2 dx dτ
4
Q−
(θ)
ρ
+ γ(N, Co )kfo k2N +2+δ̂
2
where δ is as in (1.6), and δ̂ =
2
;Q−
ρ (θ)
1− N +2 +2δ
|A±
,
k,ρ |
ε(N + 2)
. Combining these estimates
2[2(N + 2) + ε]
yields
ess sup
−θρ2 <t<0
+
≤ γ̄
ZZ
ZZ
Z
Bρ
Q−
ρ (θ)
Q−
ρ (θ)
+ γ̄ C22
ZZ
(u −
k)2± ζ 2 (x, t)dx
−
Z
Bρ
(u − k)2± ζ 2 (x, −θρ2 )dx
|∇[(u − k)± ζ]|2 dx dτ
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
Q−
ρ (θ)
2
1− N +2 +2δ
,
(u − k)2± ζ 2 dx dτ + γ∗2 |A±
k,ρ |
where δ is given by (1.6) and
γ∗2 = γ(N, Co ) kf k2N +2+ε;Q− (θ) + kfo k2N +2+δ̂
ρ
2
;Q−
ρ (θ)
.
(1.7)
If we revert to a general point (y, s) instead of considering (0, 0), we have
12 PARABOLIC DEGIORGI CLASSES
456
ess sup
s−θρ2 <t<s
+
≤ γ̄
ZZ
ZZ
Z
Bρ (y)
(u −
−
(y,s)+Qρ
(θ)
−
(y,s)+Qρ
(θ)
+ γ̄ C22
ZZ
k)2± ζ 2 (x, t)dx
−
Z
Bρ (y)
2 2
(u − k)±
ζ (x, s − θρ2 )dx
|∇[(u − k)± ζ]|2 dx dτ
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
−
(y,s)+Qρ
(θ)
(1.8)
2
1− N +2 +2δ
(u − k)2± ζ 2 dx dτ + γ∗2 |A±
,
k,ρ |
with the obvious corresponding meaning for A±
k,ρ . Analogous estimates hold
for “forward” cylinders (y, s) + Q+
ρ (θ) ⊂ ET .
1.1 Parabolic DeGiorgi Classes
Let E ⊂ RN be an open set, let ET = E × (0, T ], and let γ̄, γ∗ , C2 , and δ be
given positive constants. The parabolic DeGiorgi class PDG+ (ET , γ̄, γ∗ , C2 , δ)
1,2
is the collection of all functions u ∈ Cloc (0, T ; L2loc(E)) ∩ L2loc (0, T ; Wloc
(E))
such that (u − k)+ satisfy (1.8) for all k ∈ R, for all piecewise smooth, cutoff
function ζ vanishing on ∂Bρ (y) and such that 0 ≤ ζ ≤ 1, and for cylinders
+
(y, s) + Q±
ρ (θ) ⊂ ET . Local weak sub-solutions of (1.1) belong to PDG , for
the constants γ̄, γ∗ , C2 , and δ identified in (1.6)–(1.7).
The parabolic DeGiorgi class PDG− (ET , γ̄, γ∗ , C2 , δ) is defined similarly,
with (u − k)+ replaced by (u − k)− . Local weak super-solutions of (1.1) belong to PDG− . The parabolic DeGiorgi classes PDG(ET , γ̄, γ∗ , C2 , δ) are the
intersection of PDG+ ∩ PDG− , or equivalently the collection of all functions
1,2
u ∈ Cloc (0, T ; L2loc(E)) ∩ L2loc (0, T ; Wloc
(E)) satisfying (1.8) for all k ∈ R, for
all piecewise smooth, cutoff function ζ vanishing on ∂Bρ (y) and such that
0 ≤ ζ ≤ 1, and for cylinders (y, s) + Q±
ρ (θ) ⊂ ET . We refer to these classes as
homogeneous if γ∗ = 0. In such a case the choice of the parameter δ is immaterial. The set of parameters {N, γ̄, C2 } are the homogeneous data of the PDG
classes, whereas γ∗ and δ are the inhomogeneous parameters. This terminology stems from the structure of (1.8) versus the structure of the quasi-linear
elliptic equations in (1.1), and is evidenced by (1.7).
Functions in PDG have remarkable properties, irrespective of their connection with the quasi-linear equations (1.1). In particular, they are locally
bounded, and locally Hölder continuous in ET . Even more striking is that
non-negative functions in PDG satisfy the Harnack inequality of Section 5.1
of Chapter 2, which is typical of non-negative caloric functions.
2 Local Boundedness of Functions in the PDG Classes
We say that constants C, γ, . . . depend only on the data, and are independent of γ∗ and δ, if they can be quantitatively determined a priori only in
2 Local Boundedness of Functions in the PDG Classes
457
terms of the homogeneous parameters {N, γ̄, C2 }. The dependence on the inhomogeneous parameters {γ∗ , δ} will be traced, as a way of identifying those
additional properties afforded by inhomogeneous structures.
Theorem 2.1. Let u ∈ PDG± and σ ∈ (0, 1). There exists a constant C
depending only on the data such that for every pair of nested cylinders (y, s) +
−
Qσρ
(θ) ⊂ (y, s) + Qρ− (θ) ⊂ ET
ess sup u± ≤ max γ∗ ρδ(N +2) ;
−
(y,s)+Qσρ
(θ)
√
C θ
1
(1 − σ) 2δ
12 ZZ
1 4δ1
1
2
u± dx dτ
.
1+
θ
|Qρ (θ)|
(y,s)+Q−
ρ (θ)
For homogeneous PDG± classes, γ∗ = 0 and δ can be taken δ =
(2.1)
1
N +2 .
Proof. The proof will be given for non-negative u ∈ PDG+ (ET , γ̄, γ∗ , C2 , δ),
the proof for the remaining case being identical. Assume (y, s) = (0, 0) and
for fixed σ ∈ (0, 1) and n = 0, 1, 2, . . . set
ρn = σρ +
Bn = Bρn ,
1 − σ2 2
θρ ,
2n
Bn = Kn × (tn , 0).
1−σ
ρ,
2n
tn = −θσ 2 ρ2 −
This is a family of nested and shrinking cylinders with a common “vertex” at
(0, 0), and by construction
Qo = Bρ × (−θρ2 , 0)
and
Q∞ = Bσρ × (−θσ 2 ρ2 , 0).
Denote by ζ a non-negative, piecewise smooth cutoff function in Qn that
equals one on Qn+1 , and has the form ζ(x, t) = ζ1 (x)ζ2 (t), where
2n+1
1 in Bn+1
ζ1 =
|Dζ1 | ≤
N
0 in R − Bn
(1 − σ)ρ
ζ2 =
2n+1
0 for t ≤ tn
0 ≤ ζ2,t ≤
;
1 for t ≥ tn+1
(1 − σ)θρ2
introduce the increasing sequence of levels kn = k − 2−n k, where k > 0 is
to be chosen, stipulate that ρ ≤ C2−1 , and write (1.8) for (u − kn+1 )+ ζ 2 : we
obtain
Z
ZZ
sup
|∇[(u − kn+1 )+ ζ]|2 dx dτ
[(u − kn+1 )+ ζ]2 (x, τ )dx +
tn ≤τ ≤t
Bn
γ22n
≤
(1 − σ)2 ρ2
Qn
ZZ
1
1+
(u − kn+1 )2+ dx dτ
θ
Qn
2
1− N +2 +2δ
+ γ∗2 |A+
kn+1 ,ρn |
(2.2)
12 PARABOLIC DEGIORGI CLASSES
458
γ22n
≤
(1 − σ)2 ρ2
ZZ
1
1+
(u − kn )2+ dx dτ
θ
Qn
2
1− N +2 +2δ
+ γ∗2 |A+
,
kn+1 ,ρn |
where γ is a positive constant depending only on the data and that might be
different in different contexts.
By Hölder’s inequality and the embedding Proposition 1.1 of Chapter 11
ZZ
ZZ
NN+2
N +2
(u − kn+1 )2+ dx dτ ≤
[(u − kn+1 )+ ζ]2 N dx dτ
Qn+1
Qn
≤γ
×
ZZ
Qn
ZZ
Qn
χ[(u−kn+1 )+ >0] dx dτ
|∇[(u − kn+1 )+ ζ]|2 dx dτ
×
×
22n
≤γ
(1 − σ)2 ρ2
+γ∗2 |A+
kn+1 ,ρn
sup
tn ≤τ ≤0
ZZ
Qn
Z
NN+2
Kn
N2+2
N2+2
[(u − kn+1 )+ ζ]2 (x, τ )dx
χ[(u−kn+1 )+ >0] dx dτ
N2+2
ZZ
1
1+
(u − kn )2+ dx dτ
θ
Qn
i ZZ
2
2
χ[(u−kn+1 )+ >0] dx dτ N +2 .
|1− N +2 +2δ ×
Qn
Since
|A+
kn+1 ,ρn | =
we obtain
ZZ
Qn+1
(u −
ZZ
ZZ
2n+1
χ[(u−kn+1 )+ >0] dx dτ ≤ 2
k2
Qn
kn+1 )2+ dx dτ
≤γ
"
22n
(1 − σ)2 ρ2
Qn
(u − kn )2+ dx dτ,
ZZ
1
1+
(u − kn )2+ dx dτ
θ
Qn
1− N2+2 +2δ #
ZZ
22n+1
(u − kn )2+ dx dτ
k2
Qn
22n+1 Z Z
N2+2
2
×
(u
−
k
)
dx
dτ
.
n
+
k2
Qn
+ γ∗2
Setting
Yn =
yields
θ
2
k |Qn |
ZZ
Qn
(u − kn )2+ dx dτ =
θ
k2
ZZ
(u − kn )2+ dx dτ
Qn
3 Hölder Continuity of Functions in the PDG Classes
γbn 1+
(1 − σ)2
γbn ≤
1+
(1 − σ)2
Yn+1 ≤
where
1 1+ N2+2 γγ∗2 dn N +2 2δ 1+2δ
Yn
(ρ
) Yn
+
θ
k2
ρ2δ(N +2) 1+2δ
1
1+ 2
Yn N +2 + γ∗2
Y
,
n
θ
k2
2
b = 22(1+ N +2 ) ,
459
(2.3)
d = 22(1+2δ) .
Stipulate to take k so large that
k ≥ γ∗ ρδ(N +2) ,
k>
√ ZZ
θ
Q−
ρ (θ)
u2+ dx dτ
21
.
(2.4)
2
Then, Yn ≤ 1 for all n and YnN +2 ≤ Yn2δ . With these remarks and stipulations,
the previous recursive inequalities take the form
γbn 1 1+2δ
Yn+1 ≤
Y
for all n = 1, 2, . . .
(2.5)
1
+
(1 − σ)2
θ n
From the fast geometric convergence Lemma 15.1 of Chapter 9, it follows that
{Yn } → 0 as n → ∞, provided
ZZ
1
1
1
1 − 2δ
θ
− 1
u2+ dx dτ ≤ b (2δ)2 γ − 2δ (1 − σ) δ 1 +
.
Yo = 2
k
θ
Q−
ρ (θ)
Therefore, also taking into account (2.4), and choosing
1
ZZ
p1 1
2(2δ)2 γ 4δ √ 1 4δ1
2
δ(N +2) b
,
θ 1+
u+ dx dτ
k = max γ∗ ρ
;
1
θ
(1 − σ) 2δ
Q−
ρ (θ)
one derives
Y∞
θ
= 2
k
ZZ
(u − kn )2+ dx dτ = 0
=⇒
Q−
σρ (θ)
ess sup u+ ≤ k.
Q−
σρ (θ)
If γ∗ = 0, then (2.3) is already in the form (2.5) with δ =
1
N +2 .
3 Hölder Continuity of Functions in the PDG Classes
For a function u ∈ PDG(ET , γ̄, γ∗ , C2 , δ) and (y, s) + Q−
2ρ (θ) ⊂ ET with θ > 0,
set
µ+ = ess sup u, µ− = ess inf
u,
−
(y,s)+Q−
2ρ (θ)
+
−
ω(2ρ) = µ − µ =
(y,s)+Q2ρ (θ)
ess osc
−
u.
(3.1)
(y,s)+Q2ρ (θ)
These quantities are well defined since u ∈ L∞
loc (ET ) by the results of Section 2.
12 PARABOLIC DEGIORGI CLASSES
460
Theorem 3.1. Let u ∈ PDG(ET , γ̄, γ∗ , C2 , δ). There exist constants C > 1
and α ∈ (0, 1) depending only upon the data and independent of u, such that
for every pair of cylinders (y, s) + Qρ− (θ) ⊂ (y, s) + Q−
R (θ) ⊂ ET
ρ α
n
o
ω(ρ) ≤ C max ω(R)
; γ∗ ρδ(N +2) .
R
(3.2)
The Hölder continuity is local to ET , with Hölder exponent αo = min{α; δ(N +
2)}. An upper bound for the Hölder constant is
{Hölder constant} ≤ C max{2M R−α; γ∗ },
where
M = kuk∞;(y,s)+Q−(θ) .
R
This implies that the local Hölder estimates deteriorate near ∂p ET . Indeed,
fix (x, t), (y, s) ∈ ET and let
R = min{dist{(x, t); ∂p ET } ; dist{(y, s); ∂p ET }},
where
dist{(x, t); ∂p ET } =
inf
(xo ,to )∈∂p ET
1
2
1
|x − xo | + |t − to | 2 .
If |x − y| + |t − s| < R, then (3.2) implies
αo
1
.
|u(x, t) − u(y, s)| ≤ C max{ω(R)R−αo ; γ∗ } |x − y| + |t − s| 2
1
If |x − y| + |t − s| 2 ≥ R, then
αo
1
.
|u(x, t) − u(y, s)| ≤ 2M R−αo |x − y| + |t − s| 2
Corollary 3.1 Let u be a local weak solution of (1.1)–(1.4). Then, for every
compact subset K ⊂ ET , and for every pair (x, t) , (y, s) ∈ K
αo
2MK
1
2
|u(x, t) − u(y, s)| ≤ C max
|x
−
y|
+
|t
−
s|
;
γ
∗
dist{K; ∂p ET }α
where MK = ess supK |u| and
dist{K; ∂p ET } =
inf
(x,t)∈K
(y,s)∈∂p ET
1
|x − y| + |t − s| 2 .
3.1 On the Proof of Theorem 3.1
Although the parameter δ is fixed, in view of its value in (1.6), which naturally arises from quasi-linear equations, we will assume δ ≤ N1+2 . The value
δ = N1+2 would occur if ε → ∞ in the integrability requirements (1.3). For homogeneous PDG classes where γ∗ = 0, although immaterial, we take δ = N1+2 .
4 Estimating the Values of u
461
In what follows we assume that u ∈ PDG is given and ρ is such that
ρ ≤ C2−1 . Moreover, according to the different conditions we consider in the
+
auxiliary results, the cylinders (y, s) + Q−
2ρ (θ) ⊂ ET and (y, s) + Q2ρ (θ) ⊂ ET
±
are fixed, and µ and ω(2ρ) are defined either as in (3.1), or as
µ+ =
ess sup
µ− =
u,
(y,s)+Q+
2ρ (θ)
ω(2ρ) = µ+ − µ− =
ess inf
+
(θ)
(y,s)+Q2ρ
ess osc
+
(y,s)+Q2ρ
(θ)
u,
u.
Some statements are given in (y, s) + Qρ− (θ), others in (y, s) + Qρ+ (θ), mainly
for convenience, and also because they will be needed in Section 10. With an
obvious change of notation, they always hold both in backward and forward
cylinders. Finally, for simplicity, we denote ω(2ρ) by ω.
4 Estimating the Values of u by the Measure of the Set
Where u is Either Near µ+ or Near µ−
Proposition 4.1 For every a ∈ (0, 1), there exists ν ∈ (0, 1) depending only
on the data, θ and a, but independent of ω, such that if for some ξ ∈ (0, 1)
−
[u > µ+ − ξω] ∩ (y, s) + Q−
ρ (θ) ≤ ν|Qρ (θ)|
(4.1)+
then either ξω ≤ γ∗ ρδ(N +2) or
u ≤ µ+ − aξω
a.e. in (y, s) + Q−
1 (θ).
ρ
2
(4.2)+
Similarly, if
−
[u < µ− + ξω] ∩ (y, s) + Q−
ρ (θ) ≤ ν|Qρ (θ)|
(4.1)−
then either ξω ≤ γ∗ ρδ(N +2) or
u ≥ µ− + aξω
a.e. in (y, s) + Q−
1 (θ).
ρ
2
(4.2)−
Proof. We prove only (4.1)− –(4.2)− , the arguments for (4.1)+ –(4.2)+ being
analogous. Without loss of generality, we may assume (y, s) = (0, 0); for n =
0, 1, . . . , set
ρn = ρ +
ρ
,
2n
Bn = Bρn ,
Qn = Bn × (−θρ2n , 0].
(4.1)
Apply (1.8) over Bn and Qn to (u − kn )− , for the levels
kn = µ− + ξn ω
where
ξn = aξ +
1−a
ξ.
2n
(4.2)
12 PARABOLIC DEGIORGI CLASSES
462
The cutoff function ζ is taken of the form ζ(x, t) = ζ1 (x)ζ2 (t), where

 1 in Bn+1
1
2n+1
|∇ζ1 | ≤
ζ1 =
=

ρn − ρn+1
ρ
0 in RN − Bn
ζ2 =

 0 for t < −θρ2n

(4.3)
2(n+1)
0 ≤ ζ2,t ≤
1 for t ≥ −θρ2n+1
θ(ρ2n
1
2
≤
.
2
− ρn+1 )
θρ2
With all these stipulations, inequality (1.8) yields
Z
ZZ
2 2
ess sup
|∇[(u − kn )− ζ]|2 dx dτ
(u − kn )− ζ (x, t)dx +
−θρ2n <t<0
2n
≤γ
2
ρ2
Qn
Bn
Z Z
Qn
(u − kn )2− dx dτ +
2
1
θ
ZZ
Qn
(u − kn )2− dx dτ
1− N +2 +2δ
+ γ∗2 |A−
kn ,ρn |
22n (ξω)2
1
−
1− N2+2 +2δ
2
,
≤γ
|A−
1
+
kn ,ρn | + γ∗ |Akn ,ρn |
ρ2
θ
where
A−
kn ,ρn = [(u − kn )− > 0] ∩ Qn (θ).
By the embedding of Proposition 1.1 of Chapter 11
ZZ
ZZ
+2
2 NN
[(u − kn )− ζ]
dx dτ ≤ γ
|∇[(u − kn )− ζ]|2 dx dτ
Qn
Qn
×
ess sup
−θρ2n <t<0
Z
2
Bn
! N2
[(u − kn )− ζ(x, t)] dx
NN+2
1
−
−
1− N2+2 +2δ
2
1+
|Akn ,ρn | + γ∗ |Akn ,ρn |
θ
N +2
N +2 N +2
2
1 N
22 N n (ξω)2 N
1+
|A−
|1+ N
≤γ
N +2
k
,ρ
n
n
θ
ρ2 N
22n (ξω)2
≤γ γ
ρ2
+2
2 NN
+ γ γ∗
1+2δ
|A−
kn ,ρn |
N +2
N
.
Estimate below
2 NN+2
ZZ
N +2
(1 − a)ξω
|A−
[(u − kn )− ζ]2 N dx dτ ≥
kn+1 ,ρn+1 |
n+1
2
Qn
and set
Yn =
|A−
kn ,ρn |
|Qn |
.
(4.4)
4 Estimating the Values of u
463
Then,
Yn+1 ≤
γbn
(1 −
+
≤
N +2
a)2 N
θ
2
N
+2
2 NN
(1 − a)
γ θ̄b
n
(1 − a)2
N +2
N
where
b = 24
1
1+
θ
γ∗
ρ
NN+2
2
1+ N
Yn
2
+2
2 NN
2δ (N +2)
N
N +2
N
γdn θ2δ
+2
2 NN
(ξω)
N +2
1 N
1+
θ
N +2
N
,
d = 22
+2
1+2δ NN
Yn
1+ 2
Yn N
N +2
N
+
+2
(N +2)2
2 NN
2δ N
γ∗
ρ
(ξω)2
2
N +2
N
θ̄ = max{θ N ; θ2δ
,
N +2
N
From this, one derives
Yn+1 ≤
γ θ̄ bn
2
(1 − a)(N +2) N
+2
1+2δ NN
Yn
}.
!
,
(4.5)
N +2
+2
1 N
1+2δ NN
1+
Yn
θ
provided ξω > γ∗ ρδ(N +2) . Owing to Lemma 15.1 of Chapter 9 these recursive
inequalities imply that {Yn } → 0 as n → ∞, provided
Yo =
≤
[u < µ− + ξω] ∩ Q−
ρ (θ)
|Q−
ρ (θ)|
(1 − a)1/δ
N
N2
(γ θ̄) 2δ(N +2) b 4δ2 (N +2)2
θ
θ+1
2δ1
def
(4.6)
= ν.
Remark 4.1 Formula (4.6) provides a precise dependence of ν on the data,
θ, and a. In particular, ν is clearly independent of ξ and ω.
The proof of (4.2)+ is almost identical. One starts from the inequalities (1.8)
written for the truncated functions
(u − kn )+
with
kn = µ+ − ξn ω
for the same choice of ξn as in (4.2). By the definition of µ+ one estimates
(u − kn )+ ≤ ξω.
This validates estimates in all analogous to (4.4) with the same functional dependence on ξω. The same arguments, with the proper changes in the meaning
of the symbols, lead to (4.6) written for ν+ , and conclude the proof. Thus, ν+
depends on the data, θ, and a the same way as ν− does.
Remark 4.2 In Proposition 4.1 the statement relative to (4.1)− –(4.2)− is
given in terms of µ− and ξω. As a matter of fact, as the proof clearly shows,
when dealing with the lower truncations (u − k)− for non-negative functions,
all the estimates depend only on k ≥ 0, without any further assumption on
it.
464
12 PARABOLIC DEGIORGI CLASSES
5 Reducing the Measure of the Set Where u is Either
Near µ+ or Near µ−
Proposition 5.1 Assume that
[u(·, t) ≤ µ+ − ǫω] ∩ Bρ (y) ≥ α|Bρ |
(5.1)+
for some α, ǫ ∈ (0, 1) and for all t ∈ (s, s + θρ2 ] for some θ > 0. Then,
for every ν ∈ (0, 1) there exists ǫν ∈ (0, 1) that can be determined a priori
only in terms of the data, θ, ǫ, and α, and independent of ω, such that either
ǫν ω ≤ γ∗ ρδ(N +2) or
+
[u > µ+ − ǫν ω] ∩ (y, s) + Q+
ρ (θ) ≤ ν Qρ (θ) .
(5.2)+
[u(·, t) ≥ µ− + ǫω] ∩ Bρ (y) ≥ α|Bρ |
(5.1)−
[u < µ− + ǫν ω] ∩ (y, s) + Qρ+ (θ) ≤ ν Q+
ρ (θ) .
(5.2)−
Similarly, if
for some α, ǫ ∈ (0, 1) and for all t ∈ (s, s + θρ2 ] for some θ > 0, then for
every ν ∈ (0, 1) there exists ǫν ∈ (0, 1) depending only on the data, θ, ǫ, and
α, and independent of ω, such that either ǫν ω ≤ γ∗ ρδ(N +2) or
Remark 5.1 An analogous statement holds, if we consider cylinders (y, s) +
Qρ− (θ).
The set [u < µ− + ǫν ω] in the cylinder (y, s) + Q+
ρ (θ) can be made arbitrarily
small, provided ǫν is chosen accordingly. The main tools of the proof are the
estimate of the measure of the sets Aµ− + 12 ω,ρ (t) for all t ∈ (s, s + θρ2 ), and
the discrete isoperimetric inequality of Proposition 5.2 of Chapter 10.
5.1 Proof of Proposition 5.1
Proof. We will establish (5.2)− starting from (5.1)− . As usual, without loss
of generality, we may assume (y, s) = (0, 0).
Write down the estimates (1.8) over the cylinder
−
2
2
2
2
Q+
2ρ (θ) ∪ Q2ρ (θ) = (B2ρ × (−θρ , 0]) ∪ (B2ρ × (0, θρ ]) = B2ρ × (−θρ , θρ ]
for the truncated functions
(u − kj )−
for the levels
kj = µ− +
1
ǫω,
2j
for j = 0, 1, . . . .
(5.1)
The non-negative, piecewise smooth test function ζ is chosen so that it vanishes outside B2ρ and for t ≤ −θρ2 , and
ζ = 1 on Q+
ρ (θ),
|∇ζ| ≤
1
,
ρ
and 0 ≤ ζt ≤
1
.
θρ2
5 Reducing the Measure of the Set Where u is Either Near µ+ or Near µ−
465
The first term on the left-hand side of (1.8) is discarded since it is nonnegative, and the second vanishes because of our choice of test function. The
term involving |∇(u − kj )− | is minorized by extending the integration over
the cylinder Qρ+ (θ), which is the set where ζ = 1. These remarks give the
inequalities
ZZ
|∇(u − kj )− |2 ζ 2 dx dτ
+
Qρ
(θ)
≤γ̄
Z
θρ2
−θρ2
Z
1
2
|∇ζ|2 + ζτ + 2 dx dτ
(u − kj )−
ρ
B2ρ
2
−
1− N +2 +2δ
+ γ∗2 |[(u − kj )− > 0] ∪ Q+
2ρ (θ) ∪ Q2ρ (θ)|
ǫω 2 1
2
2
≤γθρ2
+ 2 |B2ρ | + γγ∗2 θ1− N +2 +2δ ρN +2δ(N +2)
j
2
2
ρ
θρ
ǫω 2
2 1− N2+2 +2δ N +2δ(N +2)
(θ + 1)|Bρ | + γ∗ θ
≤γ
ρ
2j
(5.2)
for a new constant γ depending only on the data {N, Co , C1 }.
Apply the discrete isoperimetric inequality of Proposition 5.2 of Chapter 10
with
ℓ = kj = µ− +
1
ǫω
2j
and
k = kj+1 = µ− +
1
2j+1
ǫω
for j = 0, 1, . . . ,
and take into account (5.1)− , to obtain
Z
γ
ǫω
A
(t)
≤
ρ
|∇u(·, t)|dx,
k
,ρ
j+1
2j+1
α Bρ ∩[kj+1 <u<kj ]
where Akj+1 ,ρ (t) = [u(·, t) < kj+1 ] ∩ Bρ . Integrate this in dt over (0, θρ2 ) and
set
Z θρ2
|Aj | = |[u < kj ] ∩ Q+
(θ)|
=
Akj ,ρ (τ ) dτ.
ρ
0
Then, the previous inequality yields
ZZ
ǫω
γ
≤
A
ρ
|∇u|dx dτ
j+1
2j+1
α
Q+
ρ (θ)∩[kj+1 <u<kj ]
ZZ
12
1
γ
2
Aj − Aj+1 2
≤ ρ
|∇(u − kj )− | dx dτ
α
Q+
ρ (θ)
"
# 21
j 2
q
2
γ ǫω
1
2
γ
∗
≤
Q+
1+
+
ρ2δ(N +2)
ρ (θ)
2
α 2j
θ
θ N +2 −2δ ǫω
1
× |Aj | − |Aj+1 | 2
466
12 PARABOLIC DEGIORGI CLASSES
"
# 12
j 2
q
θ+1
2
γ∗2
+
2δ(N +2)
Qρ (θ)
+
ρ
2
θ
θ N +2 −2δ ǫω
1
× |Aj | − |Aj+1 | 2
# 21
"
j 2
q
γ ǫω
2
2
≤ √
ρ2δ(N +2)
Q+
ρ (θ) 1 + γ∗
ǫω
α θ ∗ 2j
1
× |Aj | − |Aj+1 | 2 ,
γ ǫω
≤
α 2j
where we have used the estimates (5.2), and we have set
1
1
θ+1
.
=
max
;
2
θ∗
θ
θ N +2 −2δ
(5.3)
ǫω
, and square both sides to obtain the recursive inequalities
2j
#
"
j 2
2
(2γ)
2
2
+
2
2δ(N +2)
|Aj+1 | ≤ 2 ∗ |Qρ (θ)| 1 + γ∗
ρ
|Aj | − |Aj+1 |
α θ
ǫω
Next, divide by
for j = 0, 1 . . . . Stipulate that
γ∗2
2
∗
2j
ǫω
ρ2δ(N +2) < 1
where j∗ is a positive integer to be chosen, and add these inequalities for j =
0, 1, . . . , j∗ −1. Minorize the terms on the left-hand side by their smallest value
|Aj∗ |2 and majorize the right-hand side with the corresponding telescopic
series. The indicated estimations yield
j∗ |Aj∗ |2 ≤
j∗P
−1
j =0
|Aj+1 |2 ≤
2
∞
P
(2γ)
|Aj | − |Aj+1 |
|Q+
ρ (θ)|
2
∗
α θ
j=0
2
≤
From this
|Aj∗ | ≤
(2γ)
|Q+ (θ)|2 .
α2 θ∗ ρ
2γ
√
|Q+ (θ)|.
α θ∗ j∗ ρ
(5.4)
Thus, having fixed ν ∈ (0, 1), one can choose j∗ so large that
[u < µ− + ǫν ω] ∩ Q+
ρ (θ)
|Q+
ρ (θ)|
< ν,
for
2γ
√ ∗ ≤ ν,
α θ j∗
and
ǫν =
ǫ
.
2j∗
The proof of Proposition 5.1 shows that it can be rephrased in this equivalent
way.
6 Propagating in Time the Measure-Theoretical Information
467
Proposition 5.2 Assume that
[u(·, t) ≤ µ+ − ǫω] ∩ Bρ (y) ≥ α|Bρ |
(5.4)+
for some α, ǫ ∈ (0, 1) and for all t ∈ (s, s + θρ2 ] for some θ > 0. Then there
exists γ > 0, depending only on the data, such that for any positive integer j∗
δ(N +2)
we have either 2ǫω
or
j∗ ≤ γ∗ ρ
ǫω
2j∗ ] ∩
+
|Qρ (θ)|
[u > µ+ −
Q+
ρ (θ)
2γ
√
.
α θ∗ j∗
(5.5)+
[u(·, t) ≥ µ− + ǫω] ∩ Bρ (y) ≥ α|Bρ |
(5.4)−
≤
Similarly, if
2
for some α, ǫ ∈ (0, 1) and for all t ∈ (s, s + θρ ] for some θ > 0, then there
exists γ > 0, depending only on the data, such that for any positive integer j∗
δ(N +2)
or
we have either 2ǫω
j∗ ≤ γ∗ ρ
ǫω
2j∗ ] ∩
|Q+
ρ (θ)|
[u < µ− +
Q+
ρ (θ)
≤
2γ
√ ∗ .
α θ j∗
(5.5)−
Both in (5.5)+ and in (5.5)− θ∗ is the quantity defined above in (5.3).
Remark 5.2 As in the case of Proposition 5.1, an analogous statement holds,
if we consider cylinders (y, s) + Q−
ρ (θ).
6 Propagating in Time the Measure-Theoretical
Information
Proposition 6.1 Assume that
h
ωi
α
u(·, s) < µ+ −
∩ Bρ (y) ≥ |Bρ |.
4
2
(6.1)+
for some α ∈ (0, 1]. Then, there exist ξ ∈ (0, 1) and θ ∈ (0, 1), which depend
only on the data, such that either ω ≤ 2γ∗ ρδ(N +2) or
α
(6.2)+
u(·, t) < µ+ − ξω ∩ Bρ (y) ≥ |Bρ |
4
for all t ∈ [s, s + θρ2 ]. Analogously, if
h
α
ωi
∩ Bρ (y) ≥ |Bρ |,
u(·, s) > µ− +
4
2
(6.1)−
for some α ∈ (0, 1], then there exist ξ ∈ (0, 1) and θ ∈ (0, 1), which depend
only on the data, such that either ω ≤ 2γ∗ ρδ(N +2) or
α
(6.2)−
u(·, t) > µ− + ξω ∩ Bρ (y) ≥ |Bρ |
4
for all t ∈ [s, s + θρ2 ].
468
12 PARABOLIC DEGIORGI CLASSES
Remark 6.1 An analogous statement holds, if we consider cylinders (y, s) +
Qρ− (θ).
In the context of parabolic equations, these kinds of results are usually proven
using proper logarithmic estimates (see, for example, DiBenedetto [49], Chapter 2, Section 3). Unfortunately, in the context of PDG classes, we do not know
whether such estimates hold or not. Therefore, we give here a different proof,
which makes no use of any kind of logarithmic functions.
6.1 Proof of Proposition 6.1
Proof. We will establish (6.2)− starting from (6.1)− . As usual, without loss
of generality, we may assume (y, s) = (0, 0).
For simplicity, we set M = 14 ω. We start from (1.8) written over Q+
ρ (θ)
for (u − k)− with k = µ− + M , where θ is to be determined, and we discard
the third term on the left-hand side. Moreover, we consider a non-negative,
piecewise smooth test function ζ = ζ(x) such that
ζ = 1 in Bσρ ,
ζ = 0 in RN − Bρ ,
|∇ζ| ≤
1
.
σρ
Hence, we conclude that
Z
ess sup
(u − k)2− ζ 2 (x, t)dx
0<t<θρ2
≤
Bρ
Z
Bρ
(u − k)2− ζ 2 (x, 0)dx
ZZ
2
γ
(u − k)2− dx dτ + γ∗2 |A−
|1− N +2 +2δ
k,ρ
2
2
σ ρ
Q+
ρ (θ)
θ
α
|Bρ | + γ 2 M 2 |Bρ |
≤M 2 1 −
2
σ
2δ(N +2)
1− N2+2 +2δ
2ρ
M 2 |Bρ |.
+ γθ
γ∗
M2
+
If we stipulate taking
γ∗2
ρ2δ(N +2)
< 1,
M2
we have
ess sup
0<t<θρ2
Z
Bρ
(u − k)2− ζ 2 (x, t)dx
≤M
2
θ
α
1− N2+2 +2δ
|Bρ |.
+ γ 2 + γθ
1−
2
σ
Let ℓ = µ− + ǫM with ǫ ∈ (0, 1) to be chosen. Then,
6 Propagating in Time the Measure-Theoretical Information
Z
Bρ
2 2
(u − k)−
ζ (x, t)dx ≥
Z
B(1−σ)ρ ∩[u≤ℓ]
469
(u − k)2− (x, t)dx
≥ (1 − ǫ)2 M 2 |[u(·, t) ≤ ℓ] ∩ B(1−σ)ρ |
= (1 − ǫ)2 M 2 |Aℓ,(1−σ)ρ (t)|.
Moreover,
|Aℓ,ρ (t)| = |Aℓ,(1−σ)ρ (t) ∪ (Aℓ,ρ (t)\Aℓ,(1−σ)ρ (t))|
≤ |Aℓ,(1−σ)ρ (t)| + |Bρ \B(1−σ)ρ |
≤ |Aℓ,(1−σ)ρ (t)| + N σ|Bρ |.
Therefore, we conclude
2
2
(1 − ǫ) M |Aℓ,ρ (t)| ≤M
2
"
1−
α
θ
+γ 2
2
σ
i
2
+ γθ1− N +2 +2δ + N (1 − ǫ)2 σ |Bρ |.
Since we can take ǫ < 21 without loss of generality, we also have
"
#
1 − α2
2
θ
+ γ 2 + γθ1− N +2 +2δ + N σ |Bρ |.
|Aℓ,ρ (t)| ≤
(1 − ǫ)2
σ
Taking into account that θ ∈ (0, 1) yields
#
"
1 − α2
θ
+ 2γ 2 + N σ |Bρ |.
|Aℓ,ρ (t)| ≤
(1 − ǫ)2
σ
Finally, we choose
α3
,
2 · 323 γN 2
1 − α2
1 − α4
α
<
≤1−
2
(1 − ǫ)
(1 − ǫ)2
8
σ=
and conclude
α
,
32N
θ=
⇒
ǫ≈
3
α,
16
α
|Bρ |
|Aℓ,ρ (t)| ≤ 1 −
16
for all t ∈ [0, θρ2 ], with θ computed above.
Remark 6.2 In Section 6c of the Complements, we present a refined version
of Proposition 6.1, based on a clever idea originally introduced in Liao [171].
470
12 PARABOLIC DEGIORGI CLASSES
7 Proof of Theorem 3.1
Without loss of generality, we may assume (y, s) = (0, 0). Assume θ = 1 and
consider Q−
ρ . By the definitions in (3.1), either
|[u ≤ µ+ − 21 ω] ∩ Q−
ρ|≥
1 −
|Q |
2 ρ
(7.1)
or
1 −
|Q |.
(7.2)
2 ρ
Indeed, if both were false, we would immediately have a contradiction.
Assume the second is in force; in case (7.1) holds, the argument runs
exactly in the same way. We can equivalently rewrite (7.2) as
|[u ≥ µ− + 21 ω] ∩ Q−
ρ|≥
|[u < µ− + 21 ω] ∩ Q−
ρ|<
1 −
|Q |.
2 ρ
We claim that this implies that there exists to ∈ [−ρ2 , − 31 ρ2 ] such that
3
|Bρ |
4
|[u(·, to ) < µ− + 21 ω] ∩ Bρ | ≤
Indeed, if not, then
|[u < µ− + 12 ω] ∩ Q−
ρ| ≥
Z
− 31 ρ2
−ρ2
|[u(·, τ ) < µ− + 21 ω] ∩ Bρ |dτ >
1 −
|Q |.
2 ρ
We can equivalently conclude that there exists to ∈ [−ρ2 , 31 ρ2 ] such that
|[u(·, to ) ≥ µ− + 21 ω] ∩ Bρ | ≥
1
|Bρ |.
4
Without loss of generality, we may assume that to = −ρ2 , so that
|[u(·, −ρ2 ) ≥ µ− + 41 ω] ∩ Bρ | ≥
1
|Bρ |.
4
In such a way we have put ourselves in the worst possible situation, where
the measure theoretical information is furthest away from the origin, about
which we want to show that the oscillation reduces.
We can then apply Proposition 6.1 with α = 21 , and conclude that either
ω ≤ 2γ∗ ρδ(N +2)
or
(7.3)
1
|Bρ |
(7.4)
8
for all t ∈ [−ρ2 , −ρ2 + θρ2 ], with θ ∈ (0, 1) as computed in Proposition 6.1.
|[u(·, t) ≥ µ− + ξω] ∩ Bρ | ≥
7 Proof of Theorem 3.1
471
If (7.3) holds true, we have finished. Otherwise, assuming that (7.4) is in
force, fix ν as claimed by Proposition 4.1 in (4.6), for the choices a = 21 , with
θ as above, and the corresponding θ̄.
We point out that (7.4) is (5.1)− of Proposition 5.1 with α = 81 and ǫ = ξ;
therefore, ν being fixed, determine j∗ and hence ǫν = 2ξj∗ by the procedure of
Proposition 5.1.
Then, by Proposition 4.1, either ǫν ω ≤ γ∗ ρδ(N +2) and we have finished, or
ρ 2
1
u ≥ µ− + ǫν ω a.e. in B ρ2 × (−ρ2 + θ
, −ρ2 + θρ2 ].
2
2
In particular,
u(·, t1 ) ≥ µ− +
1
ξω
2j∗ +1
in B ρ2 ,
where t1 = −ρ2 + θρ2 . Hence,
|[u(·, t1 ) ≥ µ− +
1
2j∗ +1
ξω] ∩ Bρ | ≥ |[u(·, t1 ) ≥ µ− +
= |B ρ2 | =
1
2j∗ +1
] ∩ B ρ2 |
1
|Bρ |.
2N
We can now work as before, with α = 21N , ǫ = 2j∗1+1 ξ, and conclude that there
exist θ̃ = θ̃(data, N ) and j̃ = j̃(data, N ) such that either
1
ξω ≤ γ∗ ρδ(N +2) ,
2j∗ +1+j̃
and we have finished, or for t2 = −ρ2 + θρ2 + θ̃ρ2 we have
u(·, t2 ) ≥ µ− +
1
2j∗ +1+j̃
ξω
in B ρ2 ,
which yields that
|[u(·, t2 ) ≥ µ− +
1
2j∗ +1+j̃
ξω] ∩ Bρ | ≥ |[u(·, t2 ) ≥ µ− +
= |B ρ2 | =
1
2j∗ +1+j̃
] ∩ B ρ2 |
1
|Bρ |.
2N
The previous step can be further repeated with the same constants as before.
Let n ∈ N be such that
def
tn+1 = −ρ2 + θρ2 + nθ̃ρ2 ≥ 0.
Such a value of n is precisely determined by the previous steps. We conclude
that either
1
ξω ≤ γ∗ ρδ(N +2) ,
j
+1+n
j̃
2∗
or
12 PARABOLIC DEGIORGI CLASSES
472
u ≥ µ− +
1
in B ρ2 × (tn , 0].
ξω
2j∗ +1+nj̃
For simplicity, we let
Q̃−
= B ρ2 × (tn , 0] = B ρ2 × (−θ̄ρ2 , 0]
1
ρ
2
with θ̄ = 1 − θ − (n − 1)θ̃ and we remark that the latter implies
− ess inf u ≤ − ess inf u −
Q̃−
1
2
Q−
2ρ
ρ
1
ξ
2j∗ +1+nj̃
ess osc u.
Q−
2ρ
Now,
ess sup u ≤ ess sup u.
Q̃−
1
2
Q−
2ρ
ρ
Adding these inequalities gives
where η = 1 −
ess osc u ≤ η ess sup u,
Q̃−
1ρ
2
Q−
2ρ
1
ξ.
2j∗ +1+nj̃
2
−k
ρ
Let Q−
R. If we let Q−
R ⊂ ET be fixed and set ρk = 4
k = B 2k × (−θ̄ρk , 0],
the previous remarks imply that
δ(N +2)
ess osc u ≤ max{η ess osc u ; C̄γ∗ ρk
Q−
k−1
Q−
k
}
(7.5)
for a proper choice of C̄ that takes into account all the alternatives. By iteration
δ(N +2)
}.
osc u ; C̄γ∗ ρk
ess −
osc u ≤ max{η k ess −
QR
Qk
Compute
ρk = 4−k R =⇒ −k = ln
ρ ln14
k
R
=⇒ η k =
ρ α
k
R
for α = −
ln η
.
ln 4
Remark 7.1 We have given the proof assuming θ = 1, but a general θ > 0
is also possible, without any substantial change in the previous arguments.
After proving the Harnack inequality below, we will show how such a result
also implies the local Hölder continuity of solutions. Moreover, we will give
a third proof of the continuity result based on the same set of ideas that are
used to show the validity of the Harnack inequality.
8 Boundary Parabolic DeGiorgi Classes: Dirichlet Data
Let ∂E be the finite union of portions of (N − 1)-dimensional surfaces of class
C 1 , so that the trace of a function u ∈ W 1,p (E) can be defined on ∂E, except
possibly on an (N − 2)-dimensional subset of ∂E.
8 Boundary Parabolic DeGiorgi Classes: Dirichlet Data
Given
(
g ∈ L2 0, T ; W 1,2(E) ,
g continuous on E T with modulus of continuity ωg (·),
473
(8.1)
we are interested in the boundary behavior of solutions of the Cauchy–
Dirichlet problem

ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u) in ET ,



u(·, t)
= g(·, t) a.e. t ∈ (0, T ],
(8.2)

∂E


u(·, 0) = g(x, 0),
where A and B satisfy (1.2)–(1.3), and g is as in (8.1).
A weak sub(super)-solution of the Cauchy–Dirichlet problem (8.2) is a
measurable function
u ∈ C 0, T ; L2(E) ∩ L2 0, T ; W 1,2(E)
satisfying
Z
E
ZZ
− uϕt + A(x, t, u, ∇u) · ∇ϕ
Z
− B(x, t, u, ∇u)ϕ dxdt ≤ (≥)
gϕ(x, 0) dx
uϕ(x, T ) dx +
ET
E
Wo2 (ET ).
for all non-negative test functions ϕ ∈
In addition, we take the boundary condition u ≤ g (u ≥ g) to mean that
(u − g)+ (·, t) ∈ Wo1,2 (E) ((u − g)− (·, t) ∈ Wo1,2 (E)) for a.e. t ∈ (0, T ]. A
function u, which is both a weak sub-solution and a weak super-solution, is a
weak solution.
The formulation can be rephrased in terms of Steklov averages as in the
previous Chapter, namely
Z
uh,τ ϕ + [A(x, τ, u, ∇u)]h · ∇ϕ − [B(x, τ, u, ∇u)]h ϕ dx ≤ (≥)0, (8.3)
E×{t}
∀ 0 < t < T − h and ∀ ϕ ∈ W 1,2 (E) ∩ L∞ (E), ϕ ≥ 0.
Moreover, the initial datum is taken in the sense of L2 (E), i.e.,
(uh (·, 0) − g(·, 0))+(−) → 0 in L2 (E).
Since g ∈ C(∂p ET ), it is natural to ask whether a solution of the Dirichlet
problem, whenever it exists, is continuous up the boundary ∂p ET . The issue
can be rephrased by asking what requirements are needed on ∂E for the
interior continuity of functions in the PDG classes to extend up to ∂p ET .
For simplicity, we distinguish the behavior at the lateral boundary and at
t = 0.
474
12 PARABOLIC DEGIORGI CLASSES
8.1 Lateral Conditions
Assume that ∂E satisfies the property of positive geometric density, that is,
there exist β ∈ (0, 1) and R > 0 such that for all y ∈ ∂E, and for all 0 < ρ ≤ R
Bρ (y) ∩ (RN − E) ≥ β Bρ .
(8.4)
Fix (xo , to ) ∈ ST , and consider the cylinder (xo , to ) + Qρ− where ρ, θ > 0 are
so small that to − θρ2 > 0.
Consider a piecewise smooth, cutoff function ζ vanishing on ∂Bρ (xo ), and
such that 0 ≤ ζ ≤ 1, and let u be a local sub(super)-solution of the Dirichlet
problem associated with (1.1) for the given g. Local estimates for u near
(xo , to ) similar to (1.8) are obtained by taking, in the weak formulation (8.3),
the testing functions
2
ϕ±
h = ±(uh − k)± ζ ,
integrating over [(xo , to ) + Qρ− (θ)] and letting h → 0. Such a choice of testing
functions is admissible if for a.e. t ∈ (to − θρ2 , to ],
(u(·, t) − k)± ζ 2 (·, t) ∈ Wo1,2 (Bρ (xo ) ∩ E).
(8.5)
Since x → ζ(x, t) vanishes on the boundary of Bρ (xo ) and not on the boundary
of Bρ (xo ) ∩ E, condition (8.5) will be verified if for a.e. t ∈ (to − θρ2 , to ]
(u − k)± (·, t) = 0 in the sense of the traces on ∂Bρ (xo ) ∩ E.
In view of Lemma 1.1 of Chapter 11, this can be realized for the function
(u − k)+ if k is chosen to satisfy
k≥
sup
g.
(8.6)
−
[(xo ,to )+Qρ
(θ)]∩ST
Analogously, the functions −(u − k)− ζ 2 can be taken as testing functions if
k≤
inf
−
[(xo ,to )+Qρ
(θ)]∩ST
g.
(8.7)
With these choices of k, we may repeat calculations in all analogous to those of
Section 1, with the understanding that the various integrals are now extended
1,2
2
over [(xo , to )+Q−
ρ (θ)]∩ET . However, since ζ (u−k)± (·, t) ∈ Wo (Bρ ∩E) for
2
almost every t ∈ (to − θρ , to ], we may regard them as elements of Wo1,2 (Bρ )
by defining them to be zero outside E. Then, the same calculations lead to
the inequalities (1.8), with the same stipulations that the various functions
vanish outside ET and the various integrals are extended over the full cylinder
(xo , to ) + Q−
ρ (θ).
8 Boundary Parabolic DeGiorgi Classes: Dirichlet Data
475
8.2 Initial Conditions
Consider a weak solution of (8.2) that takes the initial datum g(·, 0) in the
sense that
Z
1 h
u(·, τ ) dτ → g(·, 0) in L2loc (E) as h → 0.
(8.8)
h 0
We point out that, as far as the initial condition is concerned, u could also be
a solution of the Cauchy–Neumann problem we are going to deal with in the
next Section. In either case the assumption
g(·, 0) is continuous in E with modulus of continuity, say ωo (·),
in force.
Fix (xo , to ) ∈ ET and consider the cylinder (xo , to ) + Q−
ρ (θ), where θ and
ρ are such that to − θρ2 = 0. Therefore, (xo , to ) + Q−
ρ (θ) lies on the bottom
of the cylindrical domain ET . Consider a piecewise smooth, cutoff function
ζ vanishing on ∂Bρ (xo ), and such that 0 ≤ ζ ≤ 1, and in addition ζ is
independent of t ∈ (0, to ).
Local estimates similar to (1.8) for u near t = 0 are derived by taking in
the weak formulation (8.3) testing functions
2
ϕ±
h = ±(uh − k)± ζ ,
integrating over (0, t), t ∈ (0, to ), and letting h → 0. The first term in (8.3)
gives
Z
Z
1
1
(uh − k)2± (x, t)ζ 2 (x) dx −
(uh − k)2± (x, 0)ζ 2 (x) dx.
2 Bρ (xo )
2 Bρ (xo )
If k is chosen so that k ≥ supBρ (xo ) g(·, 0), then in view of (8.7) we have
Z
Bρ (xo )
(uh − k)2+ (x, 0)ζ 2 (x) dx → 0 as h → 0.
Analogous considerations hold for (uh − k)2− ζ 2 , provided k ≤ inf Bρ (xo ) g(·, 0).
Therefore, summarizing, provided we choose


k ≥ sup g(·, 0) for the function (u − k)+
Bρ (xo )
(8.9)

k ≤ inf g(·, 0) for the function (u − k)− ,
Bρ (xo )
and take θ and ρ such that to − θρ2 = 0, we have
476
12 PARABOLIC DEGIORGI CLASSES
ess sup
to −θρ2 <t<to
Z
Bρ (xo )
+
≤γ
ZZ
ZZ
2
(x, t)ζ 2 (x)dx
(u − k)±
(xo ,to )+Q−
ρ (θ)
−
(xo ,to )+Qρ
(θ)
+ γC22
ZZ
|∇[(u − k)± ζ]|2 dx dτ
(u − k)2± |∇ζ|2 dx dτ
−
(xo ,to )+Qρ
(θ)
(8.10)
2 2
ζ dx dτ
(u − k)±
2
1− N +2 +2δ
+ γ∗2 |A±
.
k,ρ |
8.3 Definition of Boundary Parabolic DeGiorgi Classes
Taking into account Section 8.1 and Section 8.2, given g ∈ C(E T ), the bound±
ary parabolic DeGiorgi classes PDG±
g = PDGg (ET , γ̄, γ∗ , C2 , δ) are the collection of all
u ∈ C 0, T ; L2(E) ∩ L2 0, T ; W 1,2(E) ∩ L∞ (ET ),
which satisfy these two conditions:
•
•
For all (xo , to ) ∈ ∂p ET and all cylinders (xo , to ) + Q−
ρ (θ) such that to −
θρ2 > 0 the localized truncations (u − k)± verify (1.8) for all levels k
subject to the restrictions (8.6)–(8.7);
2
For all (xo , to ) ∈ ET and all cylinders (xo , to )+Q−
ρ (θ) such that to −θρ = 0
the localized truncations (u − k)± satisfy (8.10) for all levels k subject to
the restrictions (8.9).
−
We further define PDGg = PDG+
g ∩ PDGg and refer to these classes as
homogeneous if γ∗ = 0.
8.4 Continuity up to ∂p ET of Functions in the Boundary PDG
Classes (Dirichlet Data)
Theorem 8.1. Let u be a bounded weak solution of the Cauchy–Dirichlet
problem (8.1)–(8.2). The boundary ∂E is assumed to satisfy the property of
positive geometric density (8.4). Then, u ∈ C(E T ), and there exists a continuous, positive, nondecreasing function s → ω(s) : R+ → R+ , such that
1
|u(x1 , t1 ) − u(x2 , t2 )| ≤ ω |x1 − x2 | + |t1 − t2 | 2 ,
for every pair of points (x1 , t1 ), (x2 , t2 ) ∈ E T . In particular, if g is Hölder
continuous in ST with exponent αST , and if g is Hölder continuous at t = 0
in E with exponent αo , then u is Hölder continuous in E T , and there exist
constants γ > 1 and α ∈ (0, 1) such that
8 Boundary Parabolic DeGiorgi Classes: Dirichlet Data
477
α
1
|u(x1 , t1 ) − u(x2 , t2 )| ≤ γkuk∞;ET |x1 − x2 | + |t1 − t2 | 2
for every pair of points (x1 , t1 ), (x2 , t2 ) ∈ ET . The constants γ and α depend
only upon the data. Moreover, the constant α depends also upon the Hölder
exponents αST and αo .
Even though the arguments are fairly similar, nevertheless we will separately discuss the continuity up to the lateral boundary, and the continuity up
to t = 0. In the following, we will always assume that ρ is such that ρ < C2−1 .
Let R be the parameter in the condition of positive geometric density (8.4).
For (xo , to ) ∈ ST and for θ > 0, ρ ∈ (0, R) such that to − 4θρ2 > 0, consider
−
nested cylinders (xo , to ) + Q−
ρ (θ) ⊂ (xo , to ) + Q2ρ (θ) and set
µ+ =
ess sup
u,
µ− =
[(xo ,to )+Q−
2ρ (θ)]∩ET
ess inf
[(xo ,to )+Q−
2ρ (θ)]∩ET
u,
(8.11)
ω(2ρ) = µ+ − µ− =
ess osc
[(xo ,to )+Q−
2ρ (θ)]∩ET
u.
Moreover, let
g + (2ρ) =
ess sup
g,
g − (2ρ) =
[(xo ,to )+Q−
2ρ (θ)]∩ST
ωg (2ρ) = g + (2ρ) − g − (2ρ) =
ess inf
[(xo ,to )+Q−
2ρ (θ)]∩ST
ess osc
[(xo ,to )+Q−
2ρ (θ)]∩ST
g,
g.
As far as the behavior at the lateral boundary is concerned, Theorem 8.1 is a
straightforward consequence of the following.
Proposition 8.1 Let ∂E satisfy the positive geometric density condition
(8.4), and let g ∈ C(E T ). Then, every u ∈ PDGg is continuous up to
ST , and there exist constants C > 1 and α ∈ (0, 1), depending only on
the data defining the PDGg classes and the parameter β in (8.4), and independent of g and u, such that for all (xo , to ) ∈ ST and all cylinders
−
(xo , to ) + Q−
2ρ (θ) ⊂ (xo , to ) + QR (θ) with θ > 0 and ρ ∈ (0, R) such that
to − 4θρ2 > 0
n
ρ α
o
ω(ρ) ≤ C max ω(R)
(8.12)
; ωg (2ρ) ; γ∗ ρδ(N +2) .
R
Notice that the boundedness of u is already taken into account in the
definition of PDGg classes.
As usual, we assume (xo , to ) = (0, 0). The proof of this proposition is
almost identical to that of the interior Hölder continuity, except for a few
changes, which we outline next. First, Proposition 4.1 and its proof continue
to hold, provided the levels ξω satisfy (8.6)–(8.7). Next, Proposition 5.1 and
its proof continue to be in force, provided the levels kj in (5.1) satisfy the
478
12 PARABOLIC DEGIORGI CLASSES
restrictions (8.6)–(8.7) for all j ≥ 1. Finally, Proposition 6.1 and its proof
continue to hold, provided the levels ℓ satisfy the restrictions (8.6)–(8.7).
Now either one of the inequalities
µ+ − 14 ω ≥ g + ,
µ− + 14 ω ≤ g −
must be satisfied. Indeed, if both are violated, that is
µ+ − 14 ω ≤ g +
and
− µ− − 41 ω ≤ −g − ,
adding these inequalities gives
ω(ρ) ≤ 2ωg (2ρ)
and there is nothing to prove. Assuming the second holds, then all levels ks
as defined in (5.1) for ǫ = 21 satisfy restriction (8.7) and thus are admissible.
Moreover, (u − k1 )− vanishes outside ET , and therefore
−
[u ≤ µ− + 14 ω] ∩ Q−
ρ ≥ β|Qρ |
where β is the parameter in the positive geometric density condition (8.1).
From this, the procedure of the proof of Theorem 3.1 can be repeated with
the understanding that (u − ks )− are defined in the full cylinder Q−
ρ and are
zero outside ET .
Let us now consider the behavior at t = 0. Fix (xo , 0) ∈ E × {0}, and
ρ > 0 so that Bρ (xo ) ⊂ E. After a translation, without loss of generality, we
may assume xo = 0.
The proof of the continuity (or of the Hölder continuity) of u up to t = 0
follows from a simple variant of Theorem 3.1 and Proposition 8.1. Here, we
briefly sketch how to proceed. Set
µ+
o = ess sup g(·, 0),
Bρ
µ+ = ess sup u,
µ−
o = ess inf g(·, 0),
Bρ
µ− = ess inf u,
Bρ
ω(ρ) = µ+ − µ− = ess osc u,
Q−
ρ
Q+
ρ
ωo (ρ) = ess osc g(·, 0),
Q+
ρ
and consider the two inequalities
µ+ − 14 ω < µ+
o,
µ− + 41 ω > µ−
o .
If both hold, subtracting from one another, we obtain
ess osc u ≤ 2 ess osc g(·, 0)
Q+
ρ
Bρ
and there is nothing to prove. Let us assume, without loss of generality, that
the second one is violated, namely that
9 Boundary Parabolic DeGiorgi Classes: Neumann Data
479
µ− + 41 ω ≤ µ−
o .
ω
satisfy the second of (8.9).
2j
Therefore, we may derive estimates for the truncated functions (u − k)− as in
(8.10) with θ = 1, which take the form
Z
ess sup
(u − k)2− (x, t)ζ 2 (x)dx
Then, for all j ≥ 2, the levels k = µ− +
0<t<ρ2
Bρ
+
≤γ
ZZ
ZZ
Q+
ρ
Q+
ρ
+ γC22
|∇[(u − k)− ζ]|2 dx dτ
(u − k)2− |∇ζ|2 dx dτ
ZZ
Q+
ρ
2
1− N +2 +2δ
.
(u − k)2− ζ 2 dx dτ + γ∗2 |A−
k,2ρ |
They are phrased in terms of cylinders Q+
ρ , but are obviously equivalent to
(8.10).
Since u(·, 0) > µ− + ω4 in Bρ , we can apply Proposition 6.1 and conclude
that either ω ≤ 2γ∗ ρδ(N +2) , or
|[u(·, t) > µ− + ξω] ∩ Bρ | ≥
1
|Bρ |,
4
∀ t ∈ [0, θρ2 ],
for proper ξ and θ. These two alternatives correspond to (7.3) and (7.4);
from here on, we proceed as in the proof of Theorem 3.1 with the obvious
adjustments, which are needed in order to deal with cylinders Q+
ρ , and we
conclude.
Remark 8.1 The arguments are local in nature and as such they require
only local assumptions. For example, the positive geometric density condition
(8.4) could be satisfied on only a portion of ∂E, open in the relative topology
of ∂E, and for any t ∈ (to − 4θρ2 , to ], g(·, t) could be continuous only on that
portion of ∂E. Then, the boundary continuity of Theorem 8.1 continues to
hold only locally, on that portion of ST . Similar considerations hold for the
continuity at t = 0.
9 Boundary Parabolic DeGiorgi Classes: Neumann Data
Assume that ∂E is of class C 1,λ , λ ∈ (0, 1) so that the outward unit normal,
which we denote by n, is everywhere defined on ∂E, and consider the quasilinear Cauchy–Neumann problem


ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u) in ET
A(x, t, u, ∇u) · n = ψ(x, t, u) a.e. t ∈ (0, T ]
(9.1)


u(·, 0) = uo
480
12 PARABOLIC DEGIORGI CLASSES
where the functions A and B satisfy the structure conditions (1.2), and uo
is continuous in Ē with modulus of continuity ωo ; moreover, we assume that
ψ(·, t, u(·)) admits for a.e. t ∈ (0, T ) an extension into E, which we continue
to denote by ψ(·, t, u(·)), such that


|ψ| ≤ ψo |u| + ψ1 ,
|ψu | ≤ ψo ,
(9.2)


|ψxi | ≤ ψ1 for i = 1, 2, . . . , N,
where ψo and ψ1 are given, non-negative functions satisfying
ψo , ψ1 ∈ LN +2+ε (ET ),
(9.3)
for some ε > 0. We are interested in the boundary behavior of solutions of
this problem.
Weak solutions can be formulated by a straightforward extension of the
definition in (9.3) of Chapter 11. However, let us concentrate on local solutions.
Let K be an arbitrary compact subset of RN . A weak sub(super)-solution
of (9.1) is a measurable function
u ∈ C(0, T ; L2 (E)) ∩ L2 (0, T ; W 1,2(E))
satisfying for every compact subset K of RN , for every subinterval [t1 , t2 ] ⊂
(0, T ], and for all non-negative test functions
Z
K∩E
ϕ ∈ W 1,2 (0, T ; L2 (K)) ∩ L2 (0, T ; Wo1,2 (K)),
Z Z t2 Z
t2
uϕ dx +
[−uϕt + A(x, t, u, ∇u)∇ϕ
t1
t1
K∩E
−B(x, t, u, ∇u)ϕ] dxdt ≤ (≥) +
Z
t2
t1
Z
(9.4)
ψϕ dσdt,
K∩∂E
where dσ is the surface measure on ∂E.
All terms on the left-hand side are well defined by virtue of the structure
conditions (1.2), whereas the boundary integral on the right-hand side is well
defined by virtue of (9.2)
We point out that ϕ vanishes in the sense of traces on ∂K, and not on the
boundary of E.
A function that is both a weak sub-solution and a weak super-solution is
a weak solution.
The formulation can be rephrased in terms of Steklov averages, as in the
previous Section, namely
Z
[uh,τ ϕ + [A(x, τ, u, ∇u)]h · ∇ϕ − [B(x, τ, u, ∇u)]h ϕ] dx
(K∩E)×{t}
Z
(9.5)
≤ (≥)
[ψ(x, τ, u)]h ϕ dσ
(K∩∂E)×{t}
9 Boundary Parabolic DeGiorgi Classes: Neumann Data
481
for all 0 < t < T − h, and for all ϕ ∈ Wo1,2 (K) with ϕ ≥ 0.
Moreover, the initial datum is taken in the sense of L2 (E), i.e.,
(uh (·, t) − uo )+(−) → 0
in L2 (E).
As we did in the previous Section, also in this case it is quite natural to ask
whether a solution of the Neumann problem, provided it exists, is continuous
up to the boundary ∂p ET . In other terms, what do we need to do, for the
interior continuity of a function in the PDG classes to be extended up to the
boundary in the case of the Neumann Problem?
We concentrate on the behavior at the lateral boundary, since the case of
the initial datum was dealt with in the previous Section.
9.1 Lateral Boundary
Fix (xo , to ) ∈ ST , assume without loss of generality that it coincides with the
origin; we let KR denote the N -dimensional cube centered at the origin and
wedge 2R, i.e.,
KR = {x ∈ RN : max |xi | > R}.
1≤i≤N
In the following, mainly for simplicity, we will consider these cubes KR , instead
of balls BR . Cylinders Qρ± (θ) with the corresponding cubic cross-sections will
also be used.
We introduce a local change of coordinates by which ∂E ∩ KR for some
fixed R > 0 coincides with the hyperplane xN = 0, and E lies locally in
{xN > 0}. Setting
Kρ+ = Kρ ∩ [xN > 0]
for all 0 < ρ ≤ R
and
±
Q̃±
ρ (θ) = Qρ (θ) ∩ [xN > 0]
+
+
we require that all “concentric” 12 -cubes Kσρ
⊂ Kρ+ ⊂ KR
be contained in E.
Without loss of generality, we can assume that (9.5) is written in such a
coordinate system.
Let u be a local sub(super)-solution of (9.1) in the sense of (9.4), and in
the latter take the test functions v = ±(uh − k)± ζ 2 , where ζ is the usual
cut-off function, integrate over Q̃−
ρ (θ), and let h → 0. Moreover, we assume u
to belong to L∞ (ET ).
All the terms are treated as in the proof of (1.8), except for the boundary
integral. We arrive at
12 PARABOLIC DEGIORGI CLASSES
482
ess sup
−θρ2 <t<0
Co
+
2
ZZ
≤γ
Z
Kρ+
ZZ
Q̃−
ρ (θ)
Q̃−
ρ (θ)
+ γC22
Z
+
(u −
0
ZZ
−θρ2
Z
k)2± ζ 2 (x, t)dx
−
Z
Kρ+
2 2
ζ (x, −θρ2 )dx
(u − k)±
|∇[(u − k)± ζ]|2 dx dτ
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
(9.6)
2
Q̃−
ρ (θ)
1− N +2 +2δ
(u − k)2± ζ 2 dx dτ + γ∗2 |A±
k,ρ |
ψ(u − k)± ζ 2 dx̄dt,
K̃ρ
where K̃ρ = Kρ ∩ ∂E is the (N − 1)-dimensional cube, i.e.,
K̃ρ = {x̄ ≡ (x1 , . . . , xN −1 ) ∈ RN −1 :
max
1≤i≤N −1
|xi | < ρ}.
We estimate the boundary integral by transforming it into an interior integral
as follows.
Z 0 Z
ψ(u − k)± ζ 2 dx̄dt
−θρ2
=
Z
0
−θρ2
≤γ
ZZ
K̃ρ
Z
K̃ρ
Q̃−
ρ (θ)
+γ
ZZ
Z
ρ
0
∂
ψ(x, t, u)(u − k)± ζ 2 dxN
∂xN
dx̄dτ
|ψxN |(u − k)± ζ 2 + |ψ||∇[(u − k)± ζ]| dxdτ
Q̃−
ρ (θ)
|ψ|(u − k)± ζ|∇ζ| + |ψu ||∇(u − k)± |(u − k)± ζ 2 dxdτ
By virtue of (9.2)–(9.3), we have
ZZ
ZZ
2
|ψxN |(u − k)± ζ dxdτ ≤ kuk∞;ET
Q̃−
ρ (θ)
Q̃−
ρ (θ)
ψ1 χ[(u−k)± >0] dxdτ.
Moreover,
ZZ
|ψ| [|∇[(u − k)± ζ]| + |(u − k)± ζ|∇ζ|] dxdτ
Q̃−
ρ (θ)
≤
Co
8
ZZ
+γ
Q̃−
ρ (θ)
ZZ
|∇[(u − k)± ζ]|2 dxdτ +
Q̃−
ρ (θ)
(u −
k)2± |∇ζ|2
By (9.2) it is apparent that
ZZ
dxdτ + γ
Q̃−
ρ (θ)
ZZ
|ψ|2 χ[(u−k)± >0] dxdτ
Q̃−
ρ (θ)
ζ 2 |ψ|2 χ[(u−k)± >0] dxdτ.
9 Boundary Parabolic DeGiorgi Classes: Neumann Data
ZZ
−
Q̃ρ
(θ)
|ψ|2 χ[(u−k)± >0] dxdτ
ZZ
≤
−
(θ)
Q̃ρ
u
2
ψo2 χ[(u−k)± >0]
≤γ(data, kuk∞;ET )
ZZ
dxdτ +
Q̃−
ρ (θ)
ZZ
Q̃−
ρ (θ)
483
ψ12 χ[(u−k)± >0] dxdτ
(ψo2 + ψ12 )χ[(u−k)± >0] dxdτ
2
± 1− N +2 +2δ
≤γ(data, kuk∞;ET )kψo + ψ1 kN +2+ε;Q̃ρ− (θ) |Ak,ρ
|
,
with δ =
ZZ
Q̃−
ρ (θ)
≤
ε
(N +2)(N +2+ε) .
Finally,
|ψu ||∇(u − k)± |(u − k)± ζ 2 dxdτ
ZZ
Q̃−
ρ (θ)
ZZ
Co
≤
8
+γ
|ψu | |∇[(u − k)± ζ]|(u − k)± ζ + (u − k)2± ζ|∇ζ| dxdτ
2
|∇[(u − k)± ζ]| dxdτ + γ
Q̃−
ρ (θ)
Z
Q̃−
ρ (θ)
Z
(u − k)2± |∇ζ|2 dxdτ + γ
ZZ
Q̃−
ρ (θ)
Q̃−
ρ (θ)
|ψu |2 (u − k)2± ζ 2 dxdτ
(u − k)2± ζ 2 dxdτ,
and
ZZ
Q̃−
ρ (θ)
|ψu |2 (u − k)2± ζ 2 dxdτ
≤γ(data, kuk∞;ET )
ZZ
Q̃−
ρ (θ)
ζ 2 ψo2 χ[(u−k)± >0] dxdτ
2
1− N +2 +2δ
.
≤γ(data, kuk∞;ET )kψo k2N +2+ε;Q̃− (θ) |A±
k,ρ |
ρ
Collecting all the terms yields
Z
Z
2 2
ess sup
(u − k)± ζ (x, t)dx −
−θρ2 <t<0
+
≤γ
ZZ
ZZ
Kρ+
Q̃−
ρ (θ)
Q̃−
ρ (θ)
+ γC22
ZZ
Kρ+
(u − k)2± ζ 2 (x, −θρ2 )dx
|∇[(u − k)± ζ]|2 dx dτ
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
Q̃−
ρ (θ)
2
1− N +2 +2δ
2
,
(u − k)2± ζ 2 dx dτ + γ∗∗
|A±
k,ρ |
where the value of δ has been given before, and
2
γ∗∗
= γ∗2 + γ(data, kuk∞;ET )kψo + ψ1 k2N +2+ε;Q̃− (θ) .
ρ
484
12 PARABOLIC DEGIORGI CLASSES
9.2 Definition of Boundary Parabolic DeGiorgi Classes
Given ψ as in (9.2)–(9.3), and taking into account that u ∈ L∞ (ET ), the
boundary parabolic DeGiorgi classes
±
PDG±
ψ = PDGψ (ET , γ̄, γ∗∗ , C2 , δ)
are the collection of all
u ∈ C 0, T ; L2 (E) ∩ L2 0, T ; W 1,2 (E) ∩ L∞ (ET )
+
such that for all (xo , to ) ∈ ST , all 21 -cubes Kρ+ (xo ) ⊂ KR
(xo ) for ρ < R, and
−
2
all cylinders (xo , to ) + Q̃ρ (θ) such that to − θρ > 0, the localized truncations
(u − k)± satisfy
Z
ess sup
(u − k)2± ζ 2 (x, t)dx
Kρ+ (xo )
to −θρ2 <t<to
−
+
≤ γ̄
+
Z
Kρ+ (xo )
ZZ
ZZ
(u − k)2± ζ 2 (x, −θρ2 )dx
(xo ,to )+Q̃−
ρ (θ)
(xo ,to )+Q̃−
ρ (θ)
γ̄C22
ZZ
|∇[(u − k)± ζ]|2 dx dτ
(9.7)
(u − k)2± |∇ζ|2 + (u − k)2± |ζτ | dx dτ
2
(xo ,to )+Q̃−
ρ (θ)
1− N +2 +2δ
2
.
(u − k)2± ζ 2 dx dτ + γ∗∗
|A±
k,ρ |
−
We further define PDGψ = PDG+
ψ ∩ PDGψ and refer to these classes as
homogeneous if γ∗∗ = 0.
9.3 Continuity up to ST of Functions in the Boundary PDG
Classes (Neumann Data)
Having fixed (xo , to ) ∈ ST , let
KR (xo ) = {x ∈ RN : max |xi − xi,o | < R},
1≤i≤N
assume after a flattening of ∂E about xo that ∂E coincides with the hyperplane xN = 0 within the cube KR (xo ) for some R > 0. For θ > 0 assume
that to − 4θρ2 > 0, consider the “concentric” cylinders (xo , to ) + Q̃−
ρ (θ) ⊂
−
(xo , to ) + Q̃2ρ (θ) ⊂ ET and set
µ+ =
ess sup
u,
µ− =
(xo ,to )+Q̃−
2ρ (θ)
ω(2ρ) = µ+ − µ− =
ess osc
ess inf
(xo ,to )+Q̃−
2ρ (θ)
(xo ,to )+Q̃−
2ρ (θ)
u.
u,
(9.8)
10 The Harnack Inequality
485
Theorem 9.1. Let ∂E be of class C 1,λ , with λ ∈ (0, 1). Then, every u ∈
PDGψ is Hölder continuous up to ST , and there exist constants C > 1 and
α ∈ (0, 1), depending only on the data defining the PDGψ classes and the C 1,λ
structure of ∂E, such that for all (xo , to ) ∈ ST and all “concentric” cylinders
−
(xo , to ) + Q̃ρ− (θ) ⊂ (xo , to ) + Q̃2ρ
(θ) ⊂ ET
n
ρ α
o
ω(ρ) ≤ C max ω(R)
; γ∗∗ ρN δ .
R
(9.9)
The proof of this theorem is almost identical to that of the interior Hölder
continuity, the only difference being that we are working with “concentric” 21 cubes and corresponding cylinders, instead of full balls and related cylinders.
Proposition 4.1 and its proof continue to hold. Since (u−k)± ζ(·, t) do not vanish on ∂Kρ+ , the parabolic embedding deduced from Theorem 2.1 of Chapter 9
is used, instead of the usual multiplicative embedding. Next, Proposition 5.1
relies on the discrete isoperimetric inequality of Proposition 5.2 of Chapter 10,
which holds for convex domains, and thus in particular for 12 -cubes. The rest
of the proof is identical to the indicated change in the use of the embedding
inequalities.
Remark 9.1 The regularity of ψ enters only in the requirements (9.2)–(9.3)
through the constant γ∗∗ .
Remark 9.2 The arguments are local in nature, and as such they require
only local assumptions.
Taking into account Theorem 9.1, we conclude.
Corollary 9.1 Let ∂E be of class C 1,λ , with λ ∈ (0, 1). A bounded, weak
solution u of the Neumann problem (9.1) for a datum ψ satisfying (9.2)–
(9.3), is Hölder continuous in Ē × (0, T ). Analogous local statements are in
force, if the assumptions on ∂E and ψ hold on portions of ST .
Remark 9.3 As already remarked above, the continuity of u can be claimed
up to t = 0, provided that uo in (9.1) is continuous in Ē with modulus of
continuity ωo .
10 The Harnack Inequality
We have already seen the Harnack inequality in the context of non-negative
solutions of the heat equation (Section 13 of Chapter 5). After Pini’s and
Hadamard’s results, it was shown to hold for non-negative solutions first of
linear parabolic equations with bounded and measurable coefficients, and then
of quasi-linear parabolic equations of the type of (1.1). It is quite remarkable
that it continues to hold for non-negative functions in the PDG classes.
486
12 PARABOLIC DEGIORGI CLASSES
Theorem 10.1. [48, 187, 189, 262] Let u ≥ 0 be an element of the parabolic
DeGiorgi class PDG(ET , γ̄, γ∗ , C2 , δ). There exist positive constants c∗ and θ∗
that can be quantitatively determined a priori in terms of only the parameters
N, γ and independent of u, γ∗ , and δ such that for every radius ρ < C2−1 ,
provided that B4ρ (xo ) × (to − θ∗ (4ρ)2 , to + θ∗ (4ρ)2 ) ⊂ ET , either u(xo , to ) ≤
c∗−1 γ∗ ρδ(N +2) or
(10.1)
c∗ u(xo , to ) ≤ inf u(·, to + θ∗ ρ2 ).
Bρ (xo )
The first proof of Theorem 10.1 that relies only on the structure of the classes
is in DiBenedetto [48]. This is the proof presented here, in view of its relative
flexibility, with one fundamental difference: in the spirit of what we did in
the elliptic context in Chapter 10, we do not rely on the Hölder continuity of
solution.
10.1 Proof of Theorem 10.1. Preliminaries
Fix (xo , to ) ∈ ET , assume that u(xo , to ) > 0, and construct the cylinders
±
(xo , to ) + Q8ρ
⊂ ET . The change of variables
x→
x − xo
ρ
t→
t − to
ρ2
maps these cylinders into Q± , where
Q+ = B8 × (0, 82 ],
Q− = B8 × (−82 , 0].
Denoting again by (x, t) the transformed variables, the rescaled function
1
u xo + ρx, to + tρ2
w(x, t) =
u(xo , to )
satisfies w(0, 0) = 1 and is a bounded, non-negative, element of the PDG
classes relative to the cylinders Q± , with the same parameters as the original
PDG classes, except that γ∗ is now replaced by
γ∗
Γ∗ = (2ρ)δ(N +2)
.
(10.2)
u(xo , to )
In particular, the truncations (w − k)± satisfy
Z
Z
2 2
ess sup
(w − k)± ζ (x, t)dx −
(w − k)2± ζ 2 (x, s − θr2 )dx
s−θr 2 <t<s Br (y)
Br (y)
ZZ
|∇[(w − k)± ζ]|2 dx dτ
+
(y,s)+Q−
r (θ)
ZZ
≤ γ̄
(w − k)2± |∇ζ|2 + (w − k)2± |ζτ | dx dτ
−
(y,s)+Qr (θ)
ZZ
1− N2+2 +2δ
,
+ γ̄C22
(w − k)2± ζ 2 dx dτ + Γ∗2 |A±
k,r |
(y,s)+Q−
r (θ)
(10.3)
10 The Harnack Inequality
487
for all (y, s) + Qr− (θ) ⊂ Q± and for all k > 0. Analogous estimates hold for
“forward” cylinders (y, s) + Qr+ (θ) ⊂ Q± . By these transformations, proving
(10.1) reduces to finding positive constant γo , γ1 , γ2 that can be determined a
priori only in terms of the parameters of the original PDG classes, such that
either u(xo , to ) < γ2 γ∗ ρδ(N +2) or
inf w(·, γ1 ) ≥ γo .
B1
(10.4)
10.2 Proof of Theorem 10.1. Expansion of Positivity
Let w be a non-negative element of PDG− (Q± , γ̄, γ∗ , C2 , δ).
Proposition 10.1 Assume that for some (y, s) ∈ Q± and some r ∈ (0, C2−1 )
|[w(·, s) ≥ M ] ∩ Br (y)| ≥ α Br (y)
(10.5)
for some M > 0 and some α ∈ (0, 1). Then there exist constants η and
θ ∈ (0, 1) depending only on the data {N, Co , C1 }, and α, such that either
ηM ≤ Γ∗ rδ(N +2) or
w ≥ ηM a.e. in B2r (y) × s + 12 θr2 , s + θr2 .
(10.6)
Proof. Without loss of generality, we can assume (y, s) = (0, 0). We prove the
statement in Q+ , but it can be easily extended to other frameworks, with
minor adjustments. Since w is non-negative, we can directly assume that
µ− = inf w = 0.
Q+
2r
The assumption (10.5) can then be read as assumption (6.1)− in Proposition 6.1. If we repeat its proof, we can conclude that there exist θ and ǫ in
(0, 1), depending only on the data {N, Co , C1 }, and α, and independent of M ,
such that either
M ≤ Γ ∗ rδ(N +2)
or
|[w(·, t) > ǫM ] ∩ Br | ≥ 12 α|Br |
for all t ∈ (0, θr2 ].
(10.7)
for all t ∈ (0, θr2 ].
(10.8)
The latter conclusion implies that
|[w(·, t) > ǫM ] ∩ B4r | > 21 α4−N |B4r |,
This represents (5.1)− with µ− = 0 and ǫω substituted by ǫM . Apply Proposition 4.1 over the cylinder B4r × (0, θr2 ] with µ− = 0, ξω = ǫν M , and a = 21 ,
where ǫν is the quantity claimed in Proposition 5.1. Choose ν from (4.1) and
observe that the number ν is independent of ǫν M . It only depends on the
data {N, Co , C1 } and θ, which itself has been determined and fixed in terms
of the data {N, Co , C1 } and α. Such a ν being fixed a priori only in terms of
488
12 PARABOLIC DEGIORGI CLASSES
the data, choose j∗ ∈ N as in Proposition 5.1. Then, Proposition 4.1 implies
that
i
w(x, t) > 21 ǫν M a.e. in B2r × 12 θr2 , θr2 .
Thus, the conclusion holds with η = 12 ǫν .
Remark 10.1 Proposition 4.1 is a “shrinking” proposition, in that informa−
tion on a cylinder Q−
r , yields information on a smaller cylinder Q 1 r . Propo2
sition 10.1 is an “expanding” proposition, in the sense that information on a
ball Br (y) at time s yields information on a larger ball B2r (y) for all times in
the interval (s + 12 θr2 , s + θr2 ]. Moreover, a measure-theoretical information
is converted into a pointwise information. This “expansion of positivity” is at
the heart of the Harnack inequality (10.1).
10.3 Proof of Theorem 10.1
In the light of Theorem 3.1, we may assume that w is continuous. The only
way we are using this information, is to give unique meaning to pointwise
values of w.
For τ ∈ [0, 1), introduce the family of nested cylinders {Q−
τ } with the same
“vertex” at (0, 0), and the families of non-negative numbers {Mτ } and {Nτ },
defined by
2
Q−
τ = Kτ × (−τ , 0],
Nτ = (1 − τ )−β ,
Mτ = sup w,
Q−
τ
where β > 1 is to be chosen. The two functions [0, 1) ∋ τ → Mτ , Nτ are
increasing, and Mo = No = 1 since w(0, 0) = 1. Moreover, Nτ → ∞ as τ → 1,
whereas Mτ is bounded, since w is locally bounded. Therefore, the equation
Mτ = Nτ has roots and we let τ∗ denote the largest one. By the continuity of
w, there exists (y, s) ∈ Q̄τ∗ such that
def
w(y, s) = Mτ∗ = Nτ∗ = (1 − τ∗ )−β = M.
(10.9)
Moreover,
−
(y, s) + Q−
r ⊂ Q 1+τ∗ ⊂ Q1 ,
def
where r =
2
1
2 (1
− τ∗ ).
(10.10)
Therefore, by the definition of Mτ and Nτ
sup
(y,s)+Q−
r
def
w ≤ sup w ≤ 2β (1 − τ∗ )−β = M∗ .
Q−
1+τ∗
2
The parameter τ∗ , and hence the upper bound M∗ , is only known qualitatively,
and β has to be chosen. The arguments below have the role of eliminating the
qualitative knowledge of τ∗ by a quantitative choice of β.
10 The Harnack Inequality
489
10.3.1 Local Largeness of w Near (y, s)
The largeness of w at (y, s) as expressed by (10.9), propagates to a full spacetime neighborhood nearby (y, s). To render this quantitative, set
ξ =1−
1
2β+1
,
3 1
2 2β+1 .
a=
1
1 − β+1
2
1−
Lemma 10.1 Either Γ∗ ≥ 1, or
−
|[w > 12 M ] ∩ [(y, s) + Q−
r ]| > ν|Qr |,
where
ν=
(10.11)
1 − a N +2 1
N +2 .
γ(data)
2 2
Proof. Assume that Γ∗ < 1. If (10.11) is violated, apply Proposition 4.1 over
the cylinder
2
(y, s) + Q−
r = Br (y) × (s − r , s]
in the form (4.1)+ –(4.2)+ , for the choices µ+ = ω = M∗ and θ = 1, to conclude
that
w(y, s) ≤ M∗ (1 − aξ) = 43 (1 − τ∗ )−β ,
contradicting (10.9).
Remark 10.2 The indicated expressions of ξ, a, and ν imply that ν(β) depends on the data and β, but is independent of τ∗ . Such a constant will be
made quantitative whence β is chosen, dependent only on the data. We continue to denote by ν such a constant, keeping in mind its dependence on
β.
Corollary 10.1 Either Γ∗ ≥ 1, or there exists a time level
s − r2 ≤ s̄ ≤ s
such that
|[w(·, s̄) > 12 M ] ∩ Kr (y)| > ν|Kr |.
(10.12)
10.3.2 Expanding the Positivity of w
Starting from (10.12) apply the expansion of positivity of Proposition 10.1
to w with 21 M and r given by (10.9)–(10.10) and α = ν. Then, taking into
account the expression (10.2) of Γ∗ , either
u(xo , to ) ≤ γ̄∗ γ∗ ρδ(N +2)
rδ(N +2)
M
(10.13)∗
490
12 PARABOLIC DEGIORGI CLASSES
or
w(·, t) ≥ η∗ M
in B2r (y)
(10.14)∗
for all t in the range
s̄ + 21 θ∗ r2 ≤ t ≤ s̄ + θ∗ r2 = s∗ .
(10.15)∗
Remark 10.3 The constants {γ̄∗ , θ∗ , η∗ } in (10.13)∗ –(10.15)∗ depend on the
data {N, Co , C1 } and β, through the constant ν(β) in (10.12). However, they
are independent of the constant Γ∗ in (10.2). These constants are also independent of M and r. The parameter β is still to be chosen.
The expansion of positivity implies in particular
|[w(·, s∗ ) > η∗ M ] ∩ B2r (y)| = |B2r |.
(10.16)
Therefore, the expansion of positivity of Proposition 10.1 can be applied again,
starting at the time level s∗ , with M replaced by (η∗ M ), ρ = 2r, and α = 1.
It gives that either
u(xo , to ) ≤ γ̄γ∗ ρδ(N +2)
(2r)δ(N +2)
η∗ M
(10.13)1
in B4r (y)
(10.14)1
or
w(·, t) ≥ η(η∗ M )
for all t in the range
s∗ + 12 θ(2r)2 ≤ t ≤ s∗ + θ(2r)2 = s1 .
(10.15)1
Remark 10.4 The constants {γ̄, θ, η} in (10.13)1 –(10.15)1 are different from
the set of constants {γ̄∗ , θ∗ , η∗ } in (10.13)∗ –(10.15)∗. They depend on the
data {N, Co , C1 } but they are no longer dependent on β. By the expansion of
positivity of Proposition 10.1 these parameters depend only on {N, Co , C1 },
and the measure-theoretical lower bound α. Such a measure-theoretical lower
bound in the current context is α = 1, as provided by (10.16). The parameter
β is still to be chosen.
Starting from (10.14)1 , the expansion of positivity can now be applied again
with M replaced by η(η∗ M ), and ρ replaced by 4r, and α = 1 to yield that
either
(4r)δ(N +2)
(10.13)2
u(xo , to ) ≤ γ̄γ∗ ρδ(N +2)
η(η∗ M )
or
w(·, t) ≥ η 2 (η∗ M )
in B8r (y)
(10.14)2
for all t in the range
s1 + 12 θ(4r)2 ≤ t ≤ s1 + θ(4r)2 = s2
(10.15)2
10 The Harnack Inequality
491
for the same set of parameters {γ̄, θ, η} as in (10.13)1 –(10.15)1 . These parameters depend on {N, Co , C1 } but they are independent of β.
The process can be iterated to yield that either
(2n r)δ(N +2)
η n−1 (η∗ M )
(10.13)n
in B2n+1 r (y)
(10.14)n
u(xo , to ) ≤ γ̄γ∗ ρδ(N +2)
or
w(·, t) ≥ η n (η∗ M )
for all t in the range
sn−1 + 12 θ(2n r)2 ≤ t ≤ sn−1 + θ(2n r)2 = sn .
(10.15)n
10.3.3 Proof of Theorem 10.1 Concluded
Without loss of generality we may assume that (1 − τ∗ ) is a negative, integral
power of 2. Then, choosing n so that 2n+1 r = 2, the ball B2 (y) covers the ball
B1 centered at x = 0, and
w(·, t) ≥ η n (η∗ M )
in B1 ,
for all t in the interval (10.15)n . For the indicated choice of n, and the values
of M and r given by (10.9)–(10.10)
−β
η∗ (β)
η∗ (β)
n2
=
η
(1 − τ∗ )β
rβ
η∗ (β)
= (2β η)n n+1 β = (2β η)n γo ,
(2
r)
η n (η∗ M ) = η n
where
γo = 2−β η∗ (β).
To remove the qualitative knowledge of τ∗ and hence n, choose β from 2β η = 1.
Notice that such a choice is possible, since by Remark 10.4 the parameter η is
independent of β. This makes γo quantitative. The time level sn is computed
from
n
P
sn = s∗ + θr2
22j .
j=1
Therefore, the range of t for which (10.14)n holds, can be estimated as
s∗ + 21 θ(2n r)2 ≤ t ≤ s∗ + 2θ(2n r)2 .
From the previous choices one estimates
s∗ + γ̄1 ≤ t ≤ s∗ + 4γ̄1
where γ̄1 =
θ
.
2
492
12 PARABOLIC DEGIORGI CLASSES
By choosing η∗ even smaller if necessary, we may ensure that γ̄1 ≥ 1 so that
s∗ + γ̄1 ≥ 0 and hence
γ1 = 3γ̄1
(10.17)
is included in the times for which (10.14)n holds.
From Remark 10.4 it follows that b and η do not depend on η∗ , and hence
the assumption of possibly taking η∗ smaller is justified.
Finally, from the indicated choices of n and β the alternatives (10.13)∗ –
(10.13)n can be rewritten as u(xo , to ) ≤ γ2 γ∗ ρδ(N +2) for γ2 = γ̄γγ̄o∗ .
10.4 The Mean Value Harnack Inequality
The Harnack inequality of Theorem 10.1 can be given an equivalent formulation, which we refer to as the mean value form of the Harnack inequality for
non-negative functions in PDG classes.
Corollary 10.2 Let u ≥ 0 be an element of the parabolic DeGiorgi class
PDG(ET , γ̄, γ∗ , C2 , δ). There exist positive constants c∗ and θ∗ that can be
quantitatively determined a priori only in terms of the parameters N, γ and
independent of u, γ∗ , and δ, such that for every radius ρ < C2−1 , provided that
δ(N +2)
B4ρ (xo ) × (to − θ∗ (4ρ)2 , to + θ∗ (4ρ)2 ) ⊂ ET , either u(xo , to ) ≤ c−1
∗ γ∗ ρ
or
c∗ sup u(·, to − θ∗ ρ2 ) ≤ u(xo , to ) ≤ c−1
inf u(·, to + θ∗ ρ2 ).
∗
Bρ (xo )
Bρ (xo )
(10.18)
Remark 10.5 The terminology is suggested by the mean value property of
harmonic functions. As shown in Chapter 2, Section 5, the latter implies that
the value u(xo ) at one point xo of a non-negative harmonic function u controls
its maximum and minimum in a ball centered at xo .
Proof. Fix (xo , to ) ∈ ET , and assume u(xo , to ) > 0. Seek those values of t < to ,
if any, for which
u(xo , t) = 2c−1
(10.19)
∗ u(xo , to ).
If such a t does not exist
u(xo , t) < 2c−1
∗ u(xo , to )
for all t ∈ (to − (4ρ)2 , to ).
(10.20)
We establish by contradiction that this in turn implies
sup u(·, to − θ∗ ρ2 ) ≤ 2c−1
∗ u(xo , to ).
(10.21)
Bρ (xo )
Indeed, if not, by continuity there exists x∗ ∈ Bρ (xo ) such that
u(x∗ , to − θ∗ ρ2 ) = 2c−1
∗ u(xo , to ).
Apply the Harnack inequality in (10.1) with (xo , to ) replaced by (x∗ , to −θ∗ ρ2 ),
to get
10 The Harnack Inequality
493
u(x∗ , to −θ∗ ρ2 ) ≤ c∗−1 inf u(·, to −θ∗ ρ2 +θ∗ ρ2 ) = c−1
inf u(·, to ). (10.22)
∗
Bρ (x∗ )
Bρ (x∗ )
Now, xo ∈ Kρ (x∗ ) and therefore,
2c∗−1 u(xo , to ) = u(x∗ , to − θ∗ ρ2 ) ≤ c−1
∗ u(xo , to ).
The contradiction establishes (10.21).
10.4.1 There Exists t < to Satisfying (3.1)
Let τ < to be the first time for which (10.19) holds. For such a time
to − τ > θ ∗ ρ 2 .
(10.23)
Indeed, if such an inequality were violated, applying the Harnack inequality
in (10.1) with (xo , to ) replaced by (xo , τ ) would give
−1
2c−1
∗ u(xo , to ) = u(xo , τ ) ≤ c∗ u(xo , to ).
Set
s = to − θ ∗ ρ 2 .
From the definitions, the continuity of u and (10.5), we have
τ < s < to
u(xo , s) ≤ 2c−1
∗ u(xo , to ).
and
We claim that
u(y, s) < 2c−1
∗ u(xo , to )
for all y ∈ Bρ (xo ).
(10.24)
Proceeding by contradiction, let y ∈ Bρ (xo ) be such that
u(y, s) = 2c−1
∗ u(xo , to ).
Apply the Harnack inequality in (10.1) with (xo , to ) replaced by (y, s) to
obtain
u(y, s) ≤ c−1
inf u(·, s + θ∗ ρ2 ).
∗
Bρ (y)
Using the definition of s, since y ∈ Bρ (xo )
2c∗−1 u(xo , to ) = u(y, s) ≤ c−1
inf u(·, to ) ≤ c∗−1 u(xo , to ).
∗
Bρ (y)
The contradiction implies that (10.24) holds true. Summarizing the results of
these alternatives, either (10.21) holds or (10.24) is in force. The proof is now
concluded by using the arbitrariness of ρ and by properly redefining c−1
∗ .
494
12 PARABOLIC DEGIORGI CLASSES
In Moser [187] the Harnack inequality is given a third equivalent statement.
Fix (xo , to ) ∈ ET and ρ > 0 and construct the cylinders
(xo , to ) + Qρ+ = Bρ (xo ) × (to , to + ρ2 ]
2
(xo , to ) + Q−
ρ = Bρ (xo ) × (to − ρ , to ].
Assume that ρ is so small that (xo , to ) + Q±
4ρ ⊂ ET . Fix σ ∈ (0, 1) and inside
(xo , to ) + Qρ± construct the two subcylinders
+
= Bσρ (xo ) × (to + (σρ)2 , to + ρ2 ]
Qσρ
2
2
Q−
σρ = Bσρ (xo ) × (to − ρ , to − (σρ) ].
Assume for simplicity that u is a member of a homogeneous PDG class, and
also that C2 = 0. Then there exists a constant γ(σ) depending only on the
data {N, Co , C1 } and σ, and independent of (xo , to ) and ρ, such that
u.
sup u ≤ γ(σ) inf
+
(10.25)
Qσρ
Q−
σρ
2
The two cylinders Q±
σρ are separated along the time axis by a distance 2(σρ) ,
and the constant γ(σ) → ∞ as σ → 0. However, γ(σ) is stable as σ → 1. Other
+
than the indicated separation between Qσρ
and Q−
σρ , there is great freedom in
choosing these cylinders. For example, one could take σ ≈ 1, keep Q+
σρ fixed,
−
and choose Qσρ
with its top “vertex” at (xo , to ). This would keep it separated
by a distance (σρ)2 from Q+
σρ . By possibly modifying the form of the constant
γ, this would imply
u(xo , to ) ≤ γ inf u(·, to + ρ2 ),
Bρ (xo )
which is precisely (10.1). Likewise, one could take σ ≈ 1, keep Q−
σρ fixed, and
+
choose Qσρ
with its bottom “vertex” at (xo , to ). This would keep it separated
−
by a distance (σρ)2 from Qσρ
. By possibly modifying the form of the constant
γ, this would imply
γ −1 sup u(·, to − ρ2 ) ≤ u(xo , to ),
Bρ (xo )
which is just the left-hand side of (10.18). Hence, (10.25) implies (10.18), but
the opposite is also true (see Section 11c in the Complements).
11 The Harnack Inequality Implies the Hölder
Continuity
The Hölder continuity of a function u in the PDG classes in the form (3.2)
has been established in Theorem 3.1. However, as we have already shown in
12 A Consequence of the Harnack Inequality
495
the elliptic framework in Chapter 10, in the parabolic context the Harnack
inequality can also be used to prove the Hölder continuity [187].
Let µ± and ω(2ρ) be defined as in (3.1), with θ the quantity denoted
with θ∗ in Theorem 10.1. Again, for simplicity we assume that (y, s) = (0, 0).
Applying Theorem 10.1 to the two non-negative functions w+ = µ+ − u and
w− = u − µ− gives either
ess sup w+ = µ+ − ess −inf u ≤ c∗−1 γ∗ ρδ(N +2)
Qρ
Q−
ρ
ess sup w = ess sup u − µ− ≤ c∗−1 γ∗ ρδ(N +2)
−
Q−
ρ
or
(11.1)
−
Qρ
c∗ (µ+ − ess −inf u) ≤ µ+ − ess sup u
Qρ
−
Qρ
c∗ (ess sup u − µ ) ≤ ess inf u − µ− .
−
Q−
ρ
(11.2)
Q−
ρ
If either one of (11.1) holds, then
ω(ρ) ≤ ω(2ρ) ≤ c∗−1 γ∗ ρδ(N +2) .
(11.3)
Otherwise, both inequalities in (11.2) are in force. Adding them gives
c∗ ω(2ρ) + c∗ ω(ρ) ≤ ω(2ρ) − ω(ρ),
whence
1 − c∗
.
(11.4)
1 + c∗
The alternatives (11.3)–(11.4) yield recursive inequalities of the same form as
(7.5), from which the Hölder continuity follows.
ω(ρ) ≤ ηω(2ρ),
where
η=
12 A Consequence of the Harnack Inequality
Consider the functions
Γν (x, t; y, s) =
h
i
|x − y|2
kρν
−
ν exp
2
2
4[(t − s) + ρ ]
[(t − s) + ρ ] 2
pointwise in RN × [t > s]. For ν = N the latter are exact solutions of the heat
equation and one verifies that for ν > N they are sub-solutions.
Given the general quasi-linear structure of (1.1)–(1.2), the functions Γν are
not sub-solutions of these equations in any sense. Moreover, no comparison
principle holds for functions in PDG. Nevertheless, the “fundamental subsolutions” Γν drive, in a sense made precise by Proposition 12.1 below, the
structural behavior of non-negative functions in these homogeneous classes.
The content of Propositions 12.1 is that these functions are, locally,
bounded below by one Γν for some ν > N , and thus they do not decay
in space, faster than these “sub-potentials.”
496
12 PARABOLIC DEGIORGI CLASSES
Proposition 12.1 [187] Let E ⊂ RN and E ′ ⊂ E a convex subdomain, such
that dist(E ′ , ∂E) = d > 0. Let u > 0 be a continuous element of the homogeneous parabolic DeGiorgi class PDG(ET , γ̄, 0, 0, δ). There exists a positive
constant γ > 1, depending only on the data {N, Co , C1 }, such that for all
x, y ∈ E ′ , 0 < d2 ≤ s < t ≤ T we have
u(y, s)
t−s
|x − y|2
ln
≤γ
+ 2 +1 .
(12.1)
u(x, t)
t−s
d
Moreover, if E ≡ RN and 0 < s < t < T , we have
u(y, s)
|x − y|2
t
ln
≤γ
+ ln + 1 .
u(x, t)
t−s
s
(12.2)
Remark 12.1 The continuity of u, although a given fact, is assumed only in
order to give a unique meaning to the pointwise values of u.
Proof. First of all, we point out that the proof of Theorem 10.1 shows that,
once the constant θ∗ has been determined, there is no need to have further
room above, and it is enough to assume that to + θ∗ ρ2 < T . Hence, it suffices
to assume that B4ρ × [to − ρ2 , to + θ∗ ρ2 ) ⊂ ET .
Without loss of generality, we can assume that (y, s) = (0, 0) ∈ E ′ and
that u ∈ PDG(E × (−d2 , T ), γ̄, 0, 0, δ) is positive in E × (−d2 , T − d2 ). The
Harnack inequality can be rephrased, saying that
u(x, t) ≥ c∗ u(0, 0)
for
|x|2 ≤
t
≤ ρ2 .
θ∗
(12.3)
We take (x, t) ∈ E ′ × (0, T − d2 ) and proceed to estimate u(0, 0)/u(x, t) from
above.
We connect (0, 0) and (x, t) with a straight line and let
tj =
j
t,
n
xj =
j
x,
n
j = 0, 1, . . . , n.
By the convexity of E ′ , ∀j we have that xj ∈ E ′ . We have to determine n: we
choose 4ρ = d, so that u > 0 in
|x − xj | < 4ρ,
if n >
−ρ2 < t − tj <
t
< θ∗ ρ2
n
t
. To satisfy (12.3), we need
θ∗ ρ2
|xj+1 − xj |2 =
t
θ∗
|x|2
≤
< ρ2 ⇒ n ≥ |x|2 ,
2
n
nθ∗
t
12 A Consequence of the Harnack Inequality
497
so that
|x|2
t
n ≥ max θ∗
.
;
t θ∗ ρ 2
Under these circumstances, we have
u(xj , tj )
≤ c−1
∗ ,
u(xj+1 , tj+1 )
j = 0, 1, . . . , n − 1.
Multiplying, we have
u(0, 0)
u(x, t)
u(0, 0)
ln
u(x, t)
u(0, 0)
ln
u(x, t)
u(0, 0)
ln
u(x, t)
≤ c−n
∗
1
≤ n ln
c∗
|x|2
t
≤ γ θ∗
+
t
θ∗ ρ2
2
t
|x|
≤γ
+ 2 +1
t
d
for a constant γ > 1, which depends only on the data.
In order to prove (12.2), we still assume y = 0, and we distinguish two
possibilities, as far as t and s are concerned.
If s < t ≤ 4s, choosing s = d2 we obtain
2
u(0, s)
|x|
ln
≤ 4γ
+1
u(x, t)
t−s
and we are done.
Otherwise, we have t > 4s. In such a case, choose n ∈ N, such that
2n+1 s < t ≤ 2n+2 s, and let τ = 2n s. Since 2τ < t ≤ 4τ and t − τ > 2t > t−s
2 ,
working as in the first alternative, we obtain
2
u(0, τ )
|x|2
|x|
ln
≤ 4γ
+ 1 ≤ 8γ
+1 .
(12.4)
u(x, t)
t−τ
t−s
To estimate u(0, s)/u(0, τ ), we introduce
tj = 2j s,
uj = u(0, 2j s),
j = 0, 1, . . . , n.
From (12.1) with d2 = tj−1 we have ln(uj−1 /uj ) ≤ 2γ, so that
ln
t
uo
u(0, s)
= ln
≤ 2γn ≤ 2γ log2 .
u(0, τ )
un
s
Adding this last inequality to (12.4) yields
2
|x|
u(0, s)
t
ln
≤ 8γ
+ log2 + 1
u(x, t)
t−s
s
and we are done.
498
12 PARABOLIC DEGIORGI CLASSES
13 A More Straightforward Proof of the Hölder
Continuity
We conclude the chapter by giving a more streamlined proof of the Hölder
continuity of functions in PDG classes, based on the ideas developed for the
proof of the Harnack inequality. For simplicity, we deal only with homogeneous classes, but just minor adjustments are needed in order to cover the
general case.
−
⊂ ET ,
Take (xo , to ) ∈ ET , assume that ρ < C2−1 is such that (xo , to ) + Q2ρ
and let
µ+ =
u,
sup
µ− =
−
(xo ,to )+Q2ρ
inf
−
(xo ,to )+Q2ρ
ω = µ+ − µ− .
u,
Moreover, let ν be the quantity in (4.6) with a =
alternatives, namely either
1
2
and θ = 1. We have two
1
−
[u > µ+ − ω] ∩ (xo , to ) + Q−
ρ ≤ ν|Qρ |,
2
or
1
−
[u > µ+ − ω] ∩ (xo , to ) + Q−
ρ > ν|Qρ |.
2
If the first alternative holds, by Proposition 4.1 we have
1
u ≤ µ+ − ω
2
a.e. in (xo , to ) + Q−
1
ρ
2
and also
ess osc
(xo ,to )+Q−
1ρ
2
u=
(xo ,to )+Q−
1
2
=
u−
ess sup
ess inf
(xo ,to )+Q−
1ρ
2
ρ
1
u ≤ µ+ − ω − µ−
2
1
ess osc u.
2 (xo ,to )+Q−
2ρ
If the second alternative is satisfied, there exists s ∈ [−ρ2 , − ν2 ρ2 ] such that
1
ν
[u(·, s) > µ+ − ω] ∩ Bρ (xo ) > |Bρ (xo )|.
2
2
Indeed, if not, we would have
1
[u > µ+ − ω] ∩ (xo , to ) + Q−
ρ =
2
Z
− ν2 ρ2
−ρ2
+
Z
1
[u(·, τ ) > µ+ − ω] ∩ Bρ (xo ) dτ
2
0
− ν2 ρ2
≤ν||Q−
ρ |.
1
[u(·, τ ) > µ+ − ω] ∩ Bρ (xo ) dτ
2
2c Local Boundedness of Functions in the PDG Classes
Since
499
1
1
µ+ − ω = µ− + ω,
2
2
we have
ν
1
[u(·, s) > µ− + ω] ∩ Bρ (xo ) > |Bρ (xo )|.
2
2
If we set w = u − µ− , we have (10.5) with M = 12 ω. By a (possibly repeated)
application of Proposition 10.1, we conclude that there exists a η̄ ∈ (0, 1),
which depends only on the data {N, Co , C1 }, such that
u ≥ µ− + η̄ω
a.e. in (xo , to ) + Q−
1 ,
ρ
u=
u−
2
and also
ess osc
(xo ,to )+Q−
1
2
ρ
ess sup
(xo ,to )+Q−
1
2
= (1 − η̄)
ρ
ess inf
(xo ,to )+Q−
1
2
ρ
u ≤ µ+ − µ− − η̄ω
ess osc u.
(xo ,to )+Q−
2ρ
Combining the two alternatives, we conclude that
1
1
; (1 − η̄) ω(2ρ)
ω( ρ) ≤ max
2
2
and from here on, we conclude as in Section 7.
Problems and Complements
2c Local Boundedness of Functions in the PDG Classes
The results of Theorem 2.1 can be slightly extended and at the same time
generalized to cover a wider situation. We limit ourselves to a qualitative
statement, even though it could be phrased in a quantitative fashion, with a
rather limited extra effort.
For N ≥ 2 consider the quasi-linear parabolic equation
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u)
where the functions
weakly in ET
(2.1c)
500
12 PARABOLIC DEGIORGI CLASSES
ET ∋ (x, t) →
A x, t, u(x, t), ∇u(x, t) ∈ RN
B(x, t, u(x, t), ∇u(x, t) ∈ R
are measurable and satisfy the structure conditions
A x, t, u, ∇u · ∇u ≥ Co |∇u|2 − fo
|A x, t, u, ∇u | ≤ C1 |∇u| + f1
|B x, t, u, ∇u | ≤ C2 |∇u| + f2
(2.2c)
for given constants 0 < Co ≤ C1 and C2 > 0, and given non-negative functions
2
f1 ∈ Lloc
(ET ),
fo ∈ Lµloc (ET ) with µ > 1,
s
f1 , f2 ∈ Lloc
(ET ), s >
2(N + 2)
.
N +4
(2.3c)
Theorem 2.1c. Let u be a local, weak solution of (2.1c). Under conditions
(2.2c)–(2.3c) we have
and µ > N2+2 , then u ∈ L∞
loc (ET ).
N +2
and µ = 2 , then u ∈ Lqloc (ET ) for all q < +∞.
and µ < N2+2 , then u ∈ Lqloc (ET ) for all q < q∗ with




N +2
2N +4
2
N
;
q∗ = min
.
2
1
1 − 1 − s 1 + N 1 − 1 − 1 1 + 2 
µ
N
• If both s >
• If both s =
• If both s <
N +2
2
N +2
2
N +2
2
The result was proved in [194] for a larger class of quasi-linear parabolic
equations. Conditions (2.3c) are rather natural, as the same conditions were
imposed in Ladyzhenskaya et al. [151], Chapter 3, Sections 8–9, in order to
have the corresponding properties for the solutions of a large class of linear
equations.
The assumption f1 ∈ L2loc (ET ) may seem out of place, but its purpose is to
guarantee that the terms of the form A x, t, u, ∇u · ∇u are locally integrable.
3c Hölder Continuity of Solutions of Linear Parabolic
Equations with Bounded and Measurable Coefficients
As discussed in Section 15 of Chapter 11, the local Hölder continuity of locally bounded solutions of linear parabolic equations in divergence form with
bounded and measurable coefficients was first proved in 1958 by Nash [190].
Since his proof relies in a fundamental way on deep estimates of the fundamental solution (again, see Section 15 mentioned above), it has an intrinsic
linear feature, and it does not seem possible to extend it to the general quasilinear setting considered in this Chapter.
6c Propagating in Time the Measure-Theoretical Information
501
6c Propagating in Time the Measure-Theoretical
Information
We give a slightly stronger version of Proposition 6.1, where the measuretheoretical information is spread not just for a short time, but up to the top
of the original cylinder. It is taken from Liao [171].
Proposition 6.1c Assume that
h
ωi
α
u(·, s) < µ+ −
∩ Bρ (y) ≥ |Bρ |.
4
2
(6.1c)+
for some α ∈ (0, 1]. Then, there exist ξ ∈ (0, 1) and Cξ > 1, which depend
only on the data, such that either ω ≤ γ∗ Cξ ρδ(N +2) or
α
u(·, t) < µ+ − ξω ∩ Bρ (y) ≥ |Bρ |
4
(6.2c)+
for all t ∈ [s, s + ρ2 ]. Analogously, if
h
α
ωi
∩ Bρ (y) ≥ |Bρ |,
u(·, s) > µ− +
4
2
(6.1c)−
for some α ∈ (0, 1], then there exist ξ ∈ (0, 1) and Cξ > 1, which depend only
on the data, such that either ω ≤ γ∗ Cξ ρδ(N +2) or
α
u(·, t) > µ− + ξω ∩ Bρ (y) ≥ |Bρ |
4
(6.2c)−
for all t ∈ [s, s + ρ2 ].
Remark 6.1c An analogous statement holds, if we consider cylinders (y, s)+
Q−
ρ (θ).
6.1c Proof of Proposition 6.1c
Proof. We will establish (6.2c)− starting from (6.1c)− . As usual, without loss
of generality, we may assume (y, s) = (0, 0).
For simplicity, we set M = 14 ω. We start from (1.7) written over Q+
ρ (θ)
for (u − k)− , where θ is to be determined and k = µ− + M , we discard the
third term on the left-hand side.
Moreover, we consider a non-negative, piecewise smooth, test function ζ =
ζ(x) such that
ζ = 1 in Bσρ ,
Hence, we conclude that
ζ = 0 in RN − Bρ ,
|∇ζ| ≤
1
.
σρ
502
12 PARABOLIC DEGIORGI CLASSES
ess sup
0<t<θρ2
≤
Z
Bρ
2 2
(u − k)−
ζ (x, t)dx
Bρ
2 2
(u − k)−
ζ (x, 0)dx
Z
ZZ
γ
1− N2+2 +2δ
(u − k)2− dx dτ + γ∗2 |A−
k,ρ |
2
2
+
σ ρ
Qρ (θ)
|[u < k] ∩ Q+
α
θ
ρ (θ)|
|Bρ | + γ 2 M 2
≤M 2 1 −
|Bρ |
+
2
σ
|Qρ (θ)|
2
ρ2δ(N +2)
+ γθ1− N +2 +2δ γ∗2
M 2 |Bρ |.
M2
+
If we stipulate to take
γ∗2
we have
ess sup
0<t<θρ2
Z
Bρ
ρ2δ(N +2)
< 1,
M2
(u − k)2− ζ 2 (x, t)dx
α
θ |[u < k] ∩ Q+
ρ (θ)|
1− N2+2 +2δ
1−
≤M
+γ 2
+ γθ
|Bρ |.
2
σ
|Q+
ρ (θ)|
2
Let ℓ = µ− + ǫM with ǫ ∈ (0, 1) to be chosen. Then,
Z
Z
(u − k)2− ζ 2 (x, t)dx ≥
(u − k)2− (x, t)dx
B(1−σ)ρ ∩[u≤ℓ]
Bρ
≥ (1 − ǫ)2 M 2 |[u ≤ ℓ] ∩ B(1−σ)ρ |.
Moreover,
|Aℓ,ρ (t)| = |Aℓ,(1−σ)ρ (t) ∪ (Aℓ,ρ (t)\Aℓ,(1−σ)ρ (t))|
≤ |Aℓ,(1−σ)ρ (t)| + |Bρ \B(1−σ)ρ |
≤ |Aℓ,(1−σ)ρ (t)| + N σ|Bρ |.
Therefore, we conclude
2
2
"
θ |[u < k] ∩ Q+
α
ρ (θ)|
+γ 2
1−
+
2
σ
|Qρ (θ)|
i
2
+ γθ1− N +2 +2δ + N (1 − ǫ)2 σ |Bρ |.
(1 − ǫ) M |Aℓ,ρ (t)| ≤M
2
Since we can take ǫ < 21 without loss of generality, we also have
#
"
1 − α2
θ |[u < k] ∩ Q+
ρ (θ)|
1− N2+2 +2δ
+ N σ |Bρ |.
|Aℓ,ρ (t)| ≤
+γ 2
+ γθ
(1 − ǫ)2
σ
|Q+
ρ (θ)|
6c Propagating in Time the Measure-Theoretical Information
Choosing
σ=θ
1
3
|[u < k] ∩ Q+
ρ (θ)|
|Q+
ρ (θ)|
13
503
,
taking into account that θ ∈ (0, 1), and relabelling 2γ + N with γ yields
"
1 − α2
1
+ γθ 3
|Aℓ,ρ (t)| ≤
(1 − ǫ)2
|[u < k] ∩ Q+
ρ (θ)|
|Q+
ρ (θ)|
13 #
|Bρ |.
Finally, we choose
α
,
16
α
1− 2
1 − α4
α
<
≤1−
2
2
(1 − ǫ)
(1 − ǫ)
8
1
γθ 3 =
and conclude
⇒
ǫ≈
3
α,
16
α
|Bρ |
|Aℓ,ρ (t)| ≤ 1 −
16
for all t ∈ [0, θρ2 ], with θ computed above. We now use this estimate and we
refine it.
Let
M1
M1
M1 = ǫM, k1 = µ− + j1 , ℓ1 = µ− + j1 +n1 ,
2
2
where j1 and n1 are to be chosen. If we repeat the previous computations
with the new quantities, we have
"
1 − α2
θ |[u < k1 ] ∩ Q+
ρ (θ)|
+
γ
|Aℓ1 ,ρ (t)| ≤
+
(1 − 2−n1 )2
σ2
|Qρ (θ)|
#
2
2j1 ρδ(N +2)
1− N2+2 +2δ
+ γθ
γ∗
+ N σ |Bρ |.
M1
We now rely on Proposition 5.2 and we assume that
γ∗
1
2j1 ρδ(N +2)
≤ 1 ;
M1
j112
since in (5.3) we have θ∗ = θ because θ ∈ (0, 1), with the same choice for σ as
above, we obtain
"
31
1 − α2
1
1
3
√
+
γθ
|Aℓ1 ,ρ (t)| ≤
(1 − 2−n1 )2
α θj1
31 #
13
1
1
1− N2+2 +2δ
√
√
+ γθ
|Bρ |
+N
j1
α θj1
504
12 PARABOLIC DEGIORGI CLASSES
"
1 − α2
1
≤
+ γθ 3
(1 − 2−n1 )2
1
√
α θj1
31 #
|Bρ |,
where in the last inequality we have also relabeled γ as before. We choose j1
and n1 such that
31
1
α
√
γθ
≤θ
8
α θj1
α
1− 2
α
α
≤1− +θ .
(1 − 2−n1 )2
2
8
1
3
As a result
def
α
α
|Aℓ1 ,ρ (t)| ≤ 1 − + θ
|Bρ |
2
4
for all t ∈ [so , s1 ] = [0, θρ2 ].
We now proceed by induction. Either at every step m = 1, . . . , i − 1
γ∗
1
2jm ρδ(N +2)
≤ 1 ,
Mm
12
jm
and in such a case the sequences {Mi }, {ni } and {ji } have been chosen, and
α
α
|Bρ | ∀ t ∈ [si−1 , si ]
|Aℓi−1 ,ρ (t)| ≤ 1 − + (i − 1)θ
2
4
where
Mi−1 def −
= µ + M̂i−1 .
2ni−1 +ji−1
Otherwise, the process stops and ξ and Cξ are chosen accordingly. Suppose
we have reached step i − 1. If we now set
ℓi−1 = µ− +
ℓǫi−1 = µ− + ǫM̂i−1 ,
si+1 = si + θρ2 ,
Q+
i = Bρ × (si , si+1 ],
using the measure-theoretical information at times si , we can work exactly as
above and conclude that
"
1 #
1 − α2 + (i − 1)θ α4
|[u < k] ∩ Q+
| 3
1
i
|Bρ |.
|Aℓǫi−1 ,ρ (t)| ≤
+ γθ 3
(1 − ǫ)2
|Q+
i |
If we assume (i − 1)θ < 1, we may choose ǫ and θ as above, and conclude that
α
|Aℓǫi−1 ,ρ (t)| ≤ 1 −
|Bρ |
∀ t ∈ [si , si+1 ].
16
Now we let
Mi = ǫM̂i−1 ,
k1 = µ− +
Mi
,
2ji
ℓ i = µ− +
Mi
,
2ji +ni
7c Proof of Theorem 3.1
505
where ji and ni are still to be chosen. Using Proposition 5.2 as above and
stipulating that
2ji ρδ(N +2)
1
γ∗
≤ 1 ,
Mi
j 12
i
yields
"
1 − α2 + (i − 1)θ α4
1
+ γθ 3
|Aℓi ,ρ (t) ≤
(1 − 2−ni )2
1
√
α θji
31 #
|Bρ |
∀ t ∈ [si , si+1 ].
Finally, we choose ji and ni such that
31
1
α
√
γθ
≤θ
8
α θji
1 − α2 + (i − 1)θ α4
α
α
α
≤ 1 − + (i − 1)θ + θ .
−n
2
i
(1 − 2 )
2
4
8
1
3
As a result
α
α
|Bρ |
|Aℓi ,ρ (t)| ≤ 1 − + iθ
2
4
for all t ∈ [si , si+1 ]. The induction finishes when
either
γ∗
2ji ρδ(N +2)
1
≥ 1 ,
Mi
ji12
or
iθ ≥ 1.
Hence, we have that either
α
u(·, t) > µ− + ξω ∩ Bρ (y) ≥ |Bρ |
4
for all t ∈ [s, s + ρ2 ], with ξω =
Mi
2ni +ji
, or
ω ≤ γ∗ Cξ ρδ(N +2) ,
where Cξ > 1 takes into account at which step m, if any, the stipulation
2ji ρδ(N +2)
1
γ∗
≤ 1 is violated.
Mi
ji12
Remark 6.2c The proof shows that we can take a general θρ2 for the height
of the cylinder, with any θ > 0 and not just θ = 1.
7c Proof of Theorem 3.1
Relying on Proposition 6.1c, the proof of Theorem 3.1 is revised accordingly.
In particular, it is concluded in one stroke. The first part is as in Section 7.
506
12 PARABOLIC DEGIORGI CLASSES
Without loss of generality, we may assume (y, s) = (0, 0). We consider Qρ− ,
that is, we take θ = 1. By the definitions (3.1), either
|[u ≤ µ+ − 21 ω] ∩ Q−
ρ|≥
1 −
|Q |
2 ρ
(7.1c)
or
1 −
|Q |.
(7.2c)
2 ρ
Assume that the second is in force; if (7.1c) holds, the argument runs exactly
in the same way. We can equivalently rewrite (7.2c) as
|[u ≥ µ− + 21 ω] ∩ Q−
ρ|≥
|[u < µ− + 21 ω] ∩ Q−
ρ|<
1 −
|Q |.
2 ρ
We claim that this implies that there exists to ∈ [−ρ2 , − 31 ρ2 ] such that
|[u(·, to ) < µ− + 21 ω] ∩ Bρ | ≤
3
|Bρ |.
4
Indeed, if not, then
−
|[u < µ +
1
2 ω]
∩
Q−
ρ|
≥
Z
− 31 ρ2
−ρ2
|[u(·, τ ) < µ− + 21 ω] ∩ Bρ |dτ >
1 −
|Q |.
2 ρ
We can equivalently conclude that there exists to ∈ [−ρ2 , 31 ρ2 ] such that
|[u(·, to ) ≥ µ− + 21 ω] ∩ Bρ | ≥
1
|Bρ |.
4
Without loss of generality, we can assume that to = −ρ2 , so that
|[u(·, −ρ2 ) ≥ µ− + 41 ω] ∩ Bρ | ≥
1
|Bρ |.
4
We can then apply Proposition 6.1c with α = 21 , and conclude that either
ω ≤ γ∗ Cξ ρδ(N +2)
or
|[u(·, t) ≥ µ− + ξω] ∩ Bρ | ≥
(7.3c)
1
|Bρ |
8
(7.4c)
for all t ∈ [−ρ2 , 0].
If (7.3c) holds true, we have finished. Otherwise, assuming that (7.4c) is
in force, fix the number ν as the one claimed by Proposition 4.1 in (4.6), for
the choices a = 12 , θ = θ̄ = 1.
We point out that (7.4c) is (5.1)− of Proposition 5.1 with α = 81 and ǫ = ξ;
therefore, ν being fixed, determine j∗ and hence ǫν = 2ξj∗ by the procedure of
Proposition 5.1.
11c The Harnack Inequality
507
Then, by Proposition 4.1, either ǫν ω ≤ γρδ(N +2) , or (4.2)+ holds. The
latter implies
− ess−inf u ≤ − ess−inf − 21 ǫν ess −osc u.
Q1
2
Q2ρ
Q2ρ
ρ
Now,
ess sup u ≤ ess sup u.
Q−
1
2
ρ
Q−
2ρ
Adding these inequalities gives
ω( 12 ρ) ≤ ηω(2ρ),
1
where η = 1 − ǫν .
2
−n
Let Q−
R. The previous remarks imply that
R ⊂ ET be fixed and set ρn = 4
+2)
ω(ρn+1 ) ≤ max{ηω(ρn ) ; C̄γ∗ ρδ(N
}
n
(7.5c)
for a proper choice of C̄ that takes into account all the alternatives, and by
iteration
+2)
ω(ρn+1 ) ≤ max{η n ω(R) ; C̄γ∗ ρδ(N
}.
n
Compute
ρn = 4−n R =⇒ −n = ln
ρ ln14
n
R
=⇒ η n =
ρ α
n
R
for α = −
ln η
.
ln 4
Remark 7.1c We have given the proof assuming θ = 1, but a general θ > 0
is also possible, without any substantial change in the previous arguments.
11c The Harnack Inequality
The first to prove a Harnack inequality for linear parabolic equations in divergence form with bounded and measurable coefficient was Jürgen Moser [187],
expanding upon ideas he had previously developed for elliptic equations [186].
The proof of Lemma 4 in Moser [187] contained a faulty argument, which was
later corrected [188]. An easier proof was given later [189].
As we have shown in Section 11, the Harnack inequality can be used to
prove that weak solutions are locally Hölder continuous in ET . Even though
Moser’s method is quite different from the one we used here in Section 10,
nevertheless, the linearity assumed in (1.1) is immaterial to the proof, and
one might then expect, as in the elliptic case, an extension of these results to
quasi-linear equations of the type
ut − div A(x, t, u, ∇u) = B(x, t, u, ∇u)
in ET ,
where the structure conditions are as in (1.2) of Chapter 10, that is, with a
growth of order p for any p > 1. Surprisingly, however, Moser’s proof could
508
12 PARABOLIC DEGIORGI CLASSES
be extended only for the case p = 2, i.e., for equations whose principal part
has a linear growth with respect to ∇u. This appears in the work of Aronson
and Serrin [12] and Trudinger [262].
Also, the approach based on parabolic DeGiorgi classes cannot be simply extended, and the problem remained open for quite a number of years,
until a partial solution was first given in the late 1990s by the first author
of this monograph. For a very interesting historical perspective, we refer to
the survey paper by Kassmann [134], whereas the interested reader can look
at DiBenedetto et al. [55] for a proof of the Harnack inequality for general
operators with growth of order p > 2.
11.1. Prove that (10.18) implies (10.25).
13
PARABOLIC EQUATIONS IN
NONDIVERGENCE FORM
1 Introductory Material
The aim of this chapter is to present some of the known estimates for solutions
of parabolic PDEs in nondivergence form with only bounded and measurable
coefficients.
The time derivative of a function u will be equivalently denoted with ut ,
∂t u, and ∂u
∂t , whereas for the space derivatives of u we equivalently write Du,
∂u
uxi , ∂x
.
i
1.1 Introduction
Let E be a bounded open set in RN , N ≥ 1 with smooth boundary ∂E and
for 0 < T < ∞ let ET ≡ E × (0, T ]. The following types of equations are
considered in this chapter.
1.1.1 Linear Equations
ut − Lo (u) = 0 in ET ,
Lo = aij (x, t)uxi xj ,
(1.1)
where the summation convention is adopted. The basic assumptions on the
coefficients are:
The functions (x, t) → aij (x, t), i, j = 1, . . . , N are only bounded and
measurable, defined in ET .
The matrix (aij (x, t)) is symmetric and positive definite uniformly in ET .
Equivalently, if λ(x, t) ≤ Λ(x, t) are respectively the minimum and the
maximum eigenvalues of aij (x, t) as (x, t) ∈ ET , there exists λo ≤ Λo such
that
0 < λo ≤ λ(x, t) ≤ Λ(x, t) ≤ Λo ∀ (x, t) ∈ ET .
(1.2)(i)
In turn, this can be formulated as
λo |ξ|2 ≤ aij (x, t)ξi ξj ≤ λo |ξ|2 ,
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_14
∀ξ ∈ RN .
(1.2)(ii)
509
510
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
We assume that u ∈ C 2,1 (ET ) is a solution of (1.1), where the space C 2,1 (ET )
has been defined in Section 1 of Chapter 11.
Remark 1.1 There is no loss of generality in assuming (aij (x, t)) symmetric,
since Lo can always be transformed into a new operator Lo∗ with symmetric
coefficients
1
∗
aij
(x, t) = (aij (x, t) + aji (x, t)).
2
This follows from the fact that x → u(x, t) ∈ C 2 (E), and therefore, uxi xj =
uxj xi .
Remark 1.2 Condition (1.2)(ii) implies that
λo ≤ aij (x, t) ≤ Λo
∀ (x, t) ∈ ET , i, j = 1, . . . , N.
The equivalence of (1.2)(i) and (1.2)(ii) can be proved using the following
result, which we also state here for future reference.
Lemma 1.1 If (aij (x, t)) is a N × N symmetric matrix, then there exists a
unitary matrix X = X(x, t) that diagonalizes (aij (x, t)), i.e.,


λ1 (x, t)
0
0 ...
0
 0
λ2 (x, t) 0 . . .
0 


X−1 (x, t)(aij (x, t))X(x, t) =  .
,
..
.
.
.
.. . .
..

 ..
.
0
0
. . . 0 λN (x, t)
where λ1 (x, t), . . . , λN (x, t) are the eigenvalues of (aij (x, t)), and
X−1 = Xt ,
N
X
i=1
x2ij = 1,
(Xt being the transpose of X)
∀ j = 1, 2, . . . , N,
N
X
x2ij = 1,
j=1
∀ i = 1, 2, . . . , N.
For the proof, see Section 1.1.1c in the Complements.
For future reference, we also note the following.
Remark 1.3 Let (D2 u) = (uxi xj ) be the N × N symmetric Hessian matrix
of the second-order space derivatives of u ∈ C 2,1 (ET ). Let A be the matrix
(aij (x, t)). Then,
aij (x, t)uxi xj = tr(A · (D2 u)).
Nonhomogeneous variations of (1.1) are
ut − aij (x, t)uxi xj − b(x, t, u, Du) = 0
in ET ,
(1.3)
where Du denotes the gradient of u with respect to the space variables only,
and b is a given measurable function from ET × R × RN into R, which will be
assumed to satisfy
1 Introductory Material
511
|b(x, t, u, Du)| ≤ ϕo (x, t) + ϕ1 (x, t)|u|σ + ϕ2 (x, t)|Du|θ .
Here, (x, t) → ϕi (x, t), i = 0, 1, 2 are given, non-negative, measurable functions defined in ET , and σ, θ are given positive numbers.
The precise regularity of the ϕi and the order of growth of b with respect
to u and |Du|, i.e., the numbers σ and θ, will be specified later, depending on
the estimates we will be seeking.
The operator in (1.3) will be denoted by ut − L(u), where L(u) = Lo (u) +
b(x, t, u, Du). Thus, Lo (u) in (1.1) is the principal part of L(u).
1.1.2 Quasi-linear Equations
We consider
ut − Qo (u) = 0 in ET ,
Qo = Aij (x, t, u, Du)uxi xj .
(1.4)
Here, Aij , i, j = 1, 2, . . . , N are given measurable functions from ET × R × RN
into R.
As before, we assume that u ∈ C 2,1 (ET ) is a solution of (1.4), and with
an abuse of notation, we denote by (x, t, u, Du) points in ET × R × RN .
Moreover, for the N × N matrix (Aij (x, t, u, Du)), we assume that it is
symmetric, positive definite uniformly in ET ×R×RN and there exist numbers
0 < λo ≤ Λo such that, for some given α ∈ R
λo (1 + |Du|α )|ξ|2 ≤ Aij (x, t, u, Du)ξi ξj ≤ Λo (1 + |Du|α )|ξ|2 .
Obviously, Lo (u) is a special case of Qo (u). A nonhomogeneous variation of
(1.4) is
ut − Aij (x, t, u, Du)uxi xj − B(x, t, u, Du) = 0.
We set
Q(u) = Aij (x, t, u, Du)uxi xj + B(x, t, u, Du),
and observe that Qo (u) is the principal part of Q(u).
No smoothness is assumed on the functions Aij , i, j = 1, 2, . . . , N , and on
B : ET × R × RN → R the basic assumption is
ZZ
|B(x, t, u, Du)|N +1
dxdt ≤ C
ET det[Aij (x, t, u, Du)]
for a given constant C.
1.1.3 Fully Nonlinear Equations
Let RN ×N denote the space of all real, symmetric, N × N matrices. Such a
space has dimension N (N2+1) .
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
512
Let K be the set ET × R × RN × RN ×N and let F be a real-valued function
from K into R. Consider the evolution equation
ut − F (x, t, u, Du, D2 u) = 0
in ET .
(1.5)
We assume that u ∈ C 2,1 (ET ) is a solution of (1.5) in ET , and with an abuse
of notation we denote with (x, t, u, Du, D2 u) points in K. We also assume
that D2 u → F (x, t, u, Du, D2 u) is a.e. differentiable in RN ×N .
Equation (1.5) is parabolic in K in the sense that the matrix given by
(Fuxi xj (x, t, u, Du, D2 u)) is non-negative definite in K, i.e., ∀ ξ ∈ RN
λ(x, t, u, Du, D2 u)|ξ|2 ≤ Fuxi xj ξi ξj ≤ Λ(x, t, u, Du, D2 u)|ξ|2
(1.6)
for measurable functions λ, Λ : K → R satisfying 0 ≤ λ ≤ Λ < ∞ in K.
If there exist 0 < λo ≤ Λo < ∞ such that
0 < λo ≤ λ(x, t, u, Du, D2 u) ≤ Λ(x, t, u, Du, D2 u) ≤ Λo < ∞
(1.7)
uniformly in K, then (1.5) is uniformly parabolic in K.
The notion of parabolicity or uniform parabolicity is local in K and could
be formulated in terms of subsets K ′ of K. If (1.6) (or (1.7) respectively)
hold for all points in K ′ ⊂ K, we say that (1.5) is parabolic (or, uniformly
parabolic respectively) in K ′ .
Given the a.e. differentiability of F in RN ×N , equation (1.5) could be
written in a way that resembles the quasi-linear equations ut − Q(u) = 0,
except that now the coefficients Aij also depend on D2 u. Indeed,
Z
1
d
F (x, t, u, Du, D u) − F (x, t, u, Du, 0) =
F (x, t, u, Du, sD2 u) ds
ds
0
Z 1
2
=
Fuxi xj (x, t, u, Du, sD u) ds uxi xj .
2
0
Hence, we have
ut − F (x, t, u, Du, D2 u)
= ut − Aij (x, t, u, Du, D2 u)uxi xj − B(x, t, u, Du),
where
2
Aij (x, t, u, Du, D u) =
Z
0
1
Fuxi xj (x, t, u, Du, sD2 u) ds,
B(x, t, u, Du) = F (x, t, u, Du, 0).
Next, we discuss in some detail the fully nonlinear notion, by looking at specific
examples.
1 Introductory Material
513
1.2 The Pucci Equation
Let α ∈ (0, N1 ) and denote by Lα the class of all linear elliptic operators of
the type
N
X
∂2
aij (x)
,
L=
∂xi ∂xj
i,j=1
where
(
α|ξ|2 ≤ aij (x)ξi ξj ,
tr(aij (x)) = 1.
∀ ξ ∈ RN
(1.8)
For u ∈ C 2,1 (ET ) consider the quantities
M (u(x, t)) = sup Lu,
L∈Lα
m(u(x, t)) = inf Lu,
L∈Lα
and the associated parabolic equations
(
ut − M (u(x, t)) = 0
ut − m(u(x, t)) = 0
in ET ,
in ET .
These are called extremal operators, and are of the form (1.5) with F a.e.
differentiable in K. This will follow from the pointwise representation of M (u)
and m(u) in terms of the eigenvalues of the Hessian matrix (D2 u). In fact, we
have the following result.
Lemma 1.2 For any u ∈ C 2,1 (ET ), we have
M (u(x, t)) = α∆u + (1 − N α)CN (u),
m(u(x, t)) = α∆u + (1 − N α)C1 (u),
(1.9)
(1.10)
where C1 (u) and CN (u) are respectively the smallest and the largest eigenvalues of (D2 u).
Proof. We prove (1.9), since the proof for (1.10) is analogous. Let u ∈ C 2 (E)
and let X be the unitary N ×N matrix that diagonalizes the symmetric matrix
(D2 u), i.e.,


C1 (u) . . .
0


X−1 (D2 u)X =  0 . . .
0 ,
0
and
X−1 = Xt ,
N
X
j=1
. . . CN (u)
x2ij = 1, i = 1, 2, . . . , N.
514
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Let L ∈ Lα and let (aij (x)) be the associated matrix of the coefficients. Then,
tr[(aij (x)) · (uxi xj )] = aij (x)uxi xj
= sum of the eigenvalues of [(aij (x)) · (uxi xj )]
= sum of the eigenvalues of [X−1 · (aij (x)) · (uxi xj ) · X]
= tr[X−1 · (aij (x)) · (uxi xj ) · X].
Therefore,
aij (x)uxi xj = tr{[X−1 · (aij (x)) · X] · [X−1 · (D2 u) · X]}



C1 (u) . . .
0






−1
.
.
= tr [X · (aij (x)) · X] ·  0

.
0




0 . . . CN (u)
=
N
X
dii Ci (u),
i=1
where dii are the diagonal elements of [X−1 ·(aij (x))·X], and have the following
properties
(a)
N
X
(b) dii ≥ α, i = 1, 2, . . . , N.
dii = 1,
i=1
Property (a) follows from
N
X
i=1
dii = tr[X−1 · (aij (x)) · X] = tr(aij (x)) = 1.
Property (b) follows from (1.8). Indeed,
dii =
N
X
l,k=1
where we have used that
alk xki xli ≥ α
PN
k=1
N
X
k=1
x2ki
!
= α,
x2ki = 1. Combining these remarks, we have
(Lu)(x) = (aij uxi xj )(x)
=
N
X
dii (x)Ci (u)
i=1
=α
N
X
Ci (u) +
i=1
i=1
= α∆u +
N
X
N
X
i=1
(dii − α)Ci (u)
(dii − α)Ci (u),
1 Introductory Material
where we have taken into account that
dii − α > 0 ∀ i = 1, 2, . . . , N , we have
(Lu)(x) ≤ α∆u +
N
X
i=1
N
X
515
Ci (u) = tr(D2 u) = ∆u. Since
i=1
(dii − α)CN (u) = α∆u + (1 − N α)CN (u).
(1.11)
Inequality (1.11) holds for every L ∈ Lα , and therefore,
M [u(x)] ≤ α∆u + (1 − N α)CN (u) ∀ u ∈ C 2 (E).
Now we show that the operator on the right-hand side of (1.11) belongs to Lα .
N
X
ãii (x) = 1
In order to prove this, we have to find a matrix (ãij (x)) such that
i=1
for all x ∈ E, and the eigenvalues of (ãij ) are larger than α. Let u ∈ C 2 (E)
be fixed, construct the matrix x → X(x) that diagonalizes (D2 u), i.e.,


0
C1 (u) . . .
 −1

(D2 u) = X  0 . . .
0 X
and let

α
0

(ãij (x)) = X(x) 

0
0
. . . CN (u)
...
α
..
.
0
...



 X−1 (x).

. . . 1 − (N − 1)α
Since x → X(x) is unitary, we have
tr(ãij (x)) = 1,
and
ãij (x)ξi ξj ≥ α|ξ|2 , ∀ ξ ∈ RN .
Therefore, the operator (L̃v)(x) = tr[(ãij (x))(D2 v)] belongs to Lα , v ∈ C 2 (E).
For v = u we have
M [u(x)] ≤ α∆u + (1 − N α)CN (u)




α ...
0
0
C1 (u) . . .
0 α

...



= .
  0 ...
0 
 ..

0 . . . CN (u)
0 . . . 1 − (N − 1)α




0
α ...
0
 C1 (u) . . .
0 α
.
.
.



= tr  .
  0 ...

0
.

 .
0 . . . CN (u)
0 . . . 1 − (N − 1)α
516
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM


α



0


= tr X(x) 





0






 −1
 X (x)(D2 u)





. . . 1 − (N − 1)α
...
α
...
= tr(ãij (x))(D2 u) =
0
...
N
X
i,j=1

ãij (x)uxi xj = (L̃u)(x) ≤ M [u(x)],
whence (1.9) follows.
1.3 The Bellman–Dirichlet Equation
Let A be an arbitrary set of indices, and consider the family F of linear elliptic
operators
Lν [u] = aνij (x, t)uxi xj + bνi (x, t)uxi + cν (x, t)u,
ν ∈ A,
where (x, t) → aνij (x, t), bνi (x, t), cν (x, t) are real-valued, measurable functions
defined in E, i, j = 1, 2, . . . , N , ν ∈ A.
Let u ∈ C 2,1 (ET ) and (x, t) → f ν (x, t), ν ∈ A be a family of real-valued,
measurable functions defined in ET . Consider the quantity
F [u](x, t) = inf [Lν [u](x, t) + f ν (x, t)],
ν∈A
and the associated parabolic equation
ut − F [u] = 0
in ET .
(1.12)
Equation (1.12) is a parabolic version of the Bellman–Dirichlet equation
F [u] = 0.
The assumptions on the matrices (aνij (x, t)) are
λ(x, t)|ξ|2 ≤ aνij (x, t)ξi ξj ≤ Λ(x, t)|ξ|2 ,
∀ ξ ∈ RN ,
(1.13)
where (x, t) → λ(x, t), Λ(x, t) are given non-negative functions in ET , such
that 0 ≤ λ(x, t) ≤ Λ(x, t) for any (x, t) ∈ ET .
Besides the presence of the lower order terms in the definition of Lν [u],
ν ∈ A, the difference between the Bellman–Dirichlet equations and the Pucci
equation is that no uniform lower bound on the eigenvalues of (aνij (x, t)),
ν ∈ A, nor conditions on the trace (see (1.8)) are imposed.
For each ν ∈ A, the function y ν : RN ×N → R given by
y ν (D2 u) = aνij uxi xj + bνi uxi + cν u + f ν
is affine in RN ×N , and therefore, the function G defined by
G(D2 u) = inf y ν (D2 u)
ν∈A
1 Introductory Material
517
is a.e. differentiable in RN ×N .
Let (η) be a N × N , nontrivial, positive semi-definite matrix in RN ×N .
Then,
G(D2 u + η) − G(D2 u) = inf y ν (D2 u + η) − inf y ν (D2 u)
ν∈A
ν∈A
≥ inf (aνij (x, t)(uxi xj + ηij ) − aνij (x, t)uxi xj )
ν∈A
ν
(x, t)ηij = inf tr[(aνij (x, t)) · (η)].
≥ inf aij
ν∈A
ν∈A
Suppose now that (η) is of the form (ηij ) = (ξi ξj )ǫ2 , where ǫξ ∈ RN . Then,
for a.e. (D2 u) ∈ RN ×N
G(D2 u + η) − G(D2 u) ≥ ǫ2 |ξ|2 λ(x, t),
and
Guxi xj (D2 u + θ(ǫ)ǫ2 (ξi ξj ))ǫ2 ξi ξj ≥ ǫ2 |ξ|2 λ(x, t),
where θ(ǫ) ∈ (0, 1) and θ(ǫ) ց 0 as ǫ → 0. Letting ǫ → 0 for a.e. (D2 u) ∈
RN ×N we have
Fuxi xj (x, t, u, Du, D2 u)ξi ξj ≥ λ(x, t)|ξ|2 ,
∀ ξ ∈ RN .
Likewise,
G(D2 u + η) − G(D2 u) ≤ sup (aνij (x, t)ǫ2 ξi ξj ) ≤ Λ(x, t)ǫ2 |ξ|2 ,
ν∈A
∀ ξ ∈ RN .
Therefore, ∀ξ ∈ RN
λ(x, t)|ξ|2 ≤ Fuxi xj (x, t, u, Du, D2 u)ξi ξj ≤ Λ(x, t)|ξ|2 .
This proves that (1.12) is parabolic. These remarks suggest a way of generalizing the concept of ellipticity to nondifferentiable F .
1.4 Remarks on the Concept of Ellipticity
Definition 1.1. We say that F [u] = F (x, t, u, Du, D2 u) is increasing in
RN ×N if
F (·, D2 u + η) > F (·, D2 u),
(D2 u) ∈ RN ×N
for every nontrivial, positive semi-definite N × N matrix η ∈ RN ×N (in particular, symmetric).
We have the following.
Lemma 1.3 Suppose F [u] = F (x, t, u, Du, D2 u) is elliptic in the sense that
D2 u → F (x, t, u, Du, D2 u) is a.e. differentiable in RN ×N and (1.6) holds.
Then, D2 u → F (·, D2 u) is increasing in RN ×N .
518
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Proof. If (D2 u) ∈ RN ×N is a point where F is differentiable, there exists ǫ > 0
sufficiently small, such that
F (·, D2 u + η) − F (·, D2 u) = Fuxi xj (·, D2 u)ηij + o(ǫ)
for every matrix (η) such that |η| ≤ ǫ. Let (η) be symmetric, positive semidefinite and nontrivial in RN ×N , and let Y be the unitary matrix that diagonalizes η, i.e.,


η1 . . . 0


Yt · (η) · Y =  0 . . . 0  , ηi ≥ 0.
0 . . . ηN
If (η) is nontrivial, there exists at least one index 1 ≤ j ≤ N , such that for
the corresponding eigenvalue ηj > 0. Then,
n
o
F (·, D2 u + η) − F (·, D2 u) = tr [Yt · (Fuxi xj ) · Y] · [Yt · (η) · Y] + o(ǫ)
= gii ηi + o(ǫ),
where gii are the diagonal elements of [Yt · (Fuxi xj ) · Y], and the elements
Fuxi xi are larger than or equal to the smallest eigenvalue of (Fuxi xj ). Taking
into account that Y is unitary, it easily follows that for ǫ small enough
F (·, D2 u + η) − F (D2 u) > 0
for any nontrivial, positive semi-definite (η) ∈ RN ×N .
If D2 u → F (x, t, u, Du, D2 u) is not differentiable in RN ×N , then Definition 1.1
can be taken as the definition of ellipticity.
1.5 Equations of Mini-Max Type
Let A, B be arbitrary set of indices and consider the family F of linear elliptic
operators
α,β
α,β
Lα,β [u] = aα,β
u,
ij (x, t)uxi xj + bi (x, t)uxi + c
α,β
α,β
where (x, t) → aα,β
(x, t) are real-valued, measurable
ij (x, t), bi (x, t), c
functions defined in ET for i, j = 1, 2, . . . , N , (α, β) ∈ A × B.
Let u ∈ C 2,1 (ET ) and (x, t) → f α,β (x, t), (α, β) ∈ A × B be a family of
real-valued, bounded functions defined in ET . Consider the quantity
F [u](x, t) = inf sup [Lα,β [u](x, t) + f α,β (x, t)]
β∈B α∈A
(1.14)
and the associated evolution equation
ut − F [u] = 0
in ET .
(1.15)
2 Maximum Principles
519
α,β
If we assume that the matrices (aij
(x, t)) ∈ RN ×N and satisfy (1.13) for any
(α, β) ∈ A × B, then (1.14) is elliptic, and (1.15) is parabolic. This can be
shown as in Section 1.3.
One of the reasons to consider such mini-max type of equations is that
nearly all fully nonlinear elliptic and parabolic partial differential equations
can be written in the form (1.15). We refrain from going any further in detail
about this topic here.
2 Maximum Principles
2.1 Linear Equations
Consider the linear elliptic operator
L(u) = aij (x, t)uxi xj − ai (x, t)uxi − ao (x, t)u,
and the associated linear parabolic equation
ut − L(u) = f (x, t)
in ET .
The assumptions on (aij (x, t)), and ai (x, t), i = 0, 1, . . . , N , and f (x, t) are
H1) The functions (x, t) → aij (x, t), ai (x, t), ao (x, t), f (x, t) are only bounded
and measurable in ET .
H2) There exist 0 < λo ≤ Λo < ∞ such that
λo |ξ|2 ≤ aij (x, t)ξi ξj ≤ Λo |ξ|2
∀ ξ ∈ RN .
H3) We have
kaij k∞;ET + kai k∞;ET ≤ A
for a given constant A.
For s ∈ (0, T ] we let Es ≡ E × (0, s), Ss ≡
[
τ ∈(0,s]
∂E × {τ }, and Γs ≡
Ss ∪ (E × {0}). Clearly, Γs is the parabolic boundary of Es . If s ∈ (0, T ] is
fixed, we let
a−
Ao = ka−
(2.1)
o ≡ max{0; −ao }.
o k∞;Es ,
2.1.1 The Dirichlet Problem
Let (x, t) → h(x, t) ∈ L∞ (ST ) and x → uo (x) ∈ L∞ (E). Consider the
Cauchy–Dirichlet problem

ut − L(u) = f in ET

(D)
u(x, t) = h(x, t) on ST


u(x, 0) = uo (x) in E.
520
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
We further assume that h ∈ C(S̄T ), uo ∈ C(Ē), and uo (x) = h(x, 0) when
x ∈ ∂E. We let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D).
In order to unify the notation, we set
(
h(x, t) on ST ,
ψ(x, t) =
uo (x) in E × {0}.
Theorem 2.1. Let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D) and
let H1), H2), H3) hold. Then, ∀ s ∈ (0, T ]
1
λ(s−t)
λ(s−t)
sup min 0; min ψ e
;
min f e
Γs
λ − Ao Es
λ>Ao
(2.2)
1
λ(s−t)
λ(s−t)
≤ u(x, s) ≤ inf max 0; max ψ e
;
max f e
.
Γs
λ>Ao
λ − Ao Es
Proof. Let λ ∈ R be chosen and let (x, t) → v(x, t) be defined by the exponential shift
u(x, t) = v(x, t)eλt .
Then, v satisfies
(
vt − aij (x, t)vxi xj + ai (x, t)vxi + (ao (x, t) + λ)v = f eλt in ET ,
v = ψeλt on ΓT .
(D’)
If v(x, t) ≤ 0 in Es or if 0 ≤ maxEs v(x, t) ≤ maxΓs v(x, t), then there is
nothing to prove.
If 0 < maxEs v(x, t) = v(xo , to ) for (xo , to ) ∈ E̊s ≡ E̊ × (0, s], then
vt (xo , to ) ≥ 0,
vxi (xo , to ) = 0,
and − aij (xo , to )vxi xj (xo , to ) ≥ 0,
since (−D2 u(xo , to )) is a symmetric, positive definite N × N matrix.
Therefore, from (D’) calculated at (xo , to )
(ao (xo , to ) + λ)v(xo , to ) ≤ max f (x, t)eλs ,
Es
and ∀ (x, t) ∈ Es
u(x, t) ≤ inf
λ>Ao
1
max f (x, t)eλ(s−t) .
λ − Ao Es
The estimate below is proved analogously.
Remark 2.1 To prove the estimate above it could be enough to have a classical sub-solution of (D), i.e.,

2,1

u ∈ C (ET ) ∩ C(ĒT ),

ut − L(u) ≤ f in ET ,
(Dsub )



u
= ψ.
ΓT
2 Maximum Principles
521
Analogously, to prove the estimate below, it would suffice to have a classical
super-solution of (D), i.e.

2,1
u ∈ C (ET ) ∩ C(ĒT ),


ut − L(u) ≥ f in ET ,
(Dsuper )


u
= ψ.
ΓT
Let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D) and ∀ s ∈ (0, T ] set
M1 (s) = kψe−Ao t k∞;Γs ,
M2 (s) = kf e−Ao t k∞;Es .
Consider the functions
w± (x, t) = M1 + M2 t ± u(x, t)e−Ao t ,
(x, t) ∈ Es .
By direct computation
wt± − L(w± ) = ±f e−Aot + M2 + ao (x, t)[M1 + M2 t] ∓ Ao ue−Ao t
≥ −Ao w± ,
and therefore,
wt± − aij (x, t)wx±i xj + ai (x, t)wx±i + (ao + Ao )w± ≥ 0.
Since ao + Ao ≥ 0 (see (2.1)) and since
w±
Γs
= M1 (s) + M2 (s)t ± ψ(x, s)e−Ao s ≥ 0,
from Theorem 2.1 it follows that w± ≥ 0 in ET and we have the following
estimate.
Lemma 2.1 Let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D) in ET .
Then, ∀(x, s) ∈ ET
|u(x, s)| ≤ kψeAo (s−t) k∞;Γs + kf eAo (s−t) k∞;Es .
As a simple consequence of Theorem 2.1 we have
Corollary 2.1 Let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D).
Then, ψ ≤ 0 on ΓT and f ≤ 0 in ET imply u(x, t) ≤ 0 for all (x, t) ∈ ET .
Analogously, ψ ≥ 0 on ΓT and f ≥ 0 in ET imply u(x, t) ≥ 0 for all (x, t) ∈
ET . Finally, if f ≡ ao ≡ 0, then
min ψ ≤ u(x, t) ≤ max ψ.
Γt
Γt
All these estimates follow from (2.2); for the last one, first take λ > 0 and
then let λ tend to zero.
522
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
2.1.2 The Neumann Problem
We assume that ∂E is of class C 2 and has outer unit normal
ν = (ν1 (x), ν2 (x), . . . , νN (x))
along ∂E.
Consider the problem

ut − L(u) = f in ET

u(x, 0) = uo (x) in E


νi (x)uxi (x, t) + b(x, t)u = ψ(x, t)
(N)
on ST .
As before, we assume uo ∈ C(Ē), ψ ∈ C(ST ), and we suppose that u ∈
C 2,1 (ET ) ∩ C 1 (Ē × (0, T ]) is a classical solution of (N).
Strictly speaking, (N) is the Cauchy–Neumann problem only if b(x, t) ≡ 0.
In order to derive a priori estimates, we first consider the case b > 0, and
then the case b(x, t) ≥ −bo with bo ≥ 0, to conclude the case of the Cauchy–
Neumann problem.
H4) We assume that b(x, t) ≥ bo > 0 for all (x, t) ∈ ST and for some given
bo > 0.
Theorem 2.2. Let u ∈ C 2,1 (ET ) ∩ C 1 (Ē × (0, T ]) be a classical solution of
(N) and let H1), H2), H3), and H4) hold. Then, ∀ (x, s) ∈ ET
ψ eλ(s−t)
1
sup min 0; min
; min uo (x)eλs ;
min f eλ(s−t)
Ss
E
b(x, t)
λ − Ao Es
λ>Ao
≤ u(x, s)
1
ψ eλ(s−t)
λs
λ(s−t)
.
; max uo (x)e ;
max f e
≤ inf max 0; max
E
λ>Ao
Ss
b(x, t)
λ − Ao Es
Proof. As before, introduce
v(x, t) = u(x, t)e−λt ,
and analyze the location of the maximum and the minimum of v in ET . The
previous argument can be repeated here, except when the extremum is on Ss .
In such a case, one uses the data on Ss , observing that at a maximum point
on Ss uxi νi ≥ 0, and at a minimum point uxi νi ≤ 0.
Now we deal with the general case.
H5) We assume that b(x, t) ≥ −bo , for all (x, t) ∈ ST and for some given
bo > 0.
Theorem 2.3. Let u ∈ C 2,1 (ET ) ∩ C 1 (Ē × (0, T ]) be a classical solution of
(N) and let H1), H2), H3), and H5) hold. Then there exist constants C1 , C2
depending only on A, Ao , bo such that
kuk∞;ET ≤ C1 eC2 T max {kψk∞;ST ; kuo k∞;E ; kf k∞;ET } .
2 Maximum Principles
523
Proof. Construct a function x → ϕ(x) ∈ C 2 (Ē) such that
i) ϕ(x) ≥ 12 for all x ∈ E;
ii) ϕ ∂E = 1;
iii) − ∂ϕ
∂ν = −(ϕxi νi ) = m ≥ (bo + 1) on ∂E.
Such a function can obviously be constructed if ∂E is smooth (say, of class
C 2 ). Consider the auxiliary function
w(x, t) = u(x, t)ϕ(x).
By direct calculation we have
wt − L(w) =ϕ(x)[ut − L(u)] + aj (x, t)uϕxj + 2aij (x, t)uxi ϕxj
− aij (x, t)ϕxi xj u
=ϕ(x)f (x, t) + Φi (x, t)wxi + Φo (x, t)w,
where
Φi = 2aij
ϕxj
,
ϕ
Φo = −
aij ϕxi ϕxj
aij ϕxi xj
+
ϕ
ϕ
+
ai ϕxi
.
ϕ
It follows that w satisfies in ET
wt − aij (x, t)wxi xj + Ai (x, t)wxi + Ao (x, t)w = ϕ(x)f (x, t),
with
Ai (x, t) = [ai (x, t) − Φi (x, t)],
Moreover, on ST
∂w
+ bw = ϕ
∂ν
i.e.,
Since b −
Ao = [ao (x, t) − Φo (x, t)].
∂ϕ
w ∂ϕ
∂u
+ bu + u
= ψϕ +
,
∂ν
∂ν
ϕ ∂ν
∂ϕ
∂w
w = ψϕ
+ b−
∂ν
∂ν
on ST .
∂ϕ
≥ 1, the result follows from Theorem 2.1.
∂ν
2.2 Quasi-Linear Equations
Consider the elliptic operator
Q(u) = Aij (x, t, u, Du)uxi xj − B(x, t, u, Du)
and the associated parabolic equation
ut − Q(u) = 0
The assumptions on Q(u) are
in ET .
(2.3)
524
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Q1) (x, t, z, p) → Aij (x, t, z, p), B(x, t, z, p) are bounded and measurable in
ET × R × RN .
Q2) λo |ξ|2 ≤ Aij (x, t, z, p)ξi ξj ≤ Λo |ξ|2 , ∀ξ ∈ RN , with 0 < λo ≤ Λo < ∞,
∀ (x, t, z, p) ∈ ET × R × RN .
Q3) uB(x, t, u, 0) ≥ −Bo − B1 u2 for two given non-negative constants Bo ,
B1 .
Remark 2.2 Regarding Q2), in what follows it will be enough to assume
that the matrix (Aij (x, t, u, 0)) is non-negative definite.
Remark 2.3 Assumption Q3) would be satisfied if we had, for example,
Q3’) |B(x, t, u, Du)| ≤ Ao + A1 |u| + A2 |Du| for given non-negative constants
Al , l = 0, 1, 2.
Indeed, in such a case
1
1
uB(x, t, u, 0) ≥ −A1 u2 − Ao u ≥ − A21 +
u2 − A2o .
2
2
2.2.1 The Dirichlet Problem
Consider the Cauchy–Dirichlet problem
(
ut − Q(u) = 0
in ET ,
u=ψ
on ΓT ,
(D)
where (x, t) → ψ(x, t) ∈ C(Γ T ).
We assume that (D) has a solution u ∈ C 2,1 (ET ) ∩ C(ĒT ). The same
technique of proof of Theorem 2.1 gives the following result.
Theorem 2.4. Let u ∈ C 2,1 (ET )∩C(ĒT ) be a classical solution of (D). Then,
∀ s ∈ (0, T ]
(
)
r
Bo
λ(s−τ )
λs
sup min 0; min ψ(x, τ ) e
; −e
λ − Bo
(x,τ )∈Γs
λ>B1
)
(
r
Bo
λ(s−τ ) λs
.
≤ u(x, s) ≤ inf max 0; max ψ(x, τ ) e
;e
λ>B1
λ − Bo
(x,τ )∈Γs
Moreover,
kuk∞;ET ≤ inf e
λ>B1
λT
(
kψk∞;ΓT ;
r
Bo
λ − Bo
)
.
We have already remarked that Q3) would be implied by Q3’). Suppose now
that Q3’) is replaced by
Q3”) |B(x, t, u, Du)| ≤ Ao + A1 |u|1+α + A2 |Du| for some α > 0.
2 Maximum Principles
525
This means that the lower order terms in Q(u) grow faster than linearly with
respect to u. From Q3”) we find
1+α
1
α
uB(x, t, u, 0) ≥ − A1 +
|u|1+α |u| −
Ao α .
1+α
1+α
In general, we may consider lower order terms satisfying
uB(x, t, u, 0) ≥ −Φ(|u|)|u| − Bo ,
where Bo ≥ 0 and s → Φ(s) is a nondecreasing, positive function in R+ and
Φ(s) ≤ Cs1+α ,
for some α > 0.
In such a case the method of proof of Theorems 2.1 and 2.4 would not apply,
since at a positive interior maximum (for example) for eλt u(x, t) we would be
led to the inequality
e−λt (λ − Φ(u))u ≤ Bo ,
which would not imply an estimate for u.
On the other hand, it is conceivable that under some integrability conditions on u, an upper bound could be derived. This will be accomplished by
the Alexandrov maximum principle.
Let us now consider the case when B(x, t, u, 0) grows less than linearly
with respect to u, that is, we assume
uB(x, t, u, 0) ≥ −Φ(|u|)|u| − Bo ,
(2.4)
where s → Φ(s) is nondecreasing in R+ and
Z ∞
1
ds = ∞.
Φ(s)
0
Without loss of generality we may assume
Φ(0) > 0.
(2.5)
Let ϕ be a twice differentiable, invertible function in R+ and set u = ϕ(v).
Writing (2.3) in terms of v, we find
vt − Aij (x, t, u, Du)vxi xj
− Aij (x, t, u, Du)vxi vxj
ϕ′′ (v)
1
+ B(x, t, u, Du) ′
= 0.
ϕ′ (v)
ϕ (v)
Next we make an exponential shift
w = e−λt v,
and we find that w satisfies
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
526
wt − Aij (x, t, u, Du)wxi xj
− Aij (x, t, u, Du)eλt wxi wxj
ϕ′′ (v)
e−λt
+
λw
+
B(x,
t,
u,
Du)
= 0.
ϕ′ (v)
ϕ′ (v)
At an interior maximum point (xo , to ) for w
[B(x, t, u, 0) + λvϕ′ (v)](xo ,to ) ≤ 0.
(2.6)
Assume that
u(xo , to ) ≥ 0.
Then, multiplying (2.6) by u(xo , to ) and making use of (2.4) yields
[−Φ(u) + λvϕ′ (v)](xo ,to ) u(xo , to ) ≤ Bo .
(2.7)
Next we choose ϕ so that sϕ′ (s) = Φ(ϕ(s)), i.e.,
Z
ϕ(s)

1

dτ = ln s, s ≥ 1
Φ(τ
)
0


ϕ(1) = 0.
(2.8)
Remark 2.4 The function s → ϕ(s) is increasing and for s ∈ (0, ∞) the
range of ϕ is (−∞, ∞).
For the given choice of ϕ, we find from (2.7)
(λ − 1)u Φ(u)
(xo ,to )
≤ Bo ,
λ > 1,
and since Φ(0) > 0
u(xo , to ) ≤
Now,
Bo
.
(λ − 1)Φ(0)
max w(x, t) ≤ w(xo , to ) = ϕ−1 (u(xo , to ))e−λto ≤ ϕ−1
ET
that is,
Bo
(λ − 1)Φ(0)
e−λto ,
Bo
.
ET
(λ − 1)Φ(0)
Consequently, arguing also with −u replacing u, we have the following.
max v(x, t) ≤ max 1; eλT ϕ−1
Theorem 2.5. Let u ∈ C 2,1 (ET ) ∩ C(ĒT ) be a classical solution of (D) and
let Q1), Q2), (2.4)–(2.5) hold. Then,
kuk∞;ET ≤ inf ϕ(ξ),
λ>1
where
def
ξ = e
λT
−1
max 1; ϕ
Bo
(λ − 1)Φ(0)
−1
;ϕ
(kψk∞;ΓT ) ,
and ϕ−1 is the inverse function of s → ϕ(s) with s > 0, defined implicitly by
(2.8).
2 Maximum Principles
527
2.2.2 Variational Boundary Data
We assume that ∂E is of class C 2 , denote by ν the outer unit normal to ∂E,
and consider the Cauchy–Neumann-type problem

(2.9)
 ut − Q(u) = 0 in ET ,


Aij (x, t, u, Du)uxj cos(ν, xi ) + ψ(x, t, u)
= 0,
(2.10)
ST



(2.11)
u(x, 0) = uo (x), x ∈ E.
Here we assume that Q1), Q2), Q3’) hold and in addition on the datum
ψ(x, t, u) we impose
(
ψ is continuous in S̄T × R and
(2.12)
u ψ(x, t, u) ≥ −Co − C1 u2 for (x, t) ∈ ST
for two given constants Co , C1 . Finally, we require
uo ∈ C(Ē).
(2.13)
We have the following.
Theorem 2.6. Let u ∈ C 2,1 (ET ) ∩ C 1,0 (Ē × (0, T )) ∩ C(ĒT ) be a classical
solution of (2.9)–(2.11), and let (2.12)–(2.13) hold. Then,
q
λT
2
2
kuk∞;ET ≤ λ1 e max
(2.14)
Ao + A1 + A2 ; kuo k∞;E ,
where λ, λ1 are constants depending only upon λo , Λo , A3 , C1 .
Proof. Construct a function x → ϕ(x) ∈ C 2 (Ē) and determine a positive
number λ such that

ϕ(x) ≥ 1, ∀ x ∈ Ē
(2.15)
 − Aij (x, t, u, Du) cos(ν, xi )ϕxj (x)
≥ 2C1 ,
ST
and
ϕxi ϕxj
ϕxi xj
≥ 1.
− 2Aij (x, t, u, Du)
min λ + Aij (x, t, u, Du)
u∈R
ϕ
ϕ2
Du∈RN
(2.16)
Remark 2.5 Since the matrix (Aij ) is positive definite, the vector of jth
N
X
component
Aij cos(ν, xi ) forms an acute angle with ν.
i=1
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
528
The construction of such a ϕ is obvious if ∂E is of class C 2 . Letting
w(x, t) = e−λt ϕ(x)u(x, t),
we find by direct calculation
wt − Aij (x, t, u, Du)wxi xj
ϕx x
ϕx ϕx
+ λ + Aij (x, t, u, Du) i j − 2Aij (x, t, u, Du) i 2 j w
ϕ
ϕ
ϕxi
−λt
wxj = 0,
+ e ϕB(x, t, u, Du) + 2Aij (x, t, u, Du)
ϕ
Aij (x, t, u, Du)wxj cos(ν, xi ) − wAij (x, t, u, Du)
+ e−λt ϕ(x)ψ(x, t, u)
ST
ϕxj
cos(ν, xi )
ϕ
(2.17)
(2.18)
= 0,
and
w(x, 0) = uo (x)ϕ(x).
Let (xo , to ) ∈ ĒT be a point where w2 (x, t) achieves its maximum. If to = 0
estimate (2.14) follows. If to > 0 and xo ∈ ∂E, multiplying (2.18) by w(xo , to )
and observing that wwxj = 12 (w2 )xj , we have Aij (w2 )xj cos(ν, xi ) ≥ 0 and
using (2.15) and (2.12) we find
2C1 w2 (xo , to ) − C1 e2λto ϕ2 (xo )u2 (xo , to ) − Co e−2λto ϕ2 (xo ) ≤ 0,
that is,
|w(xo , to )| ≤
p
Co kϕk∞;E ,
and therefore, (2.14) follows with λ1 = kϕk∞;E .
If (xo , to ) ∈ E̊T , then multiply (2.17) by w(xo , to ) and observe that at
(xo , to )
wwt =
1 2
(w )t = 0,
2
wwxi =
1 2
(w )xi = 0,
2
−wAij wxi xj ≥ 0.
Moreover, using the choice of λ in (2.16)
w2 (xo , to ) − e−λto ϕ2 (xo ) [Ao + A1 |u(xo , to )| + A2 |Du(xo , to )|] ≤ 0.
(2.19)
At (xo , to ) we have
0 = Dw(xo , to ) = e−λto [u(xo , to )Dϕ(xo , to ) + ϕ(xo , to )Du(xo , to )] ,
and therefore,
|Du|(xo , to ) ≤ |u|
Consequently, (2.19) implies
|Dϕ|
ϕ
(xo ,to )
.
3 The Aleksandrov Maximum Principle
that is,
w2 (xo , to ) ≤ e−λto Ao ϕ2 + A1 ϕw + A2 |Dϕ|w
|w(xo , to )| ≤ 2
and the theorem follows.
(xo ,to )
529
,
q
Ao + A21 + A22 (kϕk∞ + kDϕk∞ ) ,
3 The Aleksandrov Maximum Principle
The main ideas presented in this section are attributed to Aleksandrov, who
introduced them in connection with elliptic equations in nondivergence form
with bounded and measurable coefficients [8]. We give here the parabolic
version of such ideas, worked out by Krylov [145]. We follow the approach of
Reye [217] (see also Nazarov and Ural’tzeva [191]).
3.1 Basic Geometric Notions
3.1.1 The Upper Contact Set
Let x → u(x) ∈ C(E). We define the upper contact set of u by
def
Γ + = {y ∈ E : u(x) ≤ u(y) + p · (x − y), ∀ x ∈ E, for some p ∈ RN }.
Let u ∈ C 1 (E). Then, y ∈ Γ + if the tangent hyperplane to the graph of u
through (y, u(y)) is all above the graph of u. Hence, if y ∈ Γ + and u ∈ C 1 in
a neighborhood of y, then p = ∇u(y).
3.1.2 The Concave Hull
We define the concave hull of u to be the smallest concave function on E lying
above u. We denote such a function by Ψu .
Using Figure 3.1 as a guideline,
one can see that
Γ + ≡ {x ∈ E : u(x) ≡ Ψu (x)}.
Remark 3.1 If u ∈ C 2 (E), then (D2 u(x)) ≤ 0 on Γ + .
530
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Ψu
E
Fig. 3.1
3.1.3 The Normal Mapping
In the definition of the upper contact set, we have that y ∈ Γ + if we can find
p ∈ RN such that
u(x) ≤ u(y) + p · (x − y) ∀ x ∈ E.
If y 6∈ Γ + , then we might set the corresponding p(y) to be the empty set. In
such a way, we have the normal mapping χ : E → RN defined by
(
{p ∈ RN : u(x) ≤ u(y) + p · (x − y), ∀x ∈ E} if y ∈ Γ + ,
χ(y) ≡
∅ if y 6∈ Γ + .
In other words, χ(y) is the set of the “slopes” of the “tangent hyperplanes”
to the graph of u at the point (y, u(y)). It is apparent that, in general, χ is
not single valued, as indicated by Figure 3.2.
E
y1
y2
Fig. 3.2
We consider the following specific example.
3 The Aleksandrov Maximum Principle
531
3.1.4 The Normal Mapping of a Cone
Let E = BR (z) ≡ {x ∈ RN : |x − z| < R}, and let x → u(x) be the function
whose graph is the cone with base BR (z) and vertex (z, a) for some positive
a, that is
|x − z|
u(x) = a 1 −
.
R
If (y, u(y)) is not the vertex (i.e., y 6= z), we have
χ(y) = Du(y) = − a y − z ,
R |y − z|
y 6= z.
At the “vertex,” there are infinitely many “tangent hyperplanes,” whose “nora
mals” fill a ball of center the origin and radius R
. Therefore,
 a y−z
−
, if y 6= z,
R |y − z|
χ(y) =

if y = z.
B Ra (0),
3.2 Increasing Concave Hull of u
Next we consider functions depending upon x and t. If (x, t) → u(x, t) ∈
C(ĒT ), for each t ∈ (0, T ), we may define the upper contact set of x → u(x, t)
and denote it by Γ + (t). Set
[
def
Γu+ =
Γ + (t).
0≤t≤T
If t → u(·, t) ∈ C 1 (0, T ), we define the increasing set of u as
def
I = {(x, t) ∈ ET : ut (x, t) ≥ 0},
and set
def
F = Γu+ ∩ I.
If u ≥ 0, to estimate the “largeness” of u in ET , it will be enough to estimate
the size of u on F .
Let u ∈ C 2,1 (ET ). The increasing concave hull of u is the smallest function
in ET , which is concave with respect to x for all t ∈ (0, T ), nondecreasing in
t, and which lies above u.
We denote such a function by ξu . The function ξu has the following properties.
2,1
Proposition 3.1 If u ∈ C 2,1 (ET ), then ξu ∈ W∞
(ET ), that is,
∂ 2 ξu
∂xi ∂xj
∞;ET
+
∂ξu
∂t
∞;ET
≤ C.
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
532
Here, the derivatives are distributional derivatives identified with L∞ (ET )
functions. The spaces Wqk,m (ET ) were defined in Section 1 of Chapter 11.
Proposition 3.2 If u ∈ C 2,1 (ET ), then
(
∂t u det(D2 u) a.e. on [u = ξu ] ≡ F,
∂t ξu det(D2 ξu ) =
0 otherwise.
3.2.1 Proof of Proposition 3.1
Define
v(x, t) = sup u(x, s),
s≤t
∀ x ∈ E, s ∈ [0, T ].
Then,
ξu (·, t) = Ψv(·,t) ,
where Ψv(·,t) is the concave hull of v(x, t). Since u ∈ C 2,1 (ET ), v is at least in
1,1
W∞
(ET ). Indeed, for any ǫ > 0 and any vector h such that x + h ∈ E, we
have
|v(x + h, t) − v(x, t)| + |v(x, t + ǫ) − v(x, t)|
≤ sup |u(x + h, s) − u(x, s)| + sup |u(x, s + ǫ) − u(x, s)|
s∈(0,t)
s∈(0,t)
≤ kDuk∞ |h| + kut k∞ ǫ.
Since Ψv(·,t) = ξu (·, t) is concave, it can be looked upon as the pointwise
infimum of all affine functions f , whose graph lies above the graph of v and
kDf k∞;E ≤ kDuk∞;E .
Consequently, Dξu (·, t) exists for a.e. x ∈ E and
kDξu k∞;ET ≤ kDuk∞;ET .
In a similar way, we have
k∂t ξu k∞;ET ≤ k∂t uk∞;ET .
∂2u
.
∂xi ∂xj
From the previous argument, it follows that for t ∈ [0, T ] fixed, for a.e. xo ∈ E,
the graph of ξu has a tangent hyperplane
We now need to obtain the bound on the second-order derivatives
π(x) = ξu (xo ) + Dξu (xo ) · (x − xo ),
(3.1)
and by the concavity of ξu we also have
ξu (x) ≤ π(x)
∀ x ∈ E.
Since t ∈ [0, T ] is fixed, we drop it from the notation.
(3.2)
3 The Aleksandrov Maximum Principle
533
Suppose now that for a.e. xo ∈ E we can find a paraboloid of the form
θ(x, xo ) = ξu (xo ) + Dξu (xo ) · (x − xo ) − C|x − xo |2
(3.3)
lying all below the graph of ξu , that is,
ξu (x) ≥ θ(x, xo ) ∀ x ∈ E.
(3.4)
This will then imply that ξu (·, t) ∈ W 2,∞ (E), where for any p ≥ 1
W 2,p (E) = {v ∈ Lp (E) : Dα v ∈ Lp (E) for |α| ≤ 2}
and the derivatives are meant in the weak sense (see Section 1.1c of the Complements of Chapter 9). We rely on the following.
Theorem 3.1. [28] Let P be the class of all polynomials of degree 1. Suppose
that a function w ∈ L2 (E) satisfies
Z
inf
|w(x) − P (x)|2 dx ≤ Cρ4
P ∈P B (x )
ρ
o
for a.e. xo and ρ > 0 such that Bρ (xo ) ⊂ E, and for a given constant C > 0.
Then, w ∈ W 2,∞ (E) and
kwxi xj k∞;E ≤ Cγ,
where γ depends only upon the dimension N .
Let xo be a point in E such that (3.1) and (3.3) both hold. Then, ∀ x ∈ E
from (3.2) and (3.4)
|ξu (x) − P (x)|2 ≤ C|x − xo |2 ,
where P (x) = ξu (xo ) + Dξu (xo ) · (x − xo ). Integrating over Bρ (xo ) and taking
the infimum over all P ∈ P the assumptions of the previous theorem are
satisfied and its conclusion proves Proposition 3.1. Thus, it remains to prove
that for a.e. xo ∈ E a paraboloid of the form (3.3) can be found. This proof
requires the following fact.
Let xo ∈ E be such that v(xo ) 6= ξu (xo ) (as before, t ∈ [0, T ] is assumed fixed and dropped from the notation). Since ξu (·, t) = Ψv(·,t) the
point (xo , ξu (xo )) in the graph of ξu is the linear combination of at most
N + 1 points in the graph of v, that is, there exist x1 , x2 , . . . , xN +1 ∈ E and
α1 , α2 , . . . , αN +1 ∈ (0, 1) such that
N
+1
X
αi = 1,
(3.5)
i=1
xo =
N
+1
X
i=1
αi xi ,
(3.6)
534
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
ξu (xo ) = Ψv (xo ) =
+1
N
X
αi v(xi ).
(3.7)
i=1
Moreover,
v(xi ) = ξu (xi ),
i = 1, 2, . . . , N + 1.
Indeed, by definition of concave hull, v(x) ≤ ξu (x) for any x ∈ Ē. If for some
i we had v(xi ) < ξu (xi ), then
ξu (xo ) =
N
+1
X
αi v(xi ) <
N
+1
X
i=1
i=1
αi ξu (xi ) ≤ ξu (xo ).
Proof of Proposition 3.1 concluded By the regularity of u, for each (xo , to ) ∈
ET , there exists a paraboloid
θu (x, xo , to ) = u(xo , to ) + Du(xo , to ) · (x − xo ) − c(xo , to )|x − xo |2 ,
with c ≤ kD2 uk∞;ET , such that
u(x, to ) ≥ θu (x, xo , to ) ∀x ∈ E.
Since the supremum in the definition of v must be attained, it follows that v
enjoys the same property with Du(xo , to ) replaced by some vector bounded
by kDuk∞;ET . Let us now refer to the N + 1 points xi . If θi is the paraboloid
+1
N
X
αi θi (x, xi , to )
lying under v(xi , to ) with θi (xi ) = v(xi , to ), it follows that
i=1
is a paraboloid with the same bound on its second-order derivatives, lying
under Ψv(·,to ) and equal to it at x. This allows us to conclude.
3.2.2 Proof of Proposition 3.2
We need some introductory tools.

Lemma 3.1 Each of the following line segments xi ,
tains xo and ξu is affine along them.
Proof. Let i = 1 and set
x̄ =
N
+1
X
1
αj xj .
1 − α1 j=2
Then from (3.6)
xo = α1 x1 + (1 − α1 )x̄.
1
1 − αi
X
j6=i

αj xj  con-
3 The Aleksandrov Maximum Principle
535
From (3.7) we have
ξu (xo ) = α1 ξu (x1 ) + (1 − α1 )
Since (1 − α1 )−1
PN +1
j=2
N
+1
X
j=2
αj = 1, we have that
1
αj v(xj ).
1 − α1
(3.8)
N
+1
X
j=2
(1 − α1 )−1 αj v(xj ) belongs
to the convex hull of the graph of v and by definition of ξu
N
+1
X
j=2
1
αj v(xj ) ≤ ξu (x̄).
1 − α1
Next, by the concavity of ξu along the line segment (x1 , x̄) we must have
ξu (xo ) ≥ α1 ξu (x1 ) + (1 − α1 )ξu (x̄).
(3.9)
Combining (3.8)–(3.9) yields
ξu (xo ) = α1 ξu (x1 ) + (1 − α1 )ξu (x̄).
(3.10)
To prove that ξu is affine along the segment (x1 , x̄), we show that ∀ τ ∈ (0, 1)
ξu (τ x1 + (1 − τ )x̄) = τ ξu (x1 ) + (1 − τ )ξu (x̄).
Since ξu is concave
ξu (τ x1 + (1 − τ )x̄) ≥ τ ξu (x1 ) + (1 − τ )ξu (x̄),
(3.11)
and we must show that equality must occur in (3.11). Suppose that for some
τ̄ ∈ (0, 1) (3.11) holds with strict inequality. Without loss of generality we
may assume that τ̄ > α1 , so that
xo = β(τ̄ x1 + (1 − τ̄ )x̄) + (1 − β)x̄,
β=
α1
∈ (0, 1).
τ̄
Then, by concavity
ξu (xo ) ≥ βξu (τ̄ x1 + (1 − τ̄ )x̄) + (1 − β)ξu (x̄)
> β τ̄ ξu (x1 ) + β(1 − τ̄ )ξu (x̄) + (1 − β)ξu (x̄)
= α1 ξu (x1 ) + (1 − α1 )ξu (x̄) = ξu (xo )
by (3.10). The contradiction proves the lemma.
Corollary 3.1 Suppose that there are at least three distinct points xj for
which (3.5)–(3.7) hold. Then, ξu is affine in a neighborhood of xo .
Proof. Indeed, the graph of ξu must contain in the neighborhood of xo , two
distinct line segments intersecting at (xo , ξu (xo )).
536
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Lemma 3.2 At a.e. xo ∈ E such that ξu (xo ) 6= v(xo ), det(D2 ξu (xo )) = 0.
Proof. If for xo ∈ E, ξu (xo ) 6= v(xo ), there exist x1 , . . . , xN +1 ∈ E,
α1 , . . . , αN +1 ∈ (0, 1) such that (3.5)–(3.7) hold. Then, by Lemma 3.1, ξu
PN +1
is affine along the segment (x1 , x̄ = (1 − α1 )−1 j=2 αj xj ) containing xo .
After a rotation and a translation, we may assume that xo = 0, and that
∂ 2 ξu
(x1 , x̄) is a portion of the x1 -axis, so that
(0) = 0, since ξu is affine on
∂x21
such a portion. Moreover, by the concavity of the matrix

0
∂ 2 ξu
∂x1 ∂x2
...
∂ 2 ξu
∂xN ∂x1
...
..
.
...
...
..
.
...
 ∂ 2 ξu
 ∂x2 ∂x1
(D ξu )(0) = 
 ..
 .
2
∂ 2 ξu
∂x1 ∂xN
...
..
.
∂ 2 ξu
∂xN ∂xN



 (0)


is negative semi-definite. Let X be a N × N unitary matrix that diagonalizes
(D2 ξu (0)), that is, such that


λ1 . . . 0


Xt · (D2 ξu )(0) · X =  0 . . . 0  ,
0 . . . λN
where λi , i = 1, . . . , N are the real eigenvalues of the
symmetric matrix
(D2 ξu )(0). By direct calculation, setting et1 = 1 0 . . . 0 , we have
0 = et1 (D2 ξu )(0)e1
= et1 XXt (D2 ξu )(0)XXt e1


λ1 . . . 0


= (Xt e1 )t  0 . . . 0  (Xt e1 ).
(3.12)
0 . . . λN
Therefore, denoting with (ηi ) the entries of the vector (Xt e1 ), we have from
(3.12)
N
X
λi ηi2 .
0=
i=1
Since at least for some index i we have ηi2 6= 0, we must have λi = 0 for some
index i, since λi ≤ 0 (the matrix (D2 ξu )(0) is negative semi-definite).
We conclude that the largest eigenvalue of (D2 ξu )(0) is zero; hence,
det(D2 ξu )(0) = λ1 · λ2 · · · · · λN = 0.
3 The Aleksandrov Maximum Principle
537
Lemma 3.3 If f ∈ W 1,∞ (E), then |Df | = 0 a.e. on the set {f = 0}.
Proof. First divide f into its positive and negative parts, to reduce the lemma
to the case f ≥ 0. Then the proof follows by a standard approximation process,
and the definition of a weak derivative.
Proof of Proposition 3.2 concluded If for some (x, t) ∈ ET , ξu (x, t) 6= v(x, t),
then by Lemma 3.2 we have that det(D2 ξu )(x, t) = 0.
Let (x, t) ∈ ET be such that ξu (x, t) = v(x, t) 6= u(x, t). Then for some
s<t
u(x, s) = v(x, s) = v(x, t).
Hence, since t → ξu (·, t) is nondecreasing, ξu (x, s) = ξu (x, t), and
0. Finally, consider those points (x, t) ∈ ET where
∂ξu
∂t (x, t)
=
ξu (x, t) = v(x, t) = u(x, t).
u −u)
= 0, and Dxi (ξu − u) = 0. Moreover,
On such a set (ξu − u) = 0, ∂(ξ∂t
1,0
since D(ξu − u) ∈ W∞ (ET ), we have Dxi Dxj (ξu − u) = 0 on [ξu = u], and
the proposition follows.
3.3 Auxiliary Lemmas
Lemma 3.4 Let w ∈ C 3 (E). Then,
N
X
∂
∂
2
det(D w) = 0,
∂xj ∂wxi xj
j=1
i = 1, . . . , N.
Proof. Fix i = 1 and for notational simplicity, set wxi xj = wij . Then, (D2 w) =
(wij ), and if Aij is the algebraic complement of wij
det(D2 w) =
N
X
(−1)1+j w1j det(A1j ),
j=1
N
X
j=1
∂
∂xj
X
N
∂
∂
2
(−1)1+j
det(D w) =
det(A1j ).
∂wij
∂xj
j=1
For j = 1, . . . , N fixed, we have
N
N
XX
Y
∂
det(A1j ) =
(−1)1+j+(h) wlhl j
wihi ,
∂xj
h∈H l=2
i6=l
where H is the set of all permutations (h2 , h3 , . . . , hN ), h2 6= h3 6= · · · 6= hN 6=
j and (h) is the index of the permutation h. Then,
538
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
X
N
N X X
N
X
Y
∂
∂
2
det(D w) =
(−1)1+j+(h) wlhl j
wihi .
∂xj ∂wij
j=1
j=1
h∈H l=2
i6=l
Since w ∈ C 3 (E), wlhl j = wljhl , and rearranging the sums, the lemma follows.
Let E be convex and bounded, and consider the cylindrical domain Q =
E × (t1 , t2 ), 0 ≤ t1 < t2 .
Lemma 3.5 Let w ∈ C ∞ (Q) be such that w = 0 on ∂E × [t1 , t2 ]. Then,
"Z
Z t2 Z
1
2
∂t w det(D w) dxdt =
w det(D2 w) dx
N
+
1
t1
E
E×{t2 }
#
(3.13)
Z
2
−
w det(D w) dx .
E×{t1 }
Proof. Integrating the left-hand side of (3.13) with respect to t, we get
ZZ
Z
2
∂t w det(D w) dxdt =
w det(D2 w) dx
Q
E×{t2 }
Z
−
w det(D2 w) dx
(3.14)
E×{t1 }
N ZZ
X
∂3w
∂
w
−
det(D2 w) dxdt.
∂t∂x
∂x
∂w
i
j
x
Q
i xj
i,j=1
Using Lemma 3.4, the last term can be integrated twice by parts in xi , xj , to
give
N ZZ
X
∂
2
−
∂t w wxi xj
det(D w) dxdt
∂wxi xj
Q
i,j=1
ZZ
= −N
∂t w det(D2 w) dxdt.
Q
Putting this in (3.14) proves the lemma.
In Lemma 3.5 consider the following special situation. Let t1 = 0, t2 = 1, and
w(x, t) = tw1 (x) + (1 − t)w2 (x),
t ∈ (0, 1),
where x → wi (x) i = 1, 2 are two concave functions in E, such that
wi = 0 on ∂E, i = 1, 2,
w1 (x) ≤ w2 (x) ∀ x ∈ E.
Since the convex combination of concave functions is concave, if w1 and w2
are smooth, we have
3 The Aleksandrov Maximum Principle
det(D2 w) ≤ 0,
539
∂t w = w1 − w2 ≤ 0,
and
∂t w det(D2 w) ≥ 0.
Putting this in (3.13) yields
Z
Z
2
w2 det(D w2 ) dx ≤
w1 det(D2 w1 ) dx.
E
(3.15)
E
We have therefore shown the following result.
Lemma 3.6 Let E be convex with smooth boundary ∂E. Let w1 , w2 ∈ C 2 (E)
be concave, satisfying w1 = w2 = 0 on ∂E and w1 (x) ≤ w2 (x) ∀ x ∈ E. Then,
(3.15) holds.
3.4 Embedding by Normal Mapping
Let E be a bounded, convex domain in RN with a smooth boundary and let
w ∈ C 2 (E) be concave, such that w = 0 on ∂E.
Let χw : E → RN be the normal mapping relative to w. If w ∈ C 2 (E),
then χw is single valued, but need not be one-to-one. For y ∈ E we have
χw (y) = Dw(y).
The approximations χǫ (y) = χw (y) − ǫy, i.e.,
χǫ (y) = Dw(y) − ǫy,
are one-to-one.
Indeed, if χǫ (E) is the image of E under χǫ , the determinant of the Jacobian of the inverse mapping is given by
det J(χǫ−1 ) = det (D2 w(y) − ǫI),
(3.16)
and since w is concave, the eigenvalues of (D2 w(y)) are nonpositive for any
y ∈ E and hence, the right-hand side of (3.16) is not zero ∀ y ∈ E. Since E is
convex, this implies that χǫ is one-to-one.
Since w ∈ C 2 (E), both χǫ and χǫ−1 are continuous, so that χǫ (E) is open
in RN .
Z
w̃(ξ) dξ, where w̃(ξ) = w̃(χǫ (y)) = w(y).
Consider the integral
χǫ (E)
Changing variables, and writing it as an integral over E we have
Z
Z
w(y) det(−D2 w + ǫI) dy.
(3.17)
w̃(ξ) dξ =
χǫ (E)
E
Z
w(y) det(−D2 w) dy. As for
Letting ǫ → 0, the right-hand side tends to
E
the left-hand side, let ξ ∈ χ̊w (E), where χ̊w (E) denotes the interior (possibly
empty) of χw (E). Then, for ǫ small enough,
540
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
χǫ (y) = Dw(y) − ǫy = χw (y) − ǫy = ξ − ǫy,
(ξ = χw (y))
belongs to a neighborhood of ξ in χ̊w (E). Therefore, if IΣ denotes the characteristic function of Σ, i.e.,
(
1 ξ∈Σ
IΣ (ξ) =
0 ξ otherwise
we see that
lim inf Iχǫ (E) (ξ) ≥ Iχ̊
ǫ→0
w (E)
(ξ).
Next, observe that for every net {ǫ′ }, there exists a subnet {ǫ′′ } such that
χ−1
χ−1
χ−1
ǫ (ξ) → y ∈ w (ξ). Indeed, since { ǫ′ (ξ)} is compact, there is a subnet
−1
′′
χ
{ǫ } such that ǫ′′ (ξ) → η ∈ E. Since χǫ′′ is continuous
χǫ′′ (χǫ−1
′′ (ξ)) = ξ
−1
χw (η) = ξ i.e. η ∈ χw
(ξ).
⇒
Since w is continuous, we must have
lim inf w(χǫ−1 (ξ)) ≥
ǫ→0
inf
{y∈E: χw (y)=ξ}
w(y) =: w̄(y).
′
Indeed, if lim inf w(χ−1
ǫ (ξ)) < w̄(ξ), there must be a net {ǫ } such that
ǫ→0
′′
w(χǫ−1
′ (ξ)) < w̄(ξ) − σ for some σ ∈ (0, 1), and then for a subnet {ǫ } we
−1
−1
−1
χ
χ
χ
¯(ξ) ≥ w̄(ξ), which is a
would have ǫ′′ (ξ) → y ∈ w (ξ) and w( ǫ′ (ξ)) → w̄
contradiction.
Using these facts we pass to the limit on the left-hand side of (3.17), using
Fatou’s Lemma to obtain
Z
Z
lim inf
lim inf w(χǫ−1 (ξ))Iχǫ (E) (ξ) dξ
w̃(ξ) dξ ≥
ǫ→0
χǫ (E)
RN ǫ→0
Z
Z
w̄(ξ) dξ.
w̄(ξ)Iχ̊ (E) (ξ) dξ =
≥
w
χ̊w (E)
RN
Combining this and (3.17), we have
Z
Z
w(y) det(−D2 w(y)) dy.
w̄(ξ) dξ ≤
χ̊w (E)
E
(3.18)
To proceed, let Cxo : E → R be the function whose graph is the cone of base
E and vertex at (xo , w(xo )), so that
max Cxo (x) = Cxo (xo ) = w(xo ).
Lemma 3.7
χCx (E) ⊂ χw (E).
o
3 The Aleksandrov Maximum Principle
541
Proof. Since χCxo (x) ⊂ χCxo (xo ) ∀ x ∈ E, it will be enough to show that for
every p ∈ χCxo (xo ) there exists y ∈ E such that Dw(y) = p = χw (y). Let
π(x) = w(xo ) + p · (x − xo )
be the hyperplane through (xo , w(xo )) and “slope” p. Since p ∈ χCxo
∀ x ∈ E.
π(x) ≥ Cxo (x)
(3.19)
The set Σ ≡ {x ∈ Ē : w(x) ≥ π(x)} is closed and convex, and by virtue of
(3.19) w(x) − π(x) = 0 on ∂Σ. By Rolle’s Theorem there exists y ∈ Σ such
that Dw(y) − Dπ(x) = 0, i.e., Dw(y) = p.
Set
k ≥ 1,
R = k diam(E),
and consider the function Dxo defined on BR (xo ), whose graph is the circular
cone of base BR (xo ) and vertex (xo , w(xo )) (see Figure 3.3).
(xo ,w(xo ))
Cx o
w(x)
xo
D xo
E
Fig. 3.3
Lemma 3.8
χDx (BR (xo )) ⊂ χCx (E).
o
o
Proof. This is obvious, once we observe that Dxo (x) ≥ Cxo (x) ∀ x ∈ E,
Dxo (xo ) = Cxo (xo ) = w(xo ). Therefore, every “tangent” hyperplane to the
graph of Dxo through (xo , w(xo )) is also a “tangent” hyperplane to the graph
of Cxo through (xo , w(xo )).
Combining the two lemmas in (3.18) we have
Z
Z
w̄(ξ) dξ ≤
w(y) det(−D2 w(y)) dy.
χ̊Dxo (BR (xo ))
E
From the calculation of the normal mapping of a cone (see Section 3.1.4) we
finally find
Z
Z
w̄(ξ) dξ ≤
w(y) det(−D2 w(y)) dy,
(3.20)
B
w(xo )
k diam(E)
(0)
E
542
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
where
w̄(ξ) =
inf
{y∈E: χw (y)=ξ}
w(y).
(3.21)
Inequality (3.20) and the definition (3.21) will be the basis of the following
embedding theorem.
Theorem 3.2. Let E be convex with a smooth boundary ∂E, and let w ∈
C 2 (E) be concave and such that w = 0 on ∂E. Then,
sup w(x) ≤
x∈E
N
− 1
2ωN N +1 (diam(E)) N +1
Z
2
w(y) det(−D w(y)) dy
E
N1+1
, (3.22)
where ωN is the volume of the unit ball in RN .
Proof. From (3.21) it follows that w̄(0) = w(xo ). We write
w̄(ξ) = w̄(0) + w̄(ξ) − w̄(0)
= w(xo ) +
inf
[w(y) − w(xo )]
{y∈E: χw (y)=ξ}
≥ w(xo ) −
sup
{y∈E: χw (y)=ξ}
(3.23)
|w(y) − w(xo )|.
In estimating the last term on the right-most side of (3.23), we first observe
that since {y ∈ E : χw (y) = ξ} is closed, the supremum is achieved at some
ȳ such that χw (ȳ) = Dw(ȳ) = ξ.
Along the line segment tȳ + (1 − t)xo , t ∈ (0, 1), the function t → w(tȳ +
(1 − t)xo ) is concave, and therefore, |wt | achieves its maximum at t = 1.
Therefore,
|w(ȳ) − w(xo )| ≤ |wt (t = 1)| = |Dw(ȳ)||ȳ − xo | = |ξ| diam(E).
Consequently, ∀ ξ ∈ B
w(xo )
k diam(E)
(0)
w(xo )
w̄(ξ) ≥ w(xo ) −
diam(E) ≥
k diam(E)
1
w(xo ).
1−
k
(3.24)
Take k = 2. Then from (3.20) and (3.24)
1
(w(xo ))N ωN
≤
w(xo ) N
2
2 (diam(E))N
Z
w(y) det(−D2 w(y)) dy,
E
which is (3.22).
By approximation, we also have the following.
Corollary 3.2 The conclusion of Theorem 3.2 holds if w ∈ Wo2,N +1 (E).
3 The Aleksandrov Maximum Principle
Now let (x, t) → w(x, t) belong to

x → w(x, t)


w(x, t) = 0
w(x, 0) = 0



wt (x, t) ≥ 0
543
C 2,1 (ET ) and such that
is concave ∀ t ∈ [0, T ],
x ∈ ∂E, ∀ t ∈ [0, T ],
∀ x ∈ E,
(H)
(x, t) ∈ ET .
Theorem 3.2 and Lemma 3.5 give
Theorem 3.3. Let w ∈ C 2,1 (ET ) satisfy (H). Then,
sup w ≤ CN (diam(E))
N
N +1
ET
Z Z
where
CN = 2
2
wt det(−D w) dxdt
ET
N +1
ωN
In particular, CN is independent of T .
N1+1
N1+1
,
(3.25)
.
(3.26)
Corollary 3.3 By approximation, Theorem 3.3 remains valid for functions
w satisfying
w ∈ W 1,N +1 (0, T ; LN +1(E)) ∩ LN +1 (0, T ; Wo2,N +1(E)),
and the conditions in (H) are meant in a weak sense.
Finally, we wish to find an embedding theorem of the type of Theorem 3.3 for
functions that are not necessarily concave in the space variables. Namely, we
assume

2,1

u ∈ C (ET ),
u(x, t) = 0 ∀ x ∈ ∂E and ∀ t ∈ (0, T ),


u(x, 0) = 0 ∀ x ∈ E,
and recall that we have denoted with ξu the increasing concave hull of u.
By virtue of Propositions 3.1 and 3.2, and Corollary 3.3, estimate (3.25)
is valid for ξu . Upon observing that
sup u = sup ξu ,
ET
ET
we have the following.
Theorem 3.4. Let u ∈ C 2,1 (ET ), such that u vanishes on the parabolic
boundary of ET , i.e., ∂ET \{t = T }. Then,
sup u ≤ CN (diam(E))
ET
N
N +1
(Z Z
where CN is given by (3.26).
2
ut det(−D u) dxdt
[u=ξu ]∩ET
) N1+1
,
(3.27)
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
544
Corollary 3.4 By approximation, Theorem 3.4 remains valid for functions
u ∈ W 1,N +1 (0, T ; LN +1(E)) ∩ LN +1 (0, T ; Wo2,N +1 (E)),
u(x, 0) = 0 (in the sense of traces) ,
provided that the domain of integration in (3.27) is replaced by ET .
3.5 Estimates of the Supremum of a Function
It is straightforward to see that
WN2,1+1 (ET ) ≡ W 1,N +1 (0, T ; LN +1(E)) ∩ LN +1 (0, T ; W 2,N +1 (E));
we let
W̊N2,1+1 (ET ) ≡ {u ∈ WN2,1+1 (ET ) : u(·, t) = 0 on ∂E ∀ t ∈ [0, T ]},
2,1
2,1
Wo,N
+1 (ET ) ≡ {u ∈ WN +1 (ET ) : u = 0 on ΓT = ∂ET \{t = T }}.
In this section, we give estimates of the supremum of a function u ∈
WN2,1+1 (ET ) in terms of a parabolic operator applied to it.
For λo > 0 we denote with
A(λo ) all the positive definite, N × N symmetric matrices with
measurable entries (aij (x, t)), (x, t) ∈ ET , i, j = 1, 2, . . . , N,
(3.28)
whose eigenvalues are larger than, or equal to λo .
For u ∈ C 2,1 (ET ), let
Lo = aij (x, t)uxi xj ,
L = aij (x, t)uxi xj + bi (x, t)uxi + c(x, t)u.
(3.29)
(3.30)
Throughout this section, (x, t) → u(x, t) is a given function satisfying
u ∈ C 2,1 (ĒT ),
u
ΓT
≤ 0,
(3.31)
where we recall that ΓT is the parabolic boundary of ET .
Denote with ξu+ the smallest, non-negative function in ET , concave in x
for all t ∈ [0, T ], nondecreasing in t, and which lies above u. This definition is
analogous to that of the increasing concave hull of u (see Section 3.2), except
that now we allow u to be negative on ΓT , and force the increasing concave
hull to vanish on the parabolic boundary of ET .
It is apparent that all the results of the previous sections apply to the
present situation upon observing that
sup u+ = sup ξu+ .
ET
ET
3 The Aleksandrov Maximum Principle
545
Lemma 3.9 Let u satisfy (3.31) and let E be convex. Then, for every matrix
(aij ) ∈ A(λo )
sup u ≤ γN
ET
where γN =
diam(E)
λo
2
N +1
NN+1 "Z Z
+
[u=ξu
]∩ET
N +1
ωN
N1+1
(ut −
+1
Lo (u))N
+
dxdt
# N1+1
,
.
Proof. Since (−D2 u) is symmetric, there exists a unitary matrix X that diagonalizes it, that is,


−C1 (u) . . .
0


..
..
..
X · (−D2 u) · Xt = 

.
.
.
0
. . . −CN (u)
where Ci (u), i = 1, . . . , N are the eigenvalues of (D2 u). At points (x, t) of
concavity [u = ξ + u], Ci (u) ≤ 0 for i = 1, . . . , N and ut ≥ 0.
If (aij (x, t)) = A ∈ A(λo ), we have
ut − aij uxi xj = ut + tr A(−D2 u)



−C1 (u) . . .
0





.
.
t
.
.
.
.
= ut + tr XAX 

.
.
.




0
. . . −CN (u)
= ut −
N
X
λi Ci (u),
i=1
where λi are the diagonal elements of XAXt . We also have (see Section 1.2)
λi ≥ λo ,
i = 1, . . . , N.
Therefore, since the geometric mean of (N + 1) non-negative numbers is less
than their arithmetic mean, we have
[(ut −Lo (u))+ (x, t)]
N +1
"
= ut −
N
X
#N +1
λi Ci (u)
i=1
ut + λo (−C1 (u)) + · · · + λo (−CN (u))
≥
N +1
N
≥ λo ut |C1 (u) · · · · · CN (u)| (N + 1)N +1
N +1
= λN
ut det(−D2 u).
o (N + 1)
From Theorem 3.4 and (3.26) we find
N +1
(N + 1)N +1
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
546
sup u ≤ γN
ET
and γN =
2
N +1
NN+1 (Z Z
diam(E)
λo
N1+1
N +1
ωN
+
[u=ξu
]∩ET
(ut −
+1
Lo (u))N
dxdt
+
) N1+1
(3.32)
.
Next, we derive an estimate like (3.32) with Lo (u) replaced by the full operator
L(u) in (3.30).
We assume that the origin 0 ∈ E and consider the ball BR (0) of radius
R = k(diam(E)),
k ≥ 1,
and the cylinder
QTR = Bk diam(E) (0) × [0, T ].
First, we extend u with ũ ∈ C 2,1 (Q̄TR ) nonpositive outside ET , and then we
work with ũ defined in the whole QTR and ũ ≤ 0 on the parabolic boundary of
QTR .
If ξũ+ is the nondecreasing, non-negative concave hull of ũ, the contact set
[ũ = ξũ+ ] occurs within ET . Therefore, for (xo , to ) ∈ [ũ = ξũ+ ]
|ũxi | = |uxi | ≤
u(xo , to )
u(xo , to )
≤
.
dist(xo , ∂BR )
(k − 1) diam(E)
(3.33)
Now consider the function
v(x, t) = ũ(x, t)e−Ct ,
(x, t) ∈ QTR ,
where C is a positive constant to be chosen.
From Lemma 3.9 we have that for every matrix (aij (x, t)) ∈ Aλo
sup v ≤ γN
QT
R
×
(Z Z
2k diam(E)
λo
[v=ξv+ ]∩QT
R
NN+1
−Ct
N +1
e
[ut − Lo (u) − Cu]+
dxdt
) N1+1
(3.34)
.
Next, write
e−Ct (ut − Lo (u) − Cu)+ = e−Ct (ut − L(u) + bi uxi − c(x, t)u − Cu)+ (3.35)
and observe that for (x, t) ∈ [v = ξv+ ] by (3.33) we have
e−Ct (ut −Lo (u))+ ≤ e−Ct (ut − L(u))+
Bo
+
+ c− (x, t) − C u(x, t)e−Ct ,
(k − 1) diam(E)
(3.36)
3 The Aleksandrov Maximum Principle
547
where with Bo we have denoted an upper bound for bi and c− (x, t) =
max{0, −c(x, t)}.
We deduce that if the negative part of c(x, t) is bounded by a constant
B1 , then we can select C so that the term in brackets on the right-hand side
of (3.36) is nonpositive. Taking k = 2 we conclude
Lemma 3.10 Let u ∈ C 2,1 (ET ), u
N
X
i=1
ΓT
≤ 0, and E convex. Assume
kc− k∞;ET ≤ B1
kbi k∞;ET ≤ Bo ,
for two given constants Bo and B1 . Then,
NN+1
B1
R
sup u+ ≤ γ̄N exp T Bo +
R
λ
o
ET
Z Z
N1+1
+1
×
(ut − L(u))N
dxdt
+
(3.37)
ET
N
for every R ≥ diam(E), where γ̄N = 4 N +1 γN .
Remark 3.2 If c(x, t) > 0 and
N
X
i=1
kbi k∞;ET ≤
1
R c(x, t),
2
(x, t) ∈ ET ,
then in (3.36) we might take C = 0 and conclude that (3.37) holds without
the exponential term. In particular, the coefficient of the integral does not
depend on T .
Assume now that b(x, t) = (b1 (x, t), . . . , bN (x, t)) is not bounded, but
N ZZ
X
i=1
ET
|bi |N +1 dxdt
! N1+1
≤ B,
for a given constant B. Then an estimate similar to (3.37) can be obtained
relying on the freedom given by the parameter k ≥ 1 in (3.35)–(3.36).
Indeed, in (3.35), taking C = kc− k∞;ET and using this in (3.34), by (3.33)
we obtain for all R ≥ diam(E)
N Z Z
N1+1
kR N +1
+1
(ut − L(u))N
dxdt
+
λo
ET
N
kR N +1
1
B sup v.
+ γ̄¯N
λo
(k − 1)R ET
sup v ≤γ̄N
ET
548
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Therefore, if k is so large that
N
kR N +1
1
1
γ̄¯N
B≤
λo
(k − 1)R
2
we obtain the following.
Lemma 3.11 Let u ∈ C 2,1 (ET ), u
ΓT
≤ 0, assume that E is convex, and
N ZZ
X
−
kc k∞;ET ≤ B1 ,
i=1
ET
|bi |
N +1
dxdt
! N1+1
≤ B,
for two given constants B1 and B.
Then there exists a constant γ = γ(N, B, R, λo ) such that
Z Z
N1+1
N +1
B1 T
sup u ≤ γe
.
(ut − L(u))+ dxdt
ET
ET
We generalize these results to the case when u
ΓT
is not required to be non-
positive, E is not required to be convex, and u is only in WN2,1+1 (ET ).
In a precise way we have the following.
Theorem 3.5. Let E be a bounded, open set in RN with boundary ∂E of class
C 2 and let u ∈ WN2,1+1 (ET ). For any matrix (aij (x, t)) ∈ A(λo ), any vector
b(x, t) and scalar c(x, t) such that
kc− k∞;ET ≤ B1 ,
kbk∞;ET ≤ Bo ,
for two given constants Bo , B1 , consider the operator
L(u) = aij (x, t)uxi xj + bi (x, t)uxi − c(x, t)u.
Then, for every R ≥ diam(E), ∀ (xo , to ) ∈ ET
NN+1
R
B1 (to −s)
u(xo , to ) ≤ ke
u+ k∞;Γto + γ̄N
·
λo
(Z Z
) N1+1
iN +1
h
Bo
[B1 + diam(E)
](to −t)
·
dxdt
e
(ut − L(u))+
,
Eto ∩Et+o
where
Eto ≡ E × (0, to ),
Et+o
Γto = ∂Eto \{t = to },
= {(x, t) ∈ Eto : u(x, t) > keB1 (t−s) u+ k∞;Γto },
and
γ̄N = 4
N
N +1
1
N +1
N +1
ωN
N1+1
.
(3.38)
3 The Aleksandrov Maximum Principle
549
Proof. First, we observe that u is not required to be the solution of a partial differential equation, and therefore, (aij (x, t)) ∈ A(λo ), b(x, t), c(x, t) ∈
L∞ (ET ) being given, and u ∈ WN2,1+1 (ET ) being fixed, the quantity (ut −
L(u))+ can be approximated a.e. by similar quantities, where
aij (x, t), b(x, t), c(x, t) ∈ C ∞ (RN +1 ), i, j = 1, . . . , N,
u ∈ C ∞ (RN +1 ).
Let to ∈ (0, T ] be fixed, and let k > 0 be a constant to be determined. Consider
the function
v(x, t) = e−kt u(x, t) − e(B1 −k)t ke−B1 s u+ k∞;Γto ,
(3.39)
where
ke−B1 s u+ k∞;Γto =
sup
e−B1 s u+ (y, s).
y∈∂E; 0<s≤to
y∈E×{0}
By direct calculation
vt − L(v) =e−kt (ut − L(u)) − kv
+ [−B1 − c(x, t)]e(B1 −k)t ke−B1 s u+ k∞;Γto .
By our choice of the constant B1
vt − L(v) ≤ e−kt (ut − L(u)) − kv
v
Γto
in Eto ,
(3.40)
≤ 0.
(3.41)
o
Since v ∈ C ∞ (RN +1 ), we may extend v in Qt2R
, R ≥ diam(E) in such a way
that
≤ 0,
v
to
∂p Q2R
to
to
o
where ∂p Q2R
\{t = to } is the parabolic boundary of Qt2R
≡ ∂Q2R
.
to
By Lemma 3.9 with ET replaced by Q2R
sup v ≤ γN
o
Qt2R
with γN =
2
N +1
2R
λo
NN+1 (Z Z
o
[v=ξv+ ]∩Qt2R
(vt −
+1
Lo (v))N
+
dxdt
) N1+1
,
(3.42)
1
N +1 N +1
.
ωN
+
ξv ] occurs
The set [v =
within Eto and on such a set v > 0. Moreover,
∀(x, t) ∈ [v = ξv+ ], arguing as in the proof of Lemma 3.10,
(vt − Lo (v))+ ≤ (vt − L(u))+ + (bi vxi − c(x, t)v)+
Bo
v.
≤ (vt − L(v))+ + B1 +
diam(E)
550
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Therefore, choosing k = B1 +
Bo
diam(E) ,
from (3.40) we find
(vt − Lo (v))+ χ[v=ξv+ ] ≤ e−kt (ut − L(u))+ χ[v>0] .
We replace this last estimate in (3.42) and use the definition (3.39) of v to
deduce
e−kto u(xo , to ) − keB1 (to −s) u+ k∞;Γto
) N1+1
NN+1 (Z Z
−kt
N +1
R
≤ γ̄N
e (ut − L(u))+
dxdt
,
λo
Eto ∩Et+
o
and this obviously implies (3.38).
The theorem can be generalized in a straightforward way to include the case
when bi ∈ LN +1 (ET ), i = 1, . . . , N .
o
First we work with the cylinder QtnR
, n ≥ 2, and v extended to be nonpositive outside Eto . Then we write (3.42) with 2R replaced by nR and estimate
on [v = ξv+ ]
(vt − Lo (v))+ ≤(vt − L(v))+ + bi vxi − cv
PN
i=1 |bi |
v
≤e−kt (ut − L(u))+ +
n diam(E)
+ [−k − c(x, t)]v.
Choosing k ≥ kc− k∞;ET we find
) N1+1
N (Z Z
nR N +1
N +1
−B1 t
e
(ut − L(u))+ dxdt
sup v ≤γ̄N
o
λo
Eto
[v=ξv+ ]∩QtnR
N PN
nR N +1 i=1 kbi kN +1;ET
¯
sup v.
+ γ̄N
λo
nR
Eto
Now choose n so large that
γ̄¯N
− NN
+1
λo
(nR)
1
N
N
X
i=1
kbi kN +1;ET ≤
1
,
2
to conclude.
Theorem 3.6. Let E be a bounded, open set in RN with C 2 boundary ∂E and
let u ∈ WN2,1+1 (ET ). Let (aij (x, t)) ∈ A(λo ), and b(x, t) and c(x, t) a vector
field and a scalar satisfying
bi ∈ LN +1 (ET ), i = 1, . . . , N,
kc− k∞;ET ≤ B1 , B1 given.
3 The Aleksandrov Maximum Principle
551
Then for every operator L(·) of the type
L(u) = aij uxi xj + bi uxi − cu,
for every (xo , to ) ∈ ET and every R ≥ diam(E) there exists a constant γ =
γ(N, diam(E), kbi kN +1;ET , B1 ) such that
u(xo , to ) ≤ keB1 (to −s) u+ k∞;Γto
# N1+1
NN+1 "Z Z
iN +1
h
R
+γ
eB1 (to −t) (ut − L(u))+
dxdt
,
λo
ETo ∩Et+
(3.43)
o
where Et+o
≡ (x, t) ∈ Eto : u(x, t) > keB1 (t−s) u+ k∞;Γto .
3.6 Maximum Principle for Nonlinear Operators
Following the discussion about nonlinear equations of Section 1, let
F [u] = F (x, t, u, Du, D2 u),
2
where F : ET ×R×RN ×RN → R is measurable in (x, t) for each (u, Du, D2 u),
and a.e. differentiable in u, uxi , uxi xj . Let
(aij ) = (Fuxi xj ),
bi = Fuxi ,
c = −Fu .
Set
aij [u] = aij (x, t, u, Du, D2 u),
and define analogously bi [u] and c[u].
Theorem 3.7 (Comparison Principle). Let u, v ∈ WN2,1+1 (ET ) satisfy
u≤v
on ∂p ET ≡ ΓT ≡ ∂ET \{t = T },
ut − F [u] ≤ vt − F [v]
in ET .
Moreover, assume that
Z 1
(ãij (x, t)) ≡
aij [su + (1 − s)v] ds ∈ A(λo ),
b̃i (x, t) ≡
c̃(x, t) =
Z
Z
(3.45)
(3.46)
0
1
0
1
0
(3.44)
bi [su + (1 − s)v] ds ∈ L∞ (ET ) and kb̃i k∞;ET ≤ Bo , (3.47)
c[su + (1 − s)v] ds satisfies kc̃− k∞;ET ≤ B1 ,
(3.48)
where Bo and B1 are given non-negative constants. Then, u(x, t) ≤ v(x, t) for
any (x, t) ∈ ĒT .
552
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Requirements (3.46)–(3.48) are formulated in terms of u and v given. If, in
particular, they hold true for all u, v ∈ WN2,1+1 (ET ), then we have the following
uniqueness result.
Corollary 3.5 Suppose (3.46)–(3.48) in Theorem 3.7 hold uniformly in u, v ∈
WN2,1+1 (ET ). Then, the Cauchy–Dirichlet problem

ut = F [u] in ET ,
u
ΓT
= ψ ∈ C(ΓT ),
has at most one solution in WN2,1+1 (ET ).
Proof of Theorem 3.7. Setting w = u−v, we have w
in ET , where
ΓT
≤ 0 and wt − L̃(w) ≤ 0
L̃(w) = ãij (x, t)wxi xj + b̃i (x, t)wxi − c̃(x, t)w.
Therefore, the theorem follows from (3.38).
Since b̃i depends upon ws = su + (1 − s)v, it is desirable to allow b̃i not to be
bounded. However, since u, v ∈ WN2,1+1 (ET ), a reasonable requirement is
b̃i ∈ LN +1 (ET ),
i = 1, . . . , N.
(3.49)
Using (3.43) and Theorem 3.5, we have that Theorem 3.7 remains true, if
(3.47) is replaced by (3.49).
4 Local Estimates and the Harnack Inequality
In this section we deal with quasi-linear equations
ut − Q[u] = 0, in ET ⊂ RN +1 ,
Q[u] = Aij (x, t, u, Du)uxi xj + B(x, t, u, Du).
The structure conditions are specified in accordance with the various results.
We derive two basic estimates. The first one is a local estimate of a subsolution only in terms of its integral average over a portion of ET and is the
object of Section 4.1.
The second one, which is more involved, is the object of the remaining
sections. In an attempt to gain in clarity, we prove first the Harnack inequality
for linear equations, and then we generalize it to quasi-linear equations.
4 Local Estimates and the Harnack Inequality
553
4.1 A Local Maximum Principle
Take R > 0; we denote with QR the cylinder
QR ≡ BR (0) × (0, R2 ).
If (xo , to ) ∈ E × (0, T − R2 ), then (xo , to ) + QR ⊂ ET denotes the cylinder
BR (xo ) × (to , to + R2 ).
We assume that u ∈ C 2,1 (ET ) satisfies
ut − Aij (x, t, u, Du)uxi xj − B(x, t, u, Du) ≤ 0
in ET ,
(4.1)
i.e., it is a sub-solution. On the matrix (Aij ) and the scalar term B we impose
the following structure conditions:
•
If Λ = Λ(x, t, u, Du) is the largest eigenvalue of Aij (x, t, u, Du), then
1
Λ(x, t, u, Du) ≤ ϕo (x, t)[det(Aij (x, t, u, Du))] N +1 ,
(4.2)
where (x, t) → ϕo (x, t) is a given non-negative function satisfying
ϕo ∈ Lq (ET )
•
(4.3)
for some q > N + 1.
As for B,
1
|B(x, t, u, Du)| ≤ [ψo |Du| + ψ1 |u| + ψ2 ][det(Aij (x, t, u, Du))] N +1 , (4.4)
•
where (x, t) → ψi (x, t), i = 0, 1, 2, are given non-negative functions satisfying (4.3).
Moreover,
1
[det(Aij (x, t, u, Du))]− N +1 ∈ Lq (ET ),
•
q > N + 1.
(4.5)
Finally, as usual,
(Aij (x, t, u, Du)) is assumed positive semi-definite and symmetric.
(4.6)
Remark 4.1 Condition (4.5) is obviously satisfied if (Aij (x, t, u, Du)) =
(aij (x, t)) ∈ A(λo ), λo > 0, where A(λo ) is defined in (3.28). In such a case,
the operator in (4.1) is uniformly parabolic.
Theorem 4.1. Let (4.1)–(4.6) hold. Then for every (xo , to ) + QR ⊂ ET , for
every σ ∈ (0, 1), for every p > 0
sup
BσR (xo )×(to +(1−σ)R2 ,to +R2 )
u+ ≤C1
ZZ
up+ dxdt
(xo ,to )+QR
N
+ C2 R N +1 ,
! p1
554
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
where C1 , C2 are two constants depending upon p, σ, N , and the quantities
ZZ
ZZ
|ϕo |q dxdt,
|ψi |q dxdt, i = 0, 1, 2.
(xo ,to )+QR
(xo ,to )+QR
Here, we have denoted with
ZZ
|f | dxdt the integral average of f ∈
(xo ,to )+QR
L1 ((xo , to ) + QR ) over (xo , to ) + QR .
Remark 4.2 Since the theorem is local in nature, conditions (4.1)–(4.6) do
not have to hold throughout ET . The theorem remains valid in a subdomain
Q of ET if (4.1)–(4.6) hold true in Q.
Let q ∗ be defined by
1
1
1
+ ∗ =
,
q
q
N +1
and let p ∈ (0, 1) be fixed. Then, C1 depends on the data as follows.
First, set
o
n
1
Φo = ϕo + ψo + ψ1 + [det(Aij (x, t, u, Du))]− N +1 ;
then
C1 = γ(N, p, q, σ)
ZZ
|Φo |q dxdt
(xo ,to )+QR
! q1
q∗
p
,
where γ(N, p, q, σ) is a constant independent of ϕo , ψi , u.
The dependence of C2 on the data is
C2 = γ(N, p, q, σ)
Z Z
ET
|ψ2 |
N +1
dxdt
N1+1
,
for a constant γ independent of u.
Proof of Theorem 4.1. After a suitable translation, we may assume that
(xo , to ) = (0, 0). In QR consider the cutoff function
η(x, t) =
R2 − |x|2
R2
β
β
t2 ,
(4.7)
where β > 0 has to be chosen, and set
v = ηu.
If ξv is the increasing concave hull of v (see Section 3.2), on the upper contact
set [v = ξv ], v > 0 and |Dv| = |Dξv | ≤ v/(R − |x|). Therefore, from ηDu =
Dv − uDη, we have
4 Local Estimates and the Harnack Inequality
|Du| ≤ u
≤u
1
|Dη|
+
R − |x|
η
(R2 − |x|2 )β−1
R + |x|
+
2β
R2 − |x|2
R(R2 − |x|2 )β
≤ 2(1 + β)
R2
555
R
u.
− |x|2
Therefore, on [v = ξv ]
1
1
|Du| ≤ 2(1 + β)t 2 η − β u.
(4.8)
Moreover, on [v = ξv ], using (4.1) we have
vt − Aij (x, t, u, Du)vxi xj =η[ut − Aij (x, t, u, Du)uxi xj ]
− 2Aij uxi ηxj + u[ηt − Aij ηxi ηxj ]
≤ηB(x, t, u, Du) + 2Λ(x, t, u, Du)|Du||Dη|
"
2
β−1 β
R − |x|2
t2
β −1
ηt + 2βAij δij
+u
2
2
R
R2
#
2
β−2
R − |x|2
xi xj β
2
− 4β(β − 1)
Aij 4 t .
R2
R
The following estimates are obvious
2
β−2
R − |x|2
xi xj β
−4β(β − 1)
Aij 4 t 2 ≤ 0
R2
R
2
β
β−1
1
1
R − |x|2
t2
2βAij δij
≤ 2βΛt 2 η 1− β
2
2
R
R
2
β −1
β
η t ≤ η 1− β .
2
2
Moreover, using (4.8)
1
1
1
1
2
2Λ|Du||Dη| ≤ 4Λ(1 + β)t 2 η − β βt 2 η 1− β u ≤ Cβ Λη 1− β u.
Combining these estimates, we find
vt − Aij (x, t, u, Du)vxi xj ≤ηB(x, t, u, Du)
2
+ γβ (1 + Λ(x, t, u, Du))η 1− β u,
where γβ is a constant depending only upon β.
Next, we use the structure conditions (4.2) and (4.4) to obtain
1
η|B(x, t, u, Du)|[det(Aij (x, t, u, Du)]− N +1
1
1
2
≤ η[2ψo (1 + β)η − β t 2 u + ψ1 u + ψ2 ] ≤ ψ̄o η − β v + ηψ2 ,
where ψ̄o = 2ψo (1 + β) + ψ1 ∈ Lq (ET ), and
(4.9)
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
556
1
γβ (1 + Λ(x, t, u, Du))[det(Aij (x, t, u, Du))]− N +1
2
2
1
≤ γβ ϕo (x, t) + [det(Aij (x, t, u, Du))]− N +1 η − β v = ϕ̄o η − β v,
1
where ϕ̄o = γβ (ϕo (x, t) + [det(Aij (x, t, u, Du))]− N +1 ) ∈ Lq (ET ).
Combining these in (4.9), we find on [v = ξv ]
1
[det(Aij (x, t, u, Du))]− N +1 (vt − Aij (x, t, u, Du)vxi xj )+
2
≤ Φo (x, t)η − β v + η ψ2 ,
where Φo (x, t) = ψ̄o + ϕ̄o ∈ Lq (ET ).
Now we apply Lemma 3.1c of the Complements, to find
sup v+
QR
≤γN R
N
N +1
(Z Z
[v=ξv ]∩QR
N
≤γ̄N R N +1
Z Z
2
QR
N
+ γ̄¯N R N +1
+1
[det(Aij )]−1 (vt − Aij vxi xj )N
dxdt
+
Z Z
+1 − β
ΦN
(η v)N +1 dxdt
o
QR
η N +1 ψ2N +1 dxdt
) N1+1
N1+1
N1+1
(4.10)
.
We estimate the first integral on the right-hand side of (4.10) as follows. By
the previous definition of q ∗ , we have
Z Z
QR
N1+1
2 N +1
2
−β
+1
≤ kΦo kq;QR kη − β vkq∗ ;QR
v
ΦN
η
dxdt
o
and
kη
− β2
∗
vkqq∗ ;QR
=
=
=
if we choose β so that 1 −
2
β
ZZ
ZZ
ZZ
2q∗
β
vq
∗
(α+1−α)
dxdt
QR
η−
2q∗
β
∗
η q uq
∗
(1−α) αq∗
u
QR
(ηu)q
∗
(1−α) αq∗
u
QR
= α, i.e.,
β=
With this choice
η−
2
,
α
∀ α ∈ (0, 1).
dxdt,
dxdt
4 Local Estimates and the Harnack Inequality
557
1−α
sup v+
ku+ kαq∗ ;QR .
2
kη − β kq∗ ;QR ≤
QR
The second integral on the right-hand side of (4.10) is easily estimated by
recalling the definition of η in (4.7). We have
N
γ̄¯N R N +1
Z Z
QR
η N +1 ψ2N +1 dxdt
N1+1
N
≤ γ̃R N +1 +β ,
where
γ̃ = γ̄¯N kψ2 kN +1;ET .
Substituting these estimates in (4.10)
sup v+ ≤ γN R
N
N +1
QR
kΦo kq;QR ku+ kα
αq∗ ;QR
1−α
N
sup v+
+ γ̃R N +1 +β .
(4.11)
QR
By Young’s inequality
γN R
N
N +1
kΦo kq;QR ku+ kα
αq∗ ;QR
1−α
sup v+
QR
N +2
N +2
1
N
1
≤ sup v+ + γ(N, α)R N +1 α + αq + αq∗
2 QR
1 Z Z
αq
αq1∗
Z Z
q
αq∗
|Φo | dxdt
|u+ |
dxdt
.
·
QR
QR
By the definition of q ∗ , we have
N
1
R N +1 α +
N +2
N +2
αq + αq∗
Setting
γ(N, α)
N
1
= R N +1 α +
Z Z
QR
N +2
α
q
( q1 + q1∗ ) = R α2 = Rβ .
|Φo | dxdt
1
αq
= C̄1 ,
we obtain from (4.11) and (4.12)
sup v+ ≤ 2C̄1 R
QR
β
Z Z
QR
|u+ |
αq∗
dxdt
αq1∗
N
+ 2γ̃R N +1 +β .
For (x, t) ∈ QσR = BσR × ((1 − σ)R2 , R2 )
sup v+ ≥ η(x, t)u+ (x, t) ≥ (1 − σ)
QR
3β
2
Rβ u+ (x, t),
and this implies the theorem for the choice αq ∗ = p.
(4.12)
558
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
4.2 A Covering Lemma
In this section we will be working with cubes in RN +1 . For x = (x1 , . . . , xN )
let kxk = max |xi |, and consider the unit cube K1 = {x ∈ RN : kxk < 1}.
i=1,...,N
Denote with Q1 the cylinder Q1 ≡ K1 ×(0, 1), and within Q1 consider cylinders
of the type (xo , to ) + QR ≡ {kx − xo k < R} × (to , to + R2 ), i.e., we will assume
that (xo , to ) ∈ Q1 and that R is so small that (xo , to ) + QR ⊂ Q1 .
If (xo , to ) = (0, 0), we write (0, 0) + QR = QR and we obviously have
|(xo , to ) + QR | = |QR |.
We are given
(a) a measurable set E ⊂ Q1 ,
(b) numbers δ, η ∈ (0, 1),
and consider the collection B(δ) of those subcubes (xo , to )+QR of Q1 satisfying
|((xo , to ) + QR ) ∩ E| ≥ δ|QR |.
For each (xo , to ) + QR ∈ B(δ), we construct two boxes as follows
Q(1) ≡ {kx − xo k < 3R} × (to − 3R2 , to + 4R2 ) ∩ Q1 ,
4
Q(2) ≡ {kx − xo k < 3R} × to + R2 , to + 1 +
R2 ,
η
as depicted in Figure 4.4.
The construction of Q(1) is self explanatory. The box Q(2) lies on top of
(xo , to ) + QR and has length η4 R2 .
By definition, Q(1) is all contained in Q1 , but Q(2) might extend beyond
the ceiling of Q1 .
Remark 4.3 The length of Q(2) in the t-direction is
Q(1) ∪ Q(2) in the t-direction is
4 2
ηR .
4
4
4R2 + R2 = (1 + η) R2 ,
η
η
i.e., length(Q(1) ∪ Q(2) ) ≤ (1 + η) length(Q(2) ).
Using Q(1) and Q(2) as building blocks, define two open sets
[
D(i) =
Q(i) , i = 1, 2.
(xo ,to )+QR ∈B(δ)
The main object of this section is to prove the following facts.
Lemma 4.1 (Krylov and Safonov [147])
The length of
4 Local Estimates and the Harnack Inequality
559
4 R2 )
(xo ,to + 1+ η
Q(2)
(xo ,to +4R2 )
(xo ,to +R2 )
(xo ,to )+QR
4 R2
η
(xo ,to )
Q(1)
(xo ,to −3R2 )
7R2
Fig. 4.4
(i) |E\D(1) | = 0;
(ii) If |E| < δ|Q1 |, then |E| < δ|D(1) |;
(iii) |D(1) | ≤ (1 + η)|D(2) |.
Corollary 4.1 Let E be a measurable subset of Q1 and let δ, η ∈ (0, 1) be
given. Then,
(I) either D(1) ≡ Q1 ,
|E|
.
(II) or |D(2) | ≥ δ(1+η)
Proof of Lemma 4.1. We first consider (i). Let χ(E) be the characteristic
function of E. Since E is measurable, χ(E) is integrable, and by the Lebesgue
theorem
ZZ
1
χ(E) dxdt = 1
lim
R→0 |QR |
(xo ,to )+QR
for a.e. (xo , to ) ∈ E. Hence, except at most for a set of measure zero, we have
lim
R→0
|((xo , to ) + QR ) ∩ E|
= 1,
|QR |
and for R small enough |((xo , to ) + QR ) ∩ E| ≥ δ|QR |. It follows that almost
every point of E belongs to some (xo , to ) + QR ∈ B(δ).
Let us now deal with (ii). We represent the open set D(1) , up to a set of
measure zero, as the union of binary boxes of the form (xo , to )+QR as follows.
Partition Q1 with hyperplanes
560
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
t=
1
1
3
1
, t = , t = , xi = 0, xi = ± , i = 1, . . . , N.
4
2
4
2
We obtain in this way 22N +2 boxes each congruent to
1
1
(x, t) : 0 < t < , 0 < xi < , i = 1, . . . , N ,
4
2
and Q1 can be represented, up to a set of measure zero, as the union of such
boxes.
We call the collection of such boxes by C1 and define
n
o
Σ1 ≡ Collection of the binary boxes in C1 that are all contained in D(1) .
If no boxes in C1 are all contained in D(1) , we set
Σ1 = ∅.
Consider next all those boxes in C1 that are not in Σ1 , and partition each of
them into 22N +2 boxes, each congruent to
1
1
(x, t) : 0 < t < 2 , 0 < xi < 2 , i = 1, . . . , N .
4
2
Let us call C2 such a collection, and define
n
o
Σ2 ≡ Collection of the binary boxes in C2 that are all contained in D(1) .
If no boxes in C2 are all contained in D(1) , we set Σ2 to be the empty set.
Proceeding in this fashion, if Cn and Σn have been defined, we partition
the boxes in Cn into 22N +2 boxes, each congruent to
1
1
(x, t) : 0 < t < n , 0 < xi < n , i = 1, . . . , N ,
4
2
and define Cn+1 as the collection of such boxes. Set
Σn+1
o
n
≡ Collection of the binary boxes in Cn+1 that are all contained in D(1) ,
and if no boxes in Cn+1 are all contained in D(1) , we set Σn+1 = ∅.
We can immediately see that distinct boxes of Σn are disjoint, and that
elements of Σn are disjoint from elements of Σm if m 6= n. Moreover,
[ [
Q| = 0
|D(1) \
n≥1 Q∈Σn
4 Local Estimates and the Harnack Inequality
|D(1) | =
∞ X
X
n=1 Q∈Σn
|Q|
∞ X
X
|E| = |E ∩ D(1) | =
561
n=1 Q∈Σn
|E ∩ Q|.
(4.13)
To prove (ii) let us start by considering Σ1 . If there exists a box in Σ1 , say
Q̃, such that
|E ∩ Q̃| ≥ δ|Q̃|, i.e., Q̃ ∈ B(δ),
then the corresponding cube Q̃(1) , constructed as we did before with Q(1) ,
would cover all Q1 , and hence,
D(1) = Q1 .
We conclude that for all cubes Q ∈ Σ1
|E ∩ Q| < δ|Q|.
Next, consider Σn+1 , n ≥ 1. Suppose that there exists a box Q∗ in Σn+1 such
that
|E ∩ Q∗ | ≥ δ|Q∗ |.
(4.14)
The box Q∗ results from a partition of a cube in Cn , which is not all contained
in D(1) . Let us call Q◦ such a cube.
By (4.14), Q∗ ∈ B(δ) and the corresponding Q∗(1) constructed starting
from Q∗ as before, belongs to D(1) and covers Q◦ (by construction). Hence,
Q◦ ⊂ D(1) , which is a contradiction. We conclude that
|E ∩ Q| < δ|Q|,
∀ Q ∈ Σn , ∀ n ∈ N.
From (4.13) now it follows that
|E| < δ
∞ X
X
n=1 Q∈Σn
|Q| = δ|D(1) |
and (ii) of the lemma follows.
We conclude with the proof of (iii). For this, we need a preliminary fact.
Lemma 4.2 Let A be the set of all subintervals of (−∞, +∞), B ⊂ A and
let g : A → A be a set-valued function satisfying
|g(I)| ≤ (1 + η)|I|,
η > 0,
for all I ∈ A, where |g(I)| and |I| denote the one-dimensional Lebesgue measure of g(I) and I. Assume that
g(I1 ) ⊂ g(I2 ),
whenever I1 ⊂ I2 .
[
[
Then,
|
I∈B
g(I)| ≤ (1 + η)|
I∈B
I|.
562
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
S
Proof. The set {I : I ∈ B} is open and it can be represented as the union
of non-intersecting intervals In . Therefore,
[ [
[
|
g(I)| ≤ |
g(I)|
n I⊂In
I∈B
≤|
[
n
g(In )| ≤
X
n
(1 + η)|In | = (1 + η)|
[
I|.
I∈B
Proof of (iii). If χ(D(1) ∪ D(2) ) is the characteristic function of D(1) ∪ D(2) ,
by Fubini’s Theorem
ZZ
χ(D(1) ∪ D(2) ) dxdt
|D(1) ∪ D(2) | =
RN +1
Z Z +∞
χ(D(1) ∪ D(2) )(x, t) dt dx.
=
RN
−∞
Let B(x) with kxk < 1 be the set of one-dimensional intervals defined by
n
o
t : (t, x) ∈ Q(2) , Q ∈ B(δ)
and define for I ∈ B(x) the function g(I) as the result of expanding I of a
factor (1 + η) by keeping the right-hand point fixed. Then, obviously,
!
Z
Z
(1)
(2)
(1)
(2)
χ D ∪D
D ∪D
=
(x, t) dt dx
S
RN
≤
≤
Z
RN
Z
I∈B(x)
|
[
g(I)
g(I)| dx
I∈B(x)
(1 + η)|
RN
[
I| dx
I∈B(x)
= (1 + η)|D(2) |.
Consequently,
and the lemma is proved.
|D(1) | ≤ (1 + η)|D(2) |
4.3 Two Technical Lemmas
To simplify the presentation, we restrict ourselves to linear operators and
prove two lemmas for non-negative super-solutions of
ut − L(u) ≥ f
in ET
L(u) = aij (x, t)uxi xj − bi (x, t)uxi + c(x, t)u.
We assume u ∈ C 2,1 (ET ) and
(4.15)
4 Local Estimates and the Harnack Inequality
563
(aij (x, t)) is positive definite ∀ (x, t) ∈ ET and if λN (x, t) ≥ λN −1 (x, t) ≥
· · · ≥ λ1 (x, t) are the ordered eigenvalues, there exists λo > 0 such that
λi (x, t) ≥ λo
∀ (x, t) ∈ ET ,
∀ i = 1, . . . , N.
(4.16)
There exist constants Bo , B1 such that
N
X
i=1
kbi k∞;ET ≤ Bo ; kck∞;ET ≤ B1 .
f ∈ LN +1 (ET ).
(4.17)
(4.18)
Remark 4.4 Because of the local nature of our estimates, assumptions
(4.16)–(4.18) need not be true in the whole ET . If they are satisfied over
compact subsets K ⊂ ET , then our estimates will be valid within K.
The case of quasi-linear operators Q[u] (see the beginning of the Section) will
be treated later.
If (xo , to ) ∈ ET , we will be working with the box (xo , to ) + Qρ , where, for
ρ>0
Qρ ≡ Kρ × (0, ρ2 ) ≡ {x ∈ RN : max |xi | < ρ} × (0, ρ2 ),
1≤i≤N
(4.19)
and will assume ρ to be so small that (xo , to ) + Qρ ⊂ ET .
Lemma 4.3 Let k > 0 be fixed. For every σ ∈ (0, 1), there exist δ ∈ (0, 1) and
a constant γ depending only upon N, Bo , B1 , kf kN +1;ET and independent of
k, such that if
|[u > k] ∩ ((xo , to ) + Qρ )| ≡ |{(x, t) ∈ (xo , to ) + Qρ : u(x, t) > k}|
≥ δ|Qρ |
(4.20)
then ∀ (x, t) ∈ ((xo , to + (1 − σ)ρ2 ) + Qσρ
N
u(x, t) ≥ θk − γρ N +1 ,
where θ =
1 −B1
e
and (see Figure 4.5)
2
(xo , to + (1 − σ)ρ2 ) + Qσρ ≡ {kx − xo k < σρ} × (to + (1 − σ)ρ2 , to + ρ2 ).
Proof. Without loss of generality, we may assume (xo , to ) = (0, 0). Since u ≥
0, the function ṽ = ueB1 t satisfies
ṽt − aij (x, t)ṽxi xj + bi (x, t)ṽxi ≥ f eB1 t
and v = k − ṽ satisfies
(c = 0),
564
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
σρ
(xo ,to +(1−σ)ρ2 )
ρ
(xo ,to )
Fig. 4.5
vt − aij (x, t)vxi xj + bi (x, t)vxi ≤ −f eB1 t .
By Theorem 4.1 with p = 1
sup v+ ≤ C1
Qσρ
ZZ
N
v+ dxdt + C2 ρ N +1 ,
Qρ
where C1 , C2 are two constants depending
upon N , σ, Bo , B1 , kf kN +1;ET .
Now the set [v+ > 0]∩Qρ coincides with (x, t) ∈ Qρ : eB1 t u(x, t) < k , which
is included in the set {(x, t) ∈ Qρ : u(x, t) < k}. Therefore, for all (x, t) ∈ Qσρ ,
using the assumption (4.20)
N
k − eB1 t u(x, t) ≤ C1 k(1 − δ) + C2 ρ N +1 ,
that is,
N
u(x, t) ≥ k [1 − C1 (1 − δ)] e−B1 − C2 ρ N +1 kf kN +1;ET .
Choosing (1 − δ) =
1
2C1 ,
θ = 12 e−B1 , γ = C2 kf kN +1;ET , the lemma follows.
In the next lemma we assume that (4.15)–(4.19) continue to hold and in
addition
0 < λo ≤ λ1 (x, t) ≤ · · · ≤ λN (x, t) ≤ B,
∀ (x, t) ∈ ET ,
(4.21)
for a given constant B. Here, as in (4.16), λi (x, t) are the eigenvalues of
(aij (x, t)).
Lemma 4.4 (Krylov and Safonov [147]) Assume u ≥ 0 and that
u(x, to ) ≥ k,
∀ x ∈ Bερ (xo )
4 Local Estimates and the Harnack Inequality
565
for k > 0 fixed and some ε ∈ (0, 1). Then for every σ ∈ (0, 1) there exist
constants ξ ∈ (0, 1), m ≫ 1 depending only upon λo , Bo , B1 , B, N , σ, and
independent of k and ε, such that
N
u(x, t) ≥ kξεm − Cρ N +1 kf kN +1;ET ,
∀ (x, t) ∈ Qρ (σ),
where
Qρ (σ) ≡ B(1−σ)ρ (xo ) × (to + σρ2 , to + σ −1 ρ2 ).
Remark 4.5 Lemma 4.4 can be seen as a statement about the expansion of
positivity. Therefore, in the framework of parabolic equations in nondivergence
form, it is analogous to Proposition 10.1 of Chapter 12 given for parabolic
DeGiorgi classes.
Proof. Without loss of generality, we may assume that (xo , to ) = (0, 0), ρ = 1,
and k = 1.
1−σ2 2
ρ
σ
to +σ −1 ρ2
u(x,t)≥ξεm
Qρ (σ)
(1−σ)ρ
to +σρ2
(xo ,to )
u(x,to )≥1, x∈Bερ (xo )
2ερ
Fig. 4.6
1 2
σ .
2
∗ ∗
We first prove the lemma for points (x , t ) ∈ Q1 (σ) satisfying
Also, without loss of generality, we may assume ε2 =
|x∗ | < (1 − σ),
t∗ = σ −1
(ρ = 1).
Let such a point be fixed and consider the two sets
2
C ≡ (x, t) : 0 < t < σ −1 , kx − σtx∗ k <
2
D ≡ (x, t) : 0 < t < σ −1 , kx − σtx∗ k <
1 2
2
σ +ε
2
1 3
2
σ t+ε .
2
566
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
(x∗ ,t∗ )≡(x∗ ,σ −1 )
σ −1
Q1 (σ)
D
σ
(0,0) ε
C
ε+ 1
σ
2
Fig. 4.7
Obviously, we have D ⊂ C ⊂ Q1 (see Figure 4.7).
The parabolic boundary of D consists of two parts
1 3
−1
∗ 2
2
∂1 D ≡ (x, t) : 0 < t < σ , kx − σtx k = σ t + ε
2
∂2 D ≡ {(x, t) : t = 0, kxk < ε}.
Consider the following functions
x + σtx∗
z(x, t) = q
,
1 3
2
σ
t
+
ε
2
z : RN +1 → RN ,
ψ(x, t) = 1 − kz(x, t)k2
2 . 1
+
2
σ 3 t + ε2
n
,
where n is a positive constant to be chosen. Notice that by definition of ∂1 D
we have kz(x, t)k = 1 on ∂1 D and
(i) ψ(x, t) ≥ 0, (x, t) ∈ D,
(ii) ψ(x, t) = 0, (x, t) ∈ ∂1 D,
(iii) 0 ≤ ψ(x, 0) ≤ ε
−2n
,
(x, t) ∈ ∂2 D.
We choose the constant n so that
ψt − L(ψ) ≤ 0 in D.
By direct computation we have
(4.22)
4 Local Estimates and the Harnack Inequality
1 3
σ t + ε2
2
n+1
[ψt − L(ψ)]
nσ 3
1 3
σ t + ε2 c(x, t) −
2
2
∗
∗
(xi − σtxi ) xj − σtxj
− 8aij
1 3
2
2σ t + ε
n
− 2 kzk2 − 1 aij δij + 2bi (xi − σtx∗i )
o
σ3
+ (xi − σtx∗i ) x∗i +
kzk2 .
2
= 1 − kzk2
Set
567
2
nσ 3
1 3
σ t + ε2 c(x, t) −
2
2
∗
∗
(xi − σtxi ) xj − σtxj
A2 = −8aij (x, t)
1 3
2
2σ t + ε
σ3
kzk2 .
A3 = −2 kzk2 − 1 aij δij + 2bi (xi − σtx∗i ) + (xi − σtx∗i ) x∗i +
2
A1 = 1 − kzk2
2
We show that we can select n large enough so that
A1 + A2 + A3 < 0.
We estimate the Ai ’s by using (4.15)–(4.19) and (4.21). We have
A2 ≤ −8λo kzk2,
A3 ≤ 2 1 − kzk
2
"
#
12
21
1
1
σ3
2
2
B + 2Bo ε + σ
+ (1 − σ) ε + σ
+
2
2
2
≤ γ 1 − kzk2 ,
nσ 3
2
2 2
B1 σ −
A1 ≤ 1 − kzk
.
2
Therefore,
2
A1 + A2 + A3 = − (8λo + γ) kzk + γ + 1 − kzk
2 2
nσ 3
γ−
,
2
where γ is a constant depending only upon the data and independent of ε.
Consider the following two cases.
Case 1. 1 > kzk2 > γ/ (8λo + γ).
In such a case the sum of the first two terms is nonpositive and
nσ 3
2
γ−
A1 + A2 + A3 ≤ 1 − kzk
.
2
568
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Therefore, selecting n ≥ 2γ/σ 3 we have
A1 + A2 + A3 ≤ 0.
Case 2. kzk2 < γ/ (8λo + γ). Then,
A1 + A2 + A3 ≤ γ + 1 −
nσ 3
γ
γ−
8λo + γ
2
nσ 3
8λo
≤ 2γ −
≤0
2
8λo + γ
4γ 8λo + γ
for the choice n > 3
.
σ
8λo
Remark 4.6 The value of n depends upon σ and it deteriorates as the modulus of ellipticity λo tends to zero.
The function ψ is in C 2,1 (D̄) and it can be extended in a C 2,1 fashion with a
function ψe defined in all Q1 and such that ψ̃ ≤ 0 in Q1 \D.
Setting
w(x, t) = ε2n ψ̃(x, t) − u(x, t) ∈ C 2,1 (Q1 )
and recalling (4.22)-(iii), we have that w ≤ 0 on the parabolic boundary of Q1 .
+
Moreover, if ξw
is the positive increasing concave hull of w (see Section 3.5),
+
the contact set [ξw
= w] occurs within D, so that
wt − L(w) ≤ −f in D.
By Theorem 3.5 with ρ = 1, (xo , to ) ≡ x∗ , σ −1
i.e.,
− N
w x∗ , σ −1 ≤ γN λo N +1 eBo +B1 kf kN +1;Q1 ,
u x∗ , σ −1 ≥ ψ x∗ , σ −1 ε2n − Ckf kN +1;Q1 .
From the definition of ψ it follows that
ψ x∗ , σ −1 ≥ σ −2n 1 − z x∗ , σ −1
2
√ √
≥ σ −2n 2( 2 − 1),
since σ ∈ (0, 1), |x∗ | < (1 − σ), and ε2 < 21 σ 2 .
Finally, to prove the lemma for all points in Q1 (σ), let (x∗ , t∗ ) ∈ Q1 (σ),
and consider the comparison function
n
. 1 2t
2
2 2
+ε
,
σ
ψ̄(x, t) = 1 − kz̄(x, t)k +
2 t∗
r
.
t
1 2 t
+ ε2 .
σ
z̄(x, t) = x + ∗ x∗
t
2 t∗
Considerations in all analogous to those above, since t∗ ∈ σ, σ −1 , yield the
lemma for all (x∗ , t∗ ) ∈ Q1 (σ).
4 Local Estimates and the Harnack Inequality
569
4.4 The Harnack Inequality for Linear Equations
We will be dealing with linear operators
L(u) = aij (x, t)uxi xj − bi (x, t)uxi + c(x, t)u,
where u ∈ C 2,1 (ET ) and
(x, t) → aij (x, t) ∈ L∞ (ET ) ,
i, j = 1, . . . , N,
(4.23)
and if λ1 (x, t) ≤ λ2 (x, t) ≤ · · · ≤ λN (x, t) are the ordered eigenvalues of
(aij (x, t)), there exist 0 < λo ≤ Λ < ∞ such that
0 < λo ≤ λ1 (x, t) ≤ λ2 (x, t) ≤ · · · ≤ λN (x, t) < Λ for a.e. (x, t) ∈ ET .
There exist positive constants Bo , B1 such that
N
X
i=1
kbi k∞;ET ≤ Bo ,
kck∞;ET ≤ B1 .
(4.24)
We are also given a function f satisfying
f ∈ LN +1 (ET ) .
(4.25)
We will prove a weak Harnack inequality for non-negative super-solutions of
vt − L(v) = f,
v ∈ C 2,1 (ET ) ,
(4.26)
and a strong Harnack inequality for non-negative solutions of (4.26).
Let (xo , to ) ∈ ET and consider the following boxes

QR ≡ {kx − xo k < R} × to − R2 , to + R2


 o

QR ≡ {kx − xo k < R} × to − R2 , to
(4.27)
+
Q R
(σ) ≡ {kx − xo k < σR} × to + (1 − σ)R2 , to + R2


 −

QR (σ) ≡ {kx − xo k < σR} × to − (1 − σ)R2 , to ,
with σ ∈ (0, 1) (see Figure 4.8).
We assume R so small that Q4R ⊂ ET . Assume also that u is a supersolution (or a solution) of (4.26) in a neighborhood of QR , say for definiteness
Q4R (by a super-solution, we mean u that satisfies ut − L(u) ≥ f ).
Theorem 4.2 (The Weak Harnack Inequality).
satisfy
u ≥ 0 in Q4R ,
ut − L(u) ≥ f, a.e. in Q4R ,
Let u ∈ C 2,1 (Q4R )
570
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
to +R2
σR2
Q+
(σ)
R
to +(1−σ)R2
(xo ,to )
−
QR (σ)
to −(1−σ)R2
σR
to −R2
Fig. 4.8
and let (4.23)–(4.25) hold. Fix σ ∈ (0, 1) and construct the boxes Q±
R (σ) as in
(4.27). There exist positive constants p ∈ (0, 1), γo ∈ (0, 1), γ1 ≫ 1, depending
upon N , Bi , i = 0, 1, λo , Λ, σ, and independent of (xo , to ), R, u, f such that
inf u + γ1 R
Q+
R (σ)
N
N +1
kf kN +1;ET ≥ γo
ZZ
up dxdt
Qo
R
! p1
.
By combining Theorem 4.2 and Theorem 4.1 we have
Theorem 4.3 (The Strong Harnack Inequality). Let u ∈ C 2,1 (Q4R )
satisfy
u ≥ 0 in Q4R ,
ut − L(u) = f, a.e. in Q4R ,
and let (4.23)–(4.25) hold. Fix σ ∈ (0, 1) and construct the boxes Q±
R (σ) as in
(4.27).
There exist positive constants γo ∈ (0, 1), γ1 ≫ 1 depending upon N , Bi ,
i = 0, 1, λo , Λ, σ, and independent of (xo , to ) , R, u, f , such that
N
inf
u + γ1 kf kN +1;ET R N +1 ≥ γo sup u
+
QR (σ)
(4.28)
Q−
R (σ)
Remark 4.7 Inequality (4.28) could be formulated in a slightly more general
−
geometrical setting. In particular, Q+
R (σ) and QR (σ) could be replaced by
2
2
,
Q+
R (σ1 , θ1 ) ≡ {kx − xo k < σ1 R} × to + (1 − θ1 ) R , to + R
−
2
QR (σ2 , θ2 ) ≡ {kx − xo k < σ2 R} × to − (1 − θ2 ) R , to ,
σi , θi ∈ (0, 1), i = 1, 2,
provided the constants γo , γ1 are suitably modified.
Remark 4.8 If f = 0, we obtain the classical version of the Harnack inequality.
4 Local Estimates and the Harnack Inequality
571
Proof of Theorem 4.2 Without loss of generality we may assume (xo , to ) =
±
o
, QR
(σ).
(0, 0) and define accordingly the boxes QR , QR
Case i): Let k > 0 be fixed and consider the set
Eo ≡ {(x, t) ∈ QoR : u(x, t) > k} .
Let δ ∈ (0, 1) be the number claimed by Lemma 4.3 for the choice σ = 12 , and
define
B(δ) the collection of sub-cubes of QoR of the form (x̄, t̄) + Qρ ⊂ QoR ,
(x̄, t̄) ∈ QoR and Qρ ≡ {kxk < ρ} × 0, ρ2 ,
such that
|Eo ∩ ((x̄, t̄) + Qρ )| ≥ δ |Qρ | .
The number δ being fixed, motivated by Corollary 4.1 (II), we fix η so that
√
(4.29)
(1 + η)δ = δ = δo .
By Lemma 4.3 (with σ = 12 ), if (x̄, t̄) + Qρ ∈ B(δ), then
N
u(x, t) > θk − γρ N +1 kf k,
∀ (x, t) ∈ (x̄, t∗ )+Q 12 ρ , where t∗ = t̄+ 21 ρ2 and kf k = kf kN +1;ET ; in particular,
N
setting v(x, t) = u(x, t) + γρ N +1 kf k, we have
v (x, t∗ ) > θk,
in kx − x̄k <
1
ρ.
2
The function v satisfies a.e. in Q4R
N
vt − L(v) ≥ f + c(x, t) γρ N +1 kf k.
Therefore, applying Lemma 4.4 with ρ replaced 6ρ, ε =
(4.29)), we obtain
1
12 ,
σ = η (η fixed in
N
N
u(x, t) ≥ θξ12−m k − γρ N +1 f + c(x, t)γρ N +1 kf k ,
in the box
Q
(2)
(4.30)
4 2
2
≡ {kx − x̄k < 3ρ} × t̄ + ρ , t̄ + ρ ,
η
where ξ, m are the constants claimed by Lemma 4.4 for the indicated choice
of σ.
Without loss of generality, we may assume that R is so small that
γR
N
N +1
ZZ
(x̄,t̄)+Q3ρ
cN +1 (x, t) dxdt
! N1+1
≤ kf k,
572
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
so that from (4.30) it follows that
N
u(x, t) ≥ λk − 2γkf kR N +1 in Q(2) ,
where λ = θξ12−m .
Let
n
o
N
E1 ≡ (x, t) ∈ QR : u(x, t) > λk − 2γR N +1 kf k
and
o
n
N
E1 ≡ (x, t) ∈ QoR : u(x, t) > λk − 2γR N +1 kf k .
For each box (x̄, t̄) + Qρ ∈ B(δ) we construct Q(1) and Q(2) as in Section 4.2,
and define accordingly the sets D(1) , D(2) .
The previous arguments show that E1 ⊂ D(2) , and by Corollary 4.1 (see
(4.29))
|E1 | ≥ |Eo | /δo .
(4.31)
Now let
α=1−
p
δo ,
δ∗ =
p
δo ,
and consider the two sets E1 and E1 \E1 . Then, in view of (4.31), we must
have either
1
1−α
|Eo | =
|Eo | ,
(4.32)
|E1 | ≥
δo
δ∗
or
α
|E1 \E1 | >
|Eo | .
(4.33)
δo
Case ii): The case when (4.33) occurs.
In such a case there must be a box Q(2) ⊂ D(2) whose intersection with
t > δαo (|Eo | / |BR |)R2 , is not empty.
By construction, Q(2) is originated by a box (x̄, t̄) + Qρ ∈ B(δ) and the
corresponding Q(2) is defined by
4
{kx − x̄k < 3ρ} × t̄ + ρ2 , to + ρ2 ,
η
when −R2 ≤ to ≤ 0. Therefore,
to + ρ 2 +
i.e.,
ρ2 ≥
α |Eo | 2
4 2
ρ ≥
R ,
η
δo |QoR |
αη
|Eo |
(4 + η)δo |QoR |
R2 .
It follows that at the level t = 0, there exists a ball Bρ (x̄) of radius
4 Local Estimates and the Harnack Inequality
ρ=β
|Eo |
|QoR |
1/2
R,
β=
αη
(4 + η)δo
21
573
,
all contained in BR , such that
N
u(x, 0) ≥ λk − 2γkf kR N +1 .
N
Setting v(x, t) = u(x, t) + 2γR N +1 kf k, we have
N
vt − L(v) ≥ f + 2c(x, t)γR N +1 kf k
in Q4R ,
and
v(x, 0) ≥ λk
in {kx − x̄k < ρ} ⊂ BR .
By Lemma 4.4 applied to v, with ρ replaced by 3R and
1
ε=
3
αη
|Eo |
(4 + η)δo |QoR |
12
we obtain
,
σ=
1
,
3
N
v(x, t) ≥ ξεm λk − CR N +1 kf k
for all (x, t) ∈ Q+
R (σ), i.e.,
u(x, t) ≥ Co k
Co = 3
−m
m2
|Eo |
|QoR |
αη
(4 + η)δo
In particular,
inf
u ≥ Co k
+
QR (σ)
N
− C1 R N +1 kf k,
m2
|Eo |
|QoR |
ξλ,
m2
C1 = (2γ + C).
N
− C1 R N +1 kf k.
(4.34)
Case iii): The iteration process.
We now repeat the process by starting with the set E1 previously defined.
Proceeding as before, we find that either
inf
u ≥ Co k
+
QR (σ)
|E1 |
|QoR |
m2
N
− (1 + Co ) C1 R N +1 kf k
for the same constants Co , C1 as in (4.34), or
n
o
N
|E2 | = (x, t) ∈ QoR : u(x, t) > λ2 k − 2γ(1 + λ)R N +1 kf k
≥
1
1−α
|E1 | =
|E1 | .
δo
δ∗
574
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Proceeding in this fashion we define sets
n
o
N
Es ≡ (x, t) ∈ QoR : u(x, t) > λs k − 2γ (1 + λ + · · · + λs ) R N +1 kf k
and have the alternative of either
m
N
|Es | 2
s
− C1 1 + Co + Co2 + · · · + Cos R N +1 kf k, (4.35)
inf u ≥ Co λ k
o|
+
|Q
QR (σ)
R
or
|Es+1 |
−(s+1) |Eo |
> δ∗
.
o
|QR |
|QoR |
(4.36)
Choose s∗ from the equation
−(s∗ +1)
δ∗
|Eo |
= δ.
|QoR |
Without loss of generality, we may assume that s∗ is an integer. Then,
c̄
∗
logλ e4
|Eo |
, c̄ =
λs = C
(4.37)
o
|QR |
ln δ
for a constant C independent of k, u, R.
Therefore, if for some s = 1, 2, . . . , s∗ , (4.35) holds, since |Es | ≥ |Eo |, we
have
q
N
|Eo |
− C̄¯ R N +1 kf k,
inf
u
≥
C̄k
o|
|Q
Q+
(σ)
R
R
where
C̄ = Co C,
m
q = c̄ + ,
2
∞
X
C̄¯ = C1
Coi
i=1
!
.
Suppose now that (4.35) is not verified for any s = 1, 2, . . . , s∗ . Then, (4.36)
must hold for all s = 1, 2, . . . , s∗ and hence,
o
n
∗
N
|Es∗ | = (x, t) ∈ QoR : u(x, t) > λs k − γ̄R N +1 kf k ≥ δ |QoR | ,
P∞ i where γ̄ = 2γ
i=1 λ .
Then, by Lemma 4.3 (with σ = 12 )
∗
N
u(x, t) ≥ θλs k − 2γ̄(1 + θ)R N +1 kf k
in the box kxk < R2 × − 21 R2 , 0 . In particular, for kxk <
∗
N
u(x, 0) > θλs k − 2γ̄(1 + θ)R N +1 kf k.
N
Define v(x, t) = u(x, t) + 2γ̄(1 + θ)R N +1 kf k, which satisfies
R
2,
4 Local Estimates and the Harnack Inequality
575
N
vt − L(v) ≥ f + 2γ̄c(x, t)(1 + θ)R N +1 kf k
a.e. in Q4R , and
∗
v(x, 0) ≥ θλs k,
in kxk <
R
.
2
By Lemma 4.4 we deduce that
∗
N
u(x, t) ≥ Ko λs k − K1 R N +1 kf k
(4.38)
for all (x, t) ∈ Q+
R (σ), for two constants Ko , K1 depending only upon the
data.
From (4.38) and (4.37) it follows that
c̄
N
|Eo |
k − K1 R N +1 kf k, K̄ = Ko C.
inf
u
≥
K̄
o
+
|QR |
QR (σ)
Case iv): Proof of Theorem 4.2 concluded.
The previous arguments show that in either case there exist constants
o
n
Ao = min{K̄, C̄}, A1 = max C̄¯ ; K1 , q ≫ 1,
all depending upon the data and independent of u, R, f , such that
q
N
|Eo |
− A1 R N +1 kf k.
inf u ≥ Ao k
o|
|Q
Q+
(σ)
R
R
Set
D=
A−1
o
inf u + A1 R
N
N +1
Q+
R (σ)
!
kf k .
Then, recalling that Eo ≡ [u > k] ∩ QoR , we have
1
1
|[u > k]|
≤ k− q D q ,
|QoR |
1
∀ k > 0.
(4.39)
We multiply both sides by k 2q −1 and integrate with respect to k over R+ to
get
ZZ
Z ∞
1
1
1
|[u > k]|
2q dxdt =
dk
u
k 2q −1
o
|QoR |
|QoR |
QR
0
Z D
Z ∞
1
1
|[u > k]|
|[u > k]|
=
k 2q −1
dk
+
k 2q −1
dk
o
|QR |
|QoR |
0
D
Z ∞
1
1
1
1
k − q −1 dk = qD 2q ,
≤ 2qD 2q + D q
D
where we have used (4.39).
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
576
If
1
2q
= p ∈ (0, 1) we have
ZZ
p
u dxdt
Qo
R
! p1
≤q
2q
A−1
o
inf u + A1 R
Q+
R (σ)
N
N +1
!
kf k .
4.5 The Harnack Inequality for Quasi-Linear Equations
In this section we adapt and modify the arguments of Section 4.4 to obtain
the weak Harnack inequality for C 2,1 (ET ) non-negative super-solutions of
ut − Q[u] ≥ 0
in ET
Q[u] = Aij (x, t, u, Du)uxi xj + B(x, t, u, Du).
(4.40)
The structural assumptions on Aij : R2N +2 → RN ×N and B : R2N +2 → R are
more restrictive than the ones in Section 4.1. Namely, we assume the following.
The matrix (Aij (x, t, u, Du)) is positive definite in ET × RN +1 and if
λ1 (x, t, u, Du) ≤ λ2 (x, t, u, Du) ≤ · · · ≤ λN (x, t, u, Du) are the (ordered)
eigenvalues, there exist numbers 0 < λo ≤ Λo < ∞, such that
0 < λo ≤ λi (x, t, u, Du) ≤ Λo ,
i = 1, . . . , N.
B(x, t, u, Du) ≥ −γ∗ |Du| − γ∗ u − f (x, t),
(4.41)
(4.42)
where γ∗ is a non-negative given constant and f ∈ LN +1 (ET ).
For notational simplicity we set
fo = kf kN +1;ET .
With this notation, the weak Harnack inequality is stated exactly as in Theorem 4.2. By combining Theorem 4.2 and Theorem 4.1, valid for sub-solutions
of quasi-linear equations, we obtain the strong Harnack inequality stated exactly as in Theorem 4.3.
We have seen in the previous sections that Theorem 4.2 (the weak Harnack
inequality) is derived only by making use of the measure-theoretical lemma of
Krylov–Safonov (see Section 4.2) and the two technical lemmas of Section 4.3.
Therefore, to extend Theorem 4.2 to super-solutions of quasi-linear equations, it will be enough to prove results analogous to Lemmas 4.3 and 4.4.
As before, we will be working with boxes (xo , to ) + Qρ ⊂ ET , congruent to
Qρ ≡ Bρ × 0, ρ2 ≡ {|x| < ρ} × 0, ρ2 .
4 Local Estimates and the Harnack Inequality
577
Lemma 4.5 Let k > 0 be fixed. For every σ ∈ (0, 1) there exist δ ∈ (0, 1) and
γ̄ > 1 depending only upon N , γ∗ , λo , Λo , and independent of k, fo , ρ, such
that if
{(x, t) ∈ ((xo , to ) + Qρ ) : u(x, t) + fo > k} > δ |Qρ | ,
then
u(x, t) ≥
N
1
k − γ̄ρ N +1 fo ,
2
where Qσρ (xo , to ) ≡ xo , to + (1 − σ)ρ
Proof. Setting
2
∀ (x, t) ∈ Qσρ (xo , to ) ,
+ Qσρ .
N
v = k − u + fo ρ N +1 ,
we have
vt − Aij (x, t, u, Du) vxi xj ≤ B̃[v]
in ET ,
where, by (4.42)
N
B̃[v] = −B x, t, k + fo ρ N +1 − v, −Dv
N
≤ γ∗ |Dv| + γ∗ v + k + fo ρ N +1 + f.
By Theorem 4.1 applied with p = 1, if σ ∈ (0, 1)
ZZ
sup v+ ≤C1
v+ (x, t) dxdt
Qσρ (xo ,to )
(xo ,to )+Qρ
+ γρ
N
N +1
(4.43)
kf + fo + kkN +1;ET .
From Remark 4.2, in view of the hypotheses (4.41)–(4.42), it follows that
C1 = C1 (N ).
Now the set [v > 0] ∩ ((xo , to ) + Qρ ) coincides with the set [u + fo > k] ∩
((xo , to ) + Qρ ). Therefore, using the assumptions of the lemma in (4.43) we
find
N
N
N
sup v+ ≤ γδ k − fo ρ N +1 + 2γfoρ N +1 + γkρ N +1 ,
Qσρ (xo ,to )
i.e.,
N
N
u(x, t) ≥ k 1 − γ(1 − δ) − γρ N +1 − fo (1 + 2γ)ρ N +1 .
N
We assume ρ to be so small that ρ N +1 < 1 − δ and then choose (1 − δ) so
small that
N
1
1 − γ(1 − δ) − γρ N +1 ≥ 1 − 2γ(1 − δ) = .
2
Setting
γ̄ = 1 + 2γ
the lemma follows.
578
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Lemma 4.6 Let u ≥ 0 be a smooth, non-negative super-solution of (4.40) in
ET . Let (xo , to ) + Qρ ⊂ ET and assume that at the level t = to
in kx − xo k < ερ
u(x, to ) ≥ k,
for k > 0 fixed, ε ∈ (0, 1) given.
Then for every σ ∈ (0, 1), there exist constants ξ ∈ (0, 1), m ≫ 1, γ > 0
depending only upon N , λo , Λo , γ∗ , and independent of f , ρ, k, u, such that
N
u(x, t) ≥ kξεm − γρ N +1 fo ,
∀ (x, t) ∈ Qρ (σ),
where Qρ (σ) ≡ {kx − xo k < (1 − σ)ρ} × to + σρ2 , to + σ −1 ρ2 , provided
N
ρ N +1 < γ −1 εm .
(4.44)
Remark 4.9 By following step by step the proof in Section 4.4, it is easily
seen that the requirement (4.44) does not affect the argument, except for
minor changes.
Proof of Lemma 4.6 As before, we may prove the lemma for (xo , to ) = (0, 0)
and momentarily make a change of variable, so that ρ = 1.
We refer to Figure 4.6 introduced in the proof of Lemma 4.4, construct
the same domains C, D, and consider the same comparison function
n
2 . 1 3
ψ̃(x, t) = 1 − kz(x, t)k2 +
σ t + ε2
2
n
. 1
with z(x, t) = (x − σtx∗ )
σ 3 t + ε2 , and n to be chosen.
2
As before, the choice of such a ψ̃ is made to prove the lemma at the point
(x∗ , t∗ ) = (x∗ , σ −1 ), with |x∗ | < 1 − σ.
Define
ψ(x, t) = ε2n k ψ̃(x, t)
and observe that


ψ ≥ 0 in D,
ψ(x, t) = 0 (x, t) ∈ ∂1 D,


ψ(x, t) ≤ k (x, t) ∈ ∂2 D.
(4.45)
We choose the constant n so that
ψt − Lo (ψ) ≤ 0
in D,
(4.46)
where
Lo (ψ) = Aij (x, t, u, Du) ψxi xj .
Since Lo (·) is linear and (Aij (x, t, u, Du)) satisfies (4.41), by repeating the
calculations of Section 4.3 we see that there exists n = n (N, σ, λo , Λo ) such
that (4.46) holds in D.
4 Local Estimates and the Harnack Inequality
579
Set
w = ψ − u,
and observe that by (4.45) w ≤ 0 on ∂1 D and ∂2 D. We consider w extended
in Q2 in a C 2,1 fashion, so that w ≤ 0 outside D.
By calculation and (4.46)
wt − Lo (w) ≤ − ut − Aij (x, t, u, Du)uxi xj
≤ B(x, t, u, Du) ≤ γ(u + |Du|) + f
i
h
+
= w+ , u ≤ ψ and |Dw| ≤ ψ, i.e.,
on the set ξw
+
|Du| ≤ ψ + |Dψ|.
i
h
+
w
Therefore, on ξw
=
+
+
wt − Lo (w) ≤ γ(ψ + |Dψ|) + |f |.
By the maximum principle, Theorem 3.5
∗
∗
w (x , t ) ≤ γfo + γ
Z Z
D
|ψ + Dψ|
N +1
dxdt
N1+1
.
Estimating the last integral we have
1 − kzk2
2n
|Dψ| ≤ ε k
g n+1/2
where
g(t) =
Moreover,
n
ψ≤ε k
+
1 − kzk2
#N +1
dxdτ.
n
≤ (ε k)
g 1/2
+
,
1 3
σ t + ε2 .
2
1 − kzk2
g 1/2
+
.
Therefore,
I=
ZZ
D
(ψ + |Dψ|)N +1 dxdt
N +1
n
≤ (ε k)
Z
σ−1
0
Z
D(τ )
"
1 − kzk2
+
g 1/2 (τ )
Notice that g 1/2 (τ ) is the radius of the ball D(τ ) so that
n
N +1
I ≤ (ε k)
Z
0
σ−1
N
[g(τ )] 2 −
N +1
2
dτ
580
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
n
N +1
= (ε k)
Z
σ−1
0
1
dτ = γ(σ)(εn k)N +1 .
( 12 σ 3 t + ε2 )1/2
Therefore, the last integral in the estimate of w (x∗ , t∗ ) is bounded by γfo +
γεn k. Scaling back to the cube of dimension ρ, we find
u (x∗ , t∗ ) ≥
ε 2n
σ
N
N
k − γρ N +1 εn k − γρ N +1 fo .
The lemma follows if ρ is so small to satisfy (4.44), for the point (x∗ , t∗ ).
For the remaining points in Qρ (σ) the argument of Lemma 4.4 applies to
the present situation.
4.6 Local Hölder Continuity of Solutions
We let u ∈ WN2,1+1 (ET ) (see Section 3.5 for the definition of such a space) be
a solution of
ut − aij (x, t)uxi xj + bi (x, t)uxi = f,
(4.47)
where (aij (x, t)) and b(x, t) satisfy (4.23)–(4.24).
We show that the Harnack inequality proved in the previous sections implies an a priori estimate of the Hölder modulus of continuity of u over compact subsets of ET , depending only upon λo , Λ, Bo , N . The proof we provide
closely resembles the one given in Section 11 of Chapter 12. We repeat it here
to keep the two chapters self-contained.
We will treat later the case when the operator in (4.47) contains the term
c(x, t)u and when u solves a quasi-linear equation.
Let (xo , to ) ∈ ET and Ro so small that
QRo ≡ BR (xo ) × to − Ro2 , to
is all contained in ET . Set

osc u = ess sup u − ess inf u

ωo = ess
QRo
QRo
Q
Ro

ω1 = ess osc u
QRo /2
Lemma 4.7 There exist constants θ ∈ (0, 1) and γ > 1 depending only upon
λo , Λ, Bo , N and independent of Ro , u, f such that
N
ω1 ≤ θωo + γfo RoN +1 ,
fo = kf kN +1;ET .
Proof. Together with QRo consider the cylinders
7 2
5 2
+
−
Q ≡ QRo /2 , Q ≡ B Ro (xo ) × to − Ro , to − Ro
2
8
8
4 Local Estimates and the Harnack Inequality
581
(xo ,to )
Q+
to − 1
R2
2
to − 5
R2
8
Q−
to − 7
R2
8
Ro
Fig. 4.9
(see Figure 4.9). Define the functions
U = ess sup u − u;
V = u − ess inf u.
QRo
QRo
Then, obviously, U ≥ 0, V ≥ 0 and they both satisfy (4.40) (with f replaced
by −f for the V ). By the Harnack inequality (see Remark 4.7)
!
N
ess sup u − ess sup u ≥γo
ess sup u − ess −inf u
ess inf u − ess inf u ≥γo
ess sup u − ess inf u
QRo
QRo /2
QRo /2
QRo
Q
QRo
Q−
QRo
!
− γ̄fo RoN +1
N
− γ̄fo RoN +1
where γo ∈ (0, 1) and γ̄ depend only upon the data.
By addition
N
ωo − ω1 ≥ γo ωo − 2γ̄fo RoN +1 ,
and the lemma follows with θ = 1 − γo , γ = 2γ̄.
Next consider the sequence of radii
Rn =
Ro
,
2n
n = 0, 1, 2, . . .
and set
ωn = ess osc u.
QRn
N
Iteration of Lemma 4.7 yields ωn+1 ≤ θωn + γfo RnN +1 and
(4.48)
582
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
ωn ≤ ωo θn + γfo
n
X
N
N +1
θi Rn−i
.
(4.49)
i=0
From (4.48) by taking logarithms in the basis 1θ
1/ ln 1 2
Ro
θ
n = ln1/θ
,
Rn
and
θi =
Ri
Ro
It follows from (4.49) that
ωn ≤ ωo
1/ ln 1
θ
2
=
Ri
Ro
η = | ln θ|/ ln 2.
Rn
Ro
η
+ γfo
η
n X
Ri
i=0
If we let
η
Ro
,
N
N +1
.
Rn−i
α = min{η; N/N + 1},
n
n X
Ri
Ro
i=0
N
N +1
Rn−i
≤
n
X
(4.50)
(4.51)
2−iα 2−(n−i)α Roα
i=0
≤ γ(α)2−nα Roα = γ(α)Rnα ,
so that (4.50) implies
ωn ≤ A
Rn
Ro
α
,
n = 0, 1, 2, . . . ,
(4.52)
where α > 0 is defined by (4.51) and
A = (ωo + γkf kN +1;ET Roα ) .
Now if 0 < ρ < Ro , there exists n ∈ N such that Rn+1 ≤ ρ < Rn . Since
Rn = 2Rn+1 ≤ 2ρ
and
ess osc u ≤ ess osc u ≤ ωn ,
Qρ
QRn
it follows from (4.52)
ess osc u ≤ 2α A
Qρ
ρ
Ro
α
,
∀ 0 < ρ < Ro .
Let K be a compact subset of ET and set
d = min {1; dist (K, ∂p ET )} ,
where ∂p ET is the parabolic boundary of ET .
(4.53)
4 Local Estimates and the Harnack Inequality
583
Lemma 4.8 For every pair of points (x, t), (y, τ ) ∈ K
o
n
|u(x, t) − u(y, τ )| ≤ C |x − y|α + |t − τ |α/2
where
C = γ ∗ {kuk∞;ET + γkf kN +1;ET dα } d−α ,
(4.54)
where γ depends only upon N , α.
Proof. Suppose (x, t), (y, τ ) ∈ K are such that
|x − y| < d,
|t − τ | < d2 .
(4.55)
Then by (4.53)
|u(x, t) − u(y, τ )| ≤|u(x, t) − u(y, t)| + |u(y, t) − u(y, τ )|
≤2α+1 Ad−α |x − y|α + |t − τ |α/2
≤2α+1 (2kuk∞;ET + γkf kN +1;ET dα ) |x − y|α + |t − τ |α/2
o
n
≤C |x − y|α + |t − τ |α/2 .
If either one of (4.55) is violated, then obviously
|x − y|α + |t − τ |α/2
|u(x, t) − u(y, τ )| ≤ 2kuk∞;ET
dα
and the lemma follows.
4.7 Hölder Continuity of Solutions of Quasi-Linear Equations
We assume here that u ∈ WN2,1+1,loc (ET ) is a local solution of
ut − Q[u] = 0
in ET ,
(4.56)
where
Q(u] = Aij (x, t, u, Du)uxi xj − B(x, t, u, Du),
the matrix (Aij (x, t, u, Du)) satisfies (4.41), and B satisfies
|B(x, t, u, Du)| ≤ γ∗ (|u| + |Du|) + |f (x, t)|,
If (xo , to ) ∈ ET and
we let
with f ∈ LN +1 (ET ) .
QRo = BRo (xo ) × to − Ro2 , to ⊂ ET ,
µ+ = ess sup u,
QRo
µ− = ess inf u.
QRo
584
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
The functions
U (x, t) = µ+ − u(x, t),
V = u(x, t) − µ−
are non-negative and satisfy an equation such as (4.56). To be specific, U
satisfies
Ut − Aij (x, t, u, Du)Uxi xj = B(x, t, u, Du)
and
|B(x, t, u, Du)| ≤ γ∗ (|U | + |DU |) + γ∗ µ+ + |f |.
Therefore, the Harnack inequality for U and V holds with fo = kf kN +1;ET
replaced by
kf kN +1;ET + γ∗ µ+ |ET |.
Consequently, the proof of the Hölder continuity of solutions, given in Lemma 4.8, also remains unchanged, except for substituting C in (4.54) with
C ′ = γ ∗ kuk∞;K (1 + γ (kf kN +1;K + 1) dα ) d−α .
Problems and Complements
1c Introductory Material
1.1c Introduction
1.1.1c Linear Equations
We show how Lemma 1.1 implies that (1.2)(i) and (1.2)(ii) are equivalent.
Indeed, employing the row by column product of matrices, we have


ξ1


aij (x, t)ξi ξj = (ξ1 , . . . , ξN )(aij (x, t))  ... 
ξN
= ξXX−1 (aij (x, t))XX−1 ξt .
Since X−1 = Xt , setting η = (η1 , . . . , ηN ) = (ξ1 , . . . , ξN )X, we have X−1 ξ t =
η t , and hence,


λ1 (x, t) . . .
0
N

 0
X
λ2 (x, t) . . . 

λi (x, t)ηi2
aij (x, t)ξi ξj = η  .
ηt =

.
.
..
..

 ..
i=1
0
. . . λN (x, t)
1c Introductory Material
Now, ηi =
N
X
585
ξj xij and
j=1
|η|2 =
However,
N
X
i=1
N
X
ηi2 =
i=1
N
X
ξj ξl xij xil =
i,j,l=1
N
X
ξj ξl
j,l=1
N
X
xij xil .
i=1
xij xil is the (j, l)th element of the row by row product of X · X.
Since X−1 = Xt , this is the same as the (l, j)th element of the row by column
product X · X = IN ; therefore,
|η|2 =
N
X
j,l=1
ξj ξl δil = |ξ|2 ,
and it follows that (1.2)(i) and (1.2)(ii) are equivalent.
1.3c The Bellman–Dirichlet Equation
The parabolic Bellman–Dirichlet equation naturally arises when finding the
minimal cost in a stochastic control problem (or equivalently, finding the optimal strategy).
Let t → Xt ≡ {X1 (t), X2 (t), . . . , XN (t)} ∈ RN be a random process subject
to the dynamics
Z t
Z t
σ(αs , Xs ) dWs
(1.1c)
b(αs , Xs ) ds +
Xt = X +
0
0
where
•
•
•
•
•
X is the initial value of the process taken, for example, in an open set
E ⊂ RN ;
Wt is a N̄ -dimensional Wiener process (N̄ ∈ N);
t → αt is a control parameter;
(α, y) → b(α, y) is a given measurable function with values in RN ;
(α, y) → σ(α, y) is a N × N̄ matrix (σij (α, y)).
Let A be the set of all admissible controls. Choosing appropriately α ∈ A,
we can determine or “pilot” the evolution of the process Xt . The choice of αs
must depend on the value of the process up to time s, i.e.,
αs = αs (X[0,s] ),
X[0,s] ≡ {(t, Xt ) : 0 ≤ t ≤ s}.
Suppose that for the optimal choice of the control to yield a desired Xt there
is a cost functional
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
586
α
ρ =
Z
∞
f αt (Xt ) dt
0
for each individual trajectory Xt starting at X. The cost ρα is itself a random
variable depending on the initial value X of the random trajectory Xt . The
“average” cost of the selected and fixed strategy αs ∈ A is given by the
expectation of ρα , i.e.,
Z ∞
v α (X) = E
f αt (Xt ) dt .
0
We seek to minimize the average cost, i.e., we seek to find
Z ∞
α
αt
v(x) = inf v (x) = inf E
f (Xt ) dt .
α∈A
α∈A
0
The Bellman–Dirichlet principle states that
Z t
αs
v(x) = inf E
f (Xs ) ds + v(Xt ) .
α∈A
0
If x → u(x) ∈ C 2 (E) by Ito’s formula, the stochastic differential du(Xt ) is
given by
N
X
N
1 X
uxi (Xt ) dXt +
du(Xt ) =
uxi xj (Xt ) dXit dXjt
2
i=1
i,j=1
=J
(1)
(t) + J
(2)
(1.2c)
(t).
By using (1.1c)
J (1) (t) = uxi bi dt +
N
X
uxi σij dWtj .
i,j=1
Moreover, by using the standard rules of stochastic differentiation, i.e.,
dWti dWtj = 0 if i 6= j; (dWti )2 = dt; dWti dt = 0; (dt)2 = 0,
we find from (1.1c)–(1.2c)
J (2) = (σik (αt , Xt )σjk (αt , Xt )) dt = σ(αt , Xt ) · σ ∗ (αt , Xt )dt.
Carrying this in (1.2c) we find
h
i
du(Xt ) = Lσ(αt ,Xt ),b(αt ,Xt ) u(Xt ) dt + uxi (Xt )σij (αt , Xt )dWti ,
(1.3c)
where
Lσ(αt ,Xt ),b(αt ,Xt ) u(Xt ) = aij (αt , Xt )uxi xj (Xt ) + bi (αt , Xt )uxi (Xt ),
(1.4c)
1c Introductory Material
and
aij (αt , Xt ) =
N
X
k=1
587
σik (αt , Xt ) · σjk (αt , Xt ) = (σσ ∗ )ij .
We integrate (1.3c) over (0, t) and take the expectation to obtain
Z t
Lσs ,bs u(Xs ) ds ,
u(x) = E u(Xt ) −
(1.5c)
0
where we have used the fact (see Krylov [146], pages 293–297) that
Z t
E
uxi (Xs )σij (αs , Xs ) dWsj = 0.
0
We now combine (1.5c) and the definition of x → v(x) given before, to obtain
Z t
0 = inf E
f α (Xs ) ds + v(Xt ) − v(x)
α∈A
0
Z t
Z t
σs ,bs
α
= inf E
u(Xs ) ds .
f (Xs ) ds +
L
α∈A
0
0
Next, we divide by t and let t → 0 to have
inf [Lα v + f α ] = 0, where Lα = Lσ(α,x),b(α,x) .
α∈A
Let (G, F ) be a measure space and {Ft , t ≥ 0} an increasing family of σalgebras (i.e., a flow of σ-algebras satisfying Ft ⊂ F for any t ≥ 0).
Now suppose we start controlling the process Xt given by (1.1c) only after
the time t. Then the cost function is
Z ∞
=
ρα
f αs (Xs ) ds,
t
t
and the “average” cost is the conditional expectation of the random variable
ω → ρα
t (ω) given Ft , i.e.,
Z ∞
α
αs
v (t, Xt ) = E
f (Xs ) ds Ft .
t
We seek to minimize the average cost, i.e., we seek to find
Z ∞
v(x, t) = inf v α (x, t) = inf E
f αs (Xs ) ds Ft .
α∈A
α∈A
t
By the Bellman–Dirichlet principle
Z τ
αs
v(x, t) = inf E
f (Xs ) ds + v(τ, Xτ ) Ft .
α∈A
t
(1.6c)
588
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
Take now (x, t) → u(x, t) ∈ C 2,1 (ET ). Then for all the random processes
(t, Xt ) that remain in ET for any T almost surely, by Ito’s formula
u(t, Xt ) = uxi (t, Xt )dXit + ut (t, Xt )dt +
N
1 X
ux x (t, Xt )dXit dXjt .
2 i,j=1 i j
Proceeding as before, we have
∂
σ(αt ,Xt ),b(αt ,Xt )
u(t, Xt ) + L
u(t, Xt ) dt
u(t, Xt ) =
∂t
+ uxi (t, Xt )σij (αt , Xt )dWti ,
where Lσ,b is defined as in (1.4c), now with
aij = aij (t, Xt ),
bi = bi (αt , Xt , t).
If ct = c(t, Xt ) is a progressively measurable process, such that
Z t
setting ϕt =
cs ds, we have
Z
∞
0
ct dt < ∞,
0
d[e−ϕt u(t, Xt )] =u(t, Xt )de−ϕt + e−ϕt du(t, Xt ) + (de−ϕt )(du(t, Xt ))
=e−ϕt uxi (t, Xt )σij (αt , Xt )dWti
∂
−ϕt
σ,b
+e
+ L − ct u(t, Xt )dt.
∂t
We integrate over (t, τ ), ∀τ ≥ t and take the conditional expectation given Ft
to find
e−ϕτ u(τ, Xτ ) =e−ϕt u(t, Xt )
Z τ
∂
+E
e−ϕs
+ Lσs ,bs − cs u(s, Xs )ds Ft .
∂s
t
This formula holds for all u ∈ C 2,1 (ET ) for all the random processes (t, Xt ) ∈
ET almost surely. Also,
u(t, Xt ) =e−
Rτ
t
−E
cs ds
Z
t
u(τ, Xτ )
τ
e−
Rs
t
cη dη
(1.7c)
∂
+ Lσs ,bs − cs u(s, Xs )ds Ft .
∂s
Next, in (1.6c), redefine the minimum v(x, t) by setting ∀τ ≥ t
v(τ, Xτ ) = e−
Rτ
t
cs ds
u(τ, Xτ ).
(1.8c)
Suppose that the processes involved are so smooth that u ∈ C 2,1 (ET ). Then,
writing (1.6c) with the change of variable (1.8c), and using (1.7c), we find
3c The Aleksandrov Maximum Principle
inf E
α∈A
Z
Z
τ
f αs (Xs )ds + e−
t
Rτ
t
cs ds
u(τ, Xτ ) − u(t, Xt ) Ft
τ
589
=0
= inf E
f αs (Xs )ds
α∈A
t
Z τ
R
∂
− ts cη dη
σs ,bs
e
+
+L
− cs u(s, Xs )ds Ft .
∂s
t
Divide by (τ − t) and let τ ց t to obtain
∂u
+ inf [(Lα − cα + f α )u] = 0
∂t α∈A
in ET .
(1.9c)
Final value problems associated with (1.9c) give rise to the parabolic equation
(1.12).
3c The Aleksandrov Maximum Principle
3.5c Estimates of the Supremum of a Function
The result of Section 3.5 generalize to the case when the matrix (aij ) is mildly
degenerate. Rather than dwelling on the best possible cases, we limit ourselves
to the following situation
u ∈ C 2,1 (ET ),
(3.1c)
(x, t) → [det(aij (x, t))]
−1
1
∈ L (ET ),
ut − Lo (u) = ut − aij (x, t)uxi xj .
(3.2c)
(3.3c)
We state and prove here a version of Lemma 3.9, since all the other results
follow from it in a rather simple fashion.
Lemma 3.1c Let (aij (x, t)) be a positive semi-definite, N × N symmetric
matrix satisfying (3.2c), and let E be convex.
Then, for every u ∈ C 2,1 (ET ) such that u
≤ 0, we have
ΓT
sup u+ ≤ γN (diam(E))
N
N +1
+
[u=ξu
]∩ET
ET
where γN =
2
N +1
N +1
ωN
N +1
(Z Z
+1
(ut − Lo (u))N
+
dxdt
det(aij (x, t))
) N1+1
,
.
For the proof of Lemma 3.1c, we need the following facts.
Lemma 3.2c Let d ∈ N and let C be a d×d positive semi-definite, symmetric
matrix. Then,
d
tr(C)
det(C) ≤
d
590
13 PARABOLIC EQUATIONS IN NONDIVERGENCE FORM
d
Proof. If {λi }i=1
are the eigenvalues of C, we have λi ≥ 0, i = 1, . . . , d and
det(C) =
d
Y
i=1
λi ≤
Pd
i=1
d
λi
!d
=
tr(C)
d
d
.
Corollary 3.1c Let A and B be two positive semi-definite, symmetric matrices. Then,
d
tr(AB)
det(AB) ≤
d
1
1
Proof. Consider the matrix A 2 BA 2 ; it is obviously symmetric, and it is
1
1
also positive semidefinite. Indeed, for any x ∈ Rd we have (A 2 BA 2 x, x) =
1
1
(BA 2 x, A 2 x) ≥ 0. The result follows from Lemma 3.2c.
Proof of Lemma 3.1c. We start from Theorem 3.4 with cN = 2
h
N +1
ωN
i N1+1
and
estimate the integrand in (3.19) by making use of Corollary 3.1c as follows.
ut det(−D2 u) = [det(aij (x, t))]−1 [det(aij (x, t))]ut det(−D2 u)
ut 0
1
0
−1
= [det(aij (x, t))] det
det
0 −D2 u
0 aij (x, t)
N +1
1
u
0
1
0
tr t
≤ [det(aij (x, t))]−1
0 −D2 u
0 aij (x, t)
N +1
1
+1
[det(aij (x, t))]−1 (ut − Lo (u))N
.
=
+
(N + 1)N +1
We leave as an exercise the task of stating the facts analogous to the ones in
Theorems 3.5 and 3.6.
14
NAVIER–STOKES EQUATIONS
1 Navier–Stokes Equations in Dimensionless Form
Let E be a physical open set in R3 filled with a fluid of dynamic viscosity µ
and constant density ρ, whose infinitesimal ideal particles at x ∈ E at time t
move with velocity v = (v1 , v2 , v3 ) function of (x, t), and are acted upon by
the pressure (x, t) → p(x, t), and by possible external force densities fe (x, t),
per unit volume. Enforcing the local, pointwise conservation of momentum
along each of the ideal Lagrangian paths t → x(t), yields the Navier–Stokes
system as in (7.1)–(7.2) of the Preliminaries,
ρ vt + (v · ∇)v − µ∆x v + ∇p = fe
in E × (0, ∞).
(1.1)
divx v = 0
Here, ∆x , ∇ and divx denote the corresponding differential operation with
respect to the physical space variables x. If fe is conservative, such as for
example gravity, then fe = ∇F for some given potential F . In such a case
(1.1) can be written in homogeneous form by redefining p as (p − F ).
The various terms in (1.1) are written in terms of pre-chosen physical unit
length [L] and time [T ] and corresponding unit velocity [V ] = [L][T ]−1, unit
pressure [P ] = ρ[V ]2 , unit force density [F ] = ρ[V ][T ]−1 and unit dynamic
viscosity [µ] = ρ[V ][L]. They can be written in dimensionless form by introducing dimensionless space variables y = x[L]−1 and time τ = t[T ]−1 and
corresponding dimensionless velocities, pressures, and force densities
ṽ(y, τ ) =
v(y[L], τ [T ])
,
[V ]
p̃(y, τ ) =
p(y[L], τ [T ])
,
[P ]
f̃ =
fe (y[L], τ [T ])
.
[F ]
Denote by Ẽ the rescaled physical domain E expressed in terms of dimensionless coordinates. Then, dividing (1.1) by ρ and formally by [V ][T ]−1 , yields
ṽτ −
1
∆y ṽ + (ṽ · ∇y )ṽ + ∇y p̃ = f˜
Re
divy ṽ = 0
© Springer Nature Switzerland AG 2023
E. DiBenedetto, U. Gianazza, Partial Differential Equations,
Cornerstones, https://doi.org/10.1007/978-3-031-46618-2_15
in Ẽ,
(1.2)
591
592
14 NAVIER–STOKES EQUATIONS
where Re is the Reynolds number1 of the system corresponding to the units
[L] and [T ] and defined by
def ρ[V ][L]
Re =
µ
.
Although Re is dimensionless, its numerical value depends on the choice of
[L] and [T ]. Indeed, the dynamic viscosity µ for a fluid of density ρ, is experimentally determined in terms of some given units, say, for example, cm2 sec−1 .
Expressing them in terms of new units [L] and [T ] changes the numerical value
of Re. The coefficient of ∆y in (1.2) is the dimensionless kinematic viscosity
ν of the rescaled fluid.
This rescaling procedure is at the basis of predicting experimentally nonaccessible fluid flows in large-scale domains, such as air past an airfoil or water
past a vessel. The physical domains are rescaled to experimentally accessible
dimensions, such as laboratory water channels or wind tunnels, with a properly
redefined Reynolds number. Information provided by the dimensionless system
(1.2) is then rescaled back to the physical domain.
To simplify the symbolism we continue to denote by x, t, v, p, and f the
rescaled, dimensionless quantities and rewrite the Navier–Stokes system (1.1)
in the dimensionless domain E, for dimensionless times t > 0 in the form
vt − ν∆v + (v · ∇)v + ∇p = f
div v = 0
in E × (0, ∞),
(1.3)
with ν = Re−1 . Typically, one prescribes the velocity field vo = v(·, 0) at
time t = 0 and v(·, t) = g(·, t) on ∂E for t > 0 and seeks to solve (1.3) subject
to these data.
If E is a rigid container at rest with respect to an inertial system, then
∂E acts as a rigid wall and g = 0, by viscosity. This is the so-called no-slip
condition. The case g 6= 0 may occur when ∂E is itself in motion with respect
to an inertial system. In the applications, other types of boundary conditions
have been considered. We talk of a kinematic condition, when the normal
component of the velocity vanishes at the boundary, that is, the velocity v is
tangent to the boundary:
v(·, t) · n = 0
on ∂E
for t > 0, where n is the outward unit normal to the boundary ∂E. In 1823
Navier proposed a more general condition, namely the so-called Navier boundary condition, which, roughly speaking, states that the tangential component
of the velocity is proportional to the tangential stress at the boundary. We do
not consider these different boundary conditions in the following.
The system (1.3) is formal since, even by prescribing smooth initial and
boundary data vo and g and the forcing term f , one cannot a priori guarantee
that v and p are so regular as to give pointwise meaning to its various terms.
1
Osborne Reynolds, 1842–1912, Irish-born physicist, made important contributions to the understanding of fluid dynamics.
2 Steady-State Flow with Homogeneous Boundary Data
593
2 Steady-State Flow with Homogeneous Boundary Data
Let E be a bounded domain in R3 with boundary ∂E, and consider, formally,
the steady-state flow in E,
−ν∆v + (v · ∇)v + ∇p = f ,
div v = 0,
v
∂E
in E.
(2.1)
=0
Introduce the space of functions
V = ϕ ∈ Co∞ (E; R3 ) such that div ϕ = 0 in E ;
H = closure of V in the norm of L2 (E; R3 ) ;
V = closure of V in the norm of Wo1,2 (E; R3 ) .
(2.2)
Formally inner-multiply the first of (2.1) by ϕ ∈ V and integrate by parts in
E. Since v and ϕ are both divergence-free, obtain formally
Z
ν∇v : ∇ϕ − v · (v · ∇)ϕ − f · ϕ dx = 0,
(2.3)
E
where
∇v : ∇ϕ =
P3
j=1 ∇vj
· ∇ϕj .
Here, we have used the relation
Z
Z
ϕ · (ζ · ∇)ψdx
ψ · (ζ · ∇)ϕdx = −
E
E
valid for any triple of solenoidal vectors ϕ, ψ, ζ ∈ W 1,2 (E; R3 ) such that at
least one of them is in V . As a consequence,
Z
ϕ · (ζ · ∇)ϕdx = 0.
E
These calculus operations are repeatedly used without specific mention.
By the embedding Theorem 2.1 of Chapter 9 there exists a constant γ
independent of E and v, such that
kvk6 ≤ γk∇vk2
for all v ∈ V.
(2.4)
Therefore, for all such v
1
kvk2 ≤ γ|E| 3 k∇vk2
and
1
kvk4 ≤ γ|E| 12 k∇vk2 .
(2.5)
As a consequence,
γo kvkV ≤ k∇vk2 ≤ kvkV ,
where γo =
1
1
γ|E| 3 + 1
(2.6)
594
14 NAVIER–STOKES EQUATIONS
for all v ∈ V , where the rigorous definition of norm in V is given in (3.1).
By these inequalities, all terms in (2.3) are well defined for all ϕ ∈ V and
6
f ∈ L 5 (E; R3 ). Thus, having prescribed one such f , we define a weak solution
of (2.1) as a function v ∈ V satisfying (2.3) for all ϕ ∈ V . The homogeneous
boundary data on ∂E are taken in the sense of the membership v ∈ V . The
same membership guarantees that div v = 0 in the weak form
Z
v · ∇ϕdx = 0
for all ϕ ∈ C ∞ (E).
E
By this definition of solution, the choice ϕ = v is admissible in (2.3) yielding
the basic energy estimate
νk∇vk2 ≤ γkf k 56 ,
(2.7)
to be satisfied by any weak solution to (2.1), where γ is the constant of the
embedding of V into L6 (E; R3 ). Thus, if f = 0, then v = 0 is the only weak
solution of (2.1).
2.1 Uniqueness of Solutions to (2.1)
Let v1 and v2 be weak solutions to (2.1) corresponding to the choice of f ∈
6
L 5 (E; R3 ). Write (2.3) for v1 and v2 , subtract the expression thus obtained
def
and in the resulting integral identity choose ϕ = w = (v1 − v2 ), to obtain
Z
2
v1 · (w · ∇)w + w · (v2 · ∇)w dx
νk∇wk2 =
E
≤ kv1 k4 + kv2 k4 kwk4 k∇wk2 .
Since vj are solutions, combining (2.5) and (2.7) gives
1
kwk4 ≤ γ|E| 12 k∇wk2
Therefore,
k∇wk2 ≤ 2γ
kvj k4 ≤
and
γ 2
γ2
1
|E| 3 kf k 56 .
ν
5
|E| 12 kf k 56 k∇wk2 .
ν
If the coefficient on the right-hand side is less than 1, then w = 0 and the
solution is unique. Such a coefficient depends on the absolute constant γ of the
embedding V ⊂ L6 (E; R3 ), on the size of E, the viscosity ν, and the nature of
the forcing term f . Given E and f uniqueness holds if the Reynolds number
of the system is sufficiently small or equivalently if the fluid is sufficiently
viscous.
It should be noted that the definition of a weak solution does not depend
on the pressure p, which itself is an unknown to be found from (2.1).
3 Existence of Solutions to (2.1)
595
3 Existence of Solutions to (2.1)
The spaces H and V introduced in (2.2) are separable Hilbert spaces by the
inner products
Z
def
H ∋ (u, v) → hu, viH =
u · vdx
E
Z
(3.1)
def
∇u : ∇vdx.
V ∋ (u, v) → u, v V = hu, viH +
E
By (2.6) the inner product (·, ·)V is equivalent to
Z
P3
V ∋ (u, v) → hu, viV =
∇u : ∇vdx = j=1 h∇uj , ∇vj iH ,
E
6
which from now on we adopt. Having fixed f ∈ L 5 (E; R3 ) and v ∈ V , return
to (2.3) and consider the two linear maps
Z
def
f · ϕdx
V ∋ϕ→ =
ZE
def
V ∋ϕ→ =
v · (v · ∇)ϕdx.
E
By Hölder’s inequality and the embedding V ⊂ L6 (E; R3 )
Z
f · ϕdx ≤ γkf k 56 k∇ϕk2 .
E
Therefore, the first is a bounded linear functional on V . By the Riesz representation theorem, there exists a unique F ∈ V such that2
Z
V ∋ϕ→
f · ϕdx = hF, ϕiV .
E
Likewise, by the same embedding and (2.5)
Z
1
v · (v · ∇)ϕdx ≤ γ 2 |E| 6 k∇vk22 k∇ϕk2 .
E
Therefore, also the second map, for every fixed v ∈ V , is a bounded linear
functional in V . By the Riesz representation theorem, there exists a unique
B(v) ∈ V such that
Z
v · (v · ∇)ϕdx = hB(v), ϕiV .
V ∋ϕ→
E
With these identifications, the weak formulation (2.3) can be recast in the
form
2
Dunford and Schwartz [62], Chap. IV, § 4
596
14 NAVIER–STOKES EQUATIONS
V ∋ ϕ → νhv, ϕiV = hB(v) + F, ϕiV .
Equivalently, in functional form
v = B(v)
in V ∗
B(v) =
where
1
B(v) + F ,
ν
(3.2)
and V ∗ denotes the dual of V identified with V itself up to an isometric
isomorphism. Thus, existence of a weak solution to (2.1) in the sense of (2.3)
is equivalent to finding a fixed point of the map V ∋ v → B(v) ∈ V ∗ .
Lemma 3.1 The map B(·) : V → V ∗ is compact.
Proof. Since V and V ∗ are separable metric spaces, compactness is equivalent
to sequential compactness. Let K be a bounded subset of V , i.e., there exists a
constant C such that kvkV ≤ C for all v ∈ K. The image B(K) is pre-compact
in V ∗ if for every sequence {vn } ⊂ K there exists a subsequence {vn′ } ⊂ {vn }
such that {B(vn′ )} is a Cauchy sequence in the operator topology of V ∗ .
By the Rellich–Kondrachov compact embedding theorem (Theorem 2.2 of
Chapter 9), the embedding V ⊃ K ֒→ Lp (E; R3 ) is compact for all 1 ≤ p < 6.
Therefore, having fixed 1 ≤ p < 6, from every sequence {vn } ⊂ K one can
extract a subsequence {vn′ } ⊂ {vn }, which is Cauchy in the topology of
Lp (E; R3 ). Hence, to show that B(K) is pre-compact in V ∗ it suffices to show
that for every sequence {vn } ⊂ K, Cauchy in L4 (E; R3 ) the corresponding
sequence {B(vn )} is Cauchy in the operator topology of V ∗ . Having fixed one
such sequence {vn } ⊂ K, the action of ν[B(vn ) − B(vm )] on elements ϕ ∈ V ,
is computed from
Z
hν[B(vn ) − B(vm )], ϕiV =
[vn · (vn · ∇) − vm · (vm · ∇)]ϕdx
ZE
(vn − vm ) · (vn · ∇)ϕdx
=
E
Z
+
vm · ((vn − vm ) · ∇)ϕdx
E
≤ kvn k4 + kvm k4 kvn − vm k4 kϕkV
1
≤ γ|E| 12 kvn kV + kvm kV kvn − vm k4 kϕkV .
Hence,
kB(vn ) − B(vm )kV ∗ =
sup h[B(vn ) − B(vm )], ϕiV
kϕkV =1
1
≤ 2Cγ|E| 12 kvn − vm k4 .
Consider next the family of variants of (3.2)
v = λB(v)
in V
for
λ ∈ (0, 1).
(3.3)
4 Nonhomogeneous Boundary Data
597
If vλ is a solution of (3.3), it is also a solution of (2.3) with ν replaced by ν/λ.
As such, the a priori estimates (2.6) and (2.7) remain in force with v replaced
by vλ and ν replaced by ν/λ, i.e.,
kvλ kV ≤
1
λ γ
k∇vλ k2 ≤
kf k 56 .
γo
ν γo
Therefore, all possible solutions of (3.3) are uniformly bounded in λ. Existence
of solutions of (3.2), and hence of (2.1), now follows from the Schauder–Leray
Fixed Point Theorem.
Theorem 3.1 (Schauder–Leray [169]). Let T be a continuous, compact
mapping from a Banach space {X; k · kX } into itself, such that all possible
solutions of x = λT (x) are equi-bounded uniformly in λ ∈ (0, 1). Then, T has
a fixed point.
4 Nonhomogeneous Boundary Data
Let E be a simply connected, bounded domain in R3 with boundary ∂E of
class C 1 and satisfying the segment property3 and consider, formally, the
steady-state flow in E,
−ν∆v + (v · ∇)v + ∇p = f ,
div v = 0,
v
∂E
in E,
(4.1)
=a
where a is a vector-valued function defined on ∂E, whose regularity will be
specified as we proceed. If v is a solution of (4.1), then, formally, by Green’s
theorem,
Z
Z
div vdx =
0=
E
∂E
a · n dσ,
(4.2)
where n is the outward unit normal to ∂E. This is then a necessary condition
to be imposed on a for the solvability of (4.1). The solvability of (4.1) hinges
on extending a with a divergence-free vector-valued function b defined in E.
The smoothness of b and the meaning of b = a on ∂E will be made precise
as we proceed. Assuming that such an extension can be found, seek a solution
to (4.1) in the form v = b + u, where formally
−ν∆u + (u · ∇)u + (b · ∇)u + (u · ∇)b + ∇p = g,
div u = 0,
u
and
3
Section 8.1 of Chapter 9.
∂E
=0
in E,
(4.3)
598
14 NAVIER–STOKES EQUATIONS
g = f + ν∆b − (b · ∇)b.
(4.4)
Solutions of (4.3) are sought in V with the equation being interpreted in its
weak form
Z
ν ∇u : ∇ϕdx
E
Z
(4.5)
=
{[u · (u · ∇) + u · (b · ∇) + b · (u · ∇)] ϕ + g · ϕ} dx
E
for all ϕ ∈ V . Taking ϕ = u gives the a priori estimate
Z
1
2
12
ν − γ|E| kbk4 k∇uk2 ≤
g · u dx ,
(4.6)
E
where γ is the constant of the embedding of V into L6 (E; R3 ). The right-hand
6
is finite if f ∈ L 5 (E; R3 ) and b ∈ W 1,2 (E; R3 ), since by the Sobolev–Nikolskii
embedding theorem 4 this implies b ∈ L6 (E; R3 ). Indeed,
Z
i
h
(4.7)
g · u dx ≤ γkf k 56 + νk∇bk2 + kbk24 k∇uk2 ,
E
where again γ is the constant of the embedding of V into L6 (E; R3 ).
If the domain E has boundary ∂E of class C 1 and satisfies in addition
the segment property, functions b ∈ W 1,2 (E; R3 ) have traces on ∂E in the
fractional Sobolev space5
b
1
∂E
= a ∈ W 2 ,2 (∂E; R3 ).
(4.8)
1
Henceforth, given a boundary datum a ∈ W 2 ,2 (∂E; R3 ), we assume that it can
be extended into a solenoidal vector field b ∈ W 1,2 (E; R3 ). A compatibility
condition for such an extension to exist is that a has zero flux across ∂E as
indicated by (4.2). We also assume that such an extension can be constructed
to satisfy
1
γ|E| 12 kbk4 ≤ 21 ν.
(4.9)
The actual construction of an extension b satisfying (4.8) is carried out in Section 4.2c of the Complements. Moreover, we assume that (4.9) can be derived
from (4.7c). Accepting it for the moment, this last requirement combined with
(4.6)–(4.7) yields the a priori estimate
k∇uk2 ≤
i
2γ h
kf k 56 + νk∇bk2 + kbk24
ν
to be satisfied by any weak solution to (4.1).
4
5
Theorem 2.1 of Chapter 9.
Theorem 8.1 of Chapter 9.
(4.10)
4 Nonhomogeneous Boundary Data
599
4.1 Uniqueness of Solutions to (4.1)
If u1 and u2 in V solve (4.1) write their weak formulations (4.5), subtract
them out and in the integral identity thus obtained take the testing function
def
ϕ = (u1 − u2 ) = w, and make use of the embedding (2.5) and the upper
bound (4.10) to be satisfied by all weak solutions to (4.1), to obtain
Z
1
k∇wk22 =
u1 · (w · ∇)w + w · (u2 · ∇)w + b · (w · ∇)w dx
ν E
1
ku1 k4 + ku2 k4 + kbk4 kwk4 k∇wk2
≤
ν
2
h
i
γ
γ
1
1
≤ |E| 12 4 |E| 12 kf k 56 + νk∇bk2 + kbk24 + kbk4 k∇wk22 .
ν
ν
If the coefficient of k∇wk22 on the right-hand side does not exceed 1 then
w = 0 and the problem admits at most one solution. The uniqueness condition
hinges on several factors including |E| and the size of the extension b through
the norms k∇bk2 and kbk4 . The key condition, however, is expressed by the
smallness of the Reynolds number Re = ν −1 . Thus, uniqueness holds if the
Reynolds number is sufficiently small or equivalently if the fluid is sufficiently
viscous.
4.2 Existence of Solutions to (4.1)
Consider the linear maps
Z
def
V ∋ϕ→ =
g · ϕ dx
ZE
def
u · (u · ∇) + u · (b · ∇) + b · (u · ∇) ϕ dx.
V ∋ϕ→ =
E
Estimate
Z
E
i
h
g · ϕ dx ≤ γ kf k 56 + νk∇bk2 + kbk24 k∇ϕk2 .
Therefore, the first is a bounded linear functional in V . By the Riesz representation theorem there exists a unique G ∈ V such that
Z
g · ϕ dx = hG, ϕiV .
V ∋ϕ→
E
Likewise, estimate
Z
u · (u · ∇) + u · (b · ∇) + b · (u · ∇) ϕ dx
E
1
1
≤ γ|E| 12 γ|E| 12 k∇uk2 + 2kbk4 k∇uk2 k∇ϕk2 ,
600
14 NAVIER–STOKES EQUATIONS
6
where γ is the constant of the embedding L 5 (E; R3 ) ⊂ V . Therefore, the second map is also a bounded linear functional in V . By the Riesz representation
theorem6 there exists B̄(u) ∈ V , such that
Z
u · (u · ∇) + u · (b · ∇) + b · (u · ∇) ϕ dx = hB̄(u), ϕiV .
E
With these identifications the weak form (4.5) reads
V ∋ ϕ → νhu, ϕiV = hB̄(u) + G, ϕiV .
Equivalently, in functional form
u = B̄(u)
in V ∗ ,
B̄(u) =
where
1
B̄(u) + G ,
ν
(4.11)
where, as before, V ∗ denotes the dual of V identified with V itself up to an
isometric isomorphism. Thus, the existence of a weak solution to (4.1) in the
sense of (4.5) is equivalent to finding a fixed point of the map V ∋ u → B̄(u) ∈
V ∗.
Lemma 4.1 The map B̄(·) : V → V ∗ is compact.
The proof is analogous to that of Lemma 3.1 with minor changes. Consider
next the family of variants of (4.11)
u = λB(u)
in V
for
λ ∈ (0, 1).
(4.12)
If uλ is a solution of (4.12), it is also a solution of (4.5) with ν replaced by
ν/λ. As such, the a priori estimate (4.10) remains in force with u replaced by
uλ and ν replaced by ν/λ, i.e.,
kuλ kV ≤
i
λ γ h
1
k∇uλ k2 ≤ 2
kf k 56 + νk∇bk2 + kbk24 .
γo
ν γo
Therefore, all possible solutions of (4.12) are uniformly bounded in λ. The
existence of the solutions of (4.11), and hence of (4.1), now follows from the
Schauder–Leray Fixed Point Theorem 3.1.
5 Recovering the Pressure
Return to the steady-state Navier–Stokes system (2.1) in its weak form (2.3).
The existence of solutions to such a system has been established in Section 3
irrespective of the pressure p appearing in the formal pointwise form (2.1).
Assume momentarily that
v ∈ W 2,2 (E; R3 )
6
and
Dunford and Schwartz [62], Chap. IV, § 4
f ∈ L2 (E; R3 ).
(5.1)
6 Steady-State Flows in Unbounded Domains
601
Then, (2.3) by back-integration by parts yields
Z
(NS) · ϕdx = 0, where (NS) = −ν∆v + (v · ∇)v − f
E
for all ϕ ∈ V. Since (NS) ∈ L2 (E; R3 ) this continues to hold for all ϕ ∈ H.
Therefore, (NS) ∈ H ⊥ . Introduce the space of functions
collection of ϕ ∈ L2 (E; R3 ) of the form
G=
ϕ = ∇p for some p ∈ W 1,2 (E)
Proposition 5.1 (Helmholtz–Weyl Decomposition [274]) Let E ⊂ R3
be open, bounded, and convex. Then, G = H ⊥ or equivalently
L2 (E; R3 ) = H ⊕ G.
Indeed, Proposition 5.1 is a special case of the Helmholtz–Weyl decomposition;
its proof will be given in Section 5c of the Complements.
The system (2.1), as such, does not provide sufficient information to determine the pressure p. However, its weak formulation (2.3) permits one to
assert that the principal part (NS) of the Navier–Stokes system has, at least
under the regularity assumptions (5.1) on v and f , and locally in E, the
1,2
form of a gradient of some pressure p ∈ Wloc
(E). This follows by applying
Proposition 5.1 to open, convex subsets of E.
6 Steady-State Flows in Unbounded Domains
Let E be an unbounded, open set in R3 filled with a fluid of dynamic viscosity
µ. The problem is particularly interesting from the physical point of view if
E is an exterior domain, that is, the complement of a bounded set; such a
situation can then be used to model the motion of a rigid body through a
viscous fluid, or the flow past an obstacle (see also Galdi [91], Chapter 1, § 2
for more details).
The domain E will be assumed to be open and simply connected, with
boundary ∂E of class C 1 , and satisfying the segment property. The fluid
velocity v is assumed to take the value a on ∂E, for a vector field a whose
regularity will be specified as we proceed, and to approach a constant vector
a∞ as |x| → ∞. The fluid is stirred in its interior by a forcing term f whose
properties are to be defined. Consider formally the steady-state flow in E,
−ν∆v + (v · ∇)v + ∇p = f ,
div v = 0,
v
∂E
= a,
in E
(6.1)
lim v(x) = a∞ .
|x|→∞
Notice that, in general, (4.2) is no longer a necessary condition on a for the
solvability of (6.1), even if a∞ = 0.
602
14 NAVIER–STOKES EQUATIONS
6.1 Assumptions on a and f
1
It is assumed that the boundary datum a ∈ W 2 ,2 (∂E; R3 ) can be extended
1,2
into a solenoidal b ∈ Wloc
(E; R3 ), satisfying
b=a
on ∂E as traces of functions in W 1,2 (E; R3 ),
b − a∞ ∈ L2 (E; R3 ),
Mo
|b(x) − a∞ | ≤ p
,
1 + |x|2
M1
and |∇b| ≤
1 + |x|2
(6.2)
in E,
for two given constants Mo and M1 . For exterior domains and smooth a with
zero flux on ∂E, such an extension can always be realized. Indeed, we have
the following.
Proposition 6.1 Let E be an exterior domain, complement of a bounded,
simply connected domain E c = R3 \Ē. Then, any a ∈ C 2 (∂E; R3 ) satisfying
(4.2) admits a solenoidal extension b ∈ C 2 (R3 ; R3 ) satisfying (6.2).
Proof. For δ > 0, consider the set Eδ = [dist(·, ∂E) < δ] and construct the
vector field ψ a ∈ C 3 (R3 ; R3 ) corresponding to a, compactly supported in Eδ ,
such that the solenoidal extension ba of a is realized by ba = curl ψ. Such a
construction is guaranteed by Proposition 4.3c of the Complements.
Let R > 1 be sufficiently large, such that BR−1 ⊃ E c , and set
b′ = curl x3 a∞,2 , x1 a∞,3 , x2 a∞,1 ζ ,
(6.3)
where


1 outside a ball of radius R,
ζ = 0 inside a ball of radius R − 1,


smooth, 0 ≤ ζ ≤ 1 otherwise.
Finally, let b(x) = ba (x) + b′ (x). One verifies that such a b is solenoidal, and
satisfies the requirements (6.2).
For general vector fields with the regularity assumed in (6.2), again one
relies on Proposition 4.3c of the Complements for the construction of ba ,
whereas b′ is built as in (6.3).
By the previous construction, it is also apparent that supp ∇b is a compact
set in R3 .
The forcing term f is taken in L2loc (E; R3 ) and decreasing sufficiently fast
as |x| → ∞, in the sense
|x| f ∈ L2 (E; R3 ).
(6.4)
7 Existence of Solutions to (6.1)
603
6.2 Toward a Notion of a Solution to (6.1)
Proceeding as in the case of bounded domains, solutions are sought of the
form v = b + u, for some u ∈ V , where formally u satisfies (4.3)–(4.5), the
latter holding for all ϕ ∈ V. The membership u ∈ V provides weak forms of
the last two conditions in (6.1), whereas (4.5) interprets weakly the Navier–
Stokes system. The next step is in deriving a priori estimates for u, by taking
ϕ = u in (4.5).
For bounded domains E, the inner product (·, ·)V introduced in (3.1) is
equivalent to the inner product h∇·, ∇·iH . This follows from the embedding
inequalities (2.5)–(2.6). If E is unbounded this is, in general, no longer the case
and the topology generated by (·, ·)V cannot be related to the norm k∇ · k2 .
Nevertheless, the first of (2.5) has a weaker counterpart in V .
Proposition 6.2 Let u ∈ V and 0 ∈ R3 \ supp u. Then,
Z
Z
|u|2
dx
≤
4
|∇u|2 dx
2
|x|
E
E
(6.5)
Proof. Since u has a vanishing trace on ∂E, by extending it with zero outside
E, regard u as an element of V in R3 . Assume momentarily that u ∈ V, and
for ε ∈ (0, 1) compute and estimate
Z
Z
|u|2
(∆ ln |x|)|u|2 dx
dx =
2
ε<|x|<ε−1 |x|
ε<|x|<ε−1
Z
Z
x
x
2
|u| dσ −
∇ ln |x| ·
|u|2 dσ
=
∇ ln |x| ·
|x|
|x|
|x|=ε
|x|=ε−1
Z
u ∇u −2
dx.
· x·
|x|
ε<|x|<ε−1 |x|
Letting ε → 0 gives
Z
Z
21
Z |u|2 12 Z
|u|2
|u|
2
,
|∇u|
dx
dx
≤
2
dx
|∇u|dx
≤
4
2
2
E |x|
E
E |x|
E |x|
which yields (6.5). The proof is then concluded by density.
Notice that in the proof u being solenoidal is not used.
7 Existence of Solutions to (6.1)
7.1 Approximating Solutions and A Priori Estimates
For n > diam(E c ) let Bn be the ball of radius n about the origin of R3 , set
En = E ∩ Bn , and let Vn and Vn be the spaces introduced in (2.2) for En .
In each En we consider the problem
604
14 NAVIER–STOKES EQUATIONS
−ν∆vn + (vn · ∇)vn + ∇p = f ,
div vn = 0,
vn
vn
∂E
∂Bn
= a,
=b
∂Bn
.
Let vn = b + un , where un ∈ Vn is a weak solution in En of
−ν∆un + (un · ∇)un + (b · ∇)un + (un · ∇)b + ∇p = g,
div un = 0,
un
un
∂E
∂Bn
= 0,
(7.1)
= 0,
where g = f + ν∆b − (b · ∇)b. Since En is bounded, such a un exists, by the
construction in Section 4.
Proposition 7.1 Let un ∈ Vn be a solution of (7.1) in En , with b and f
satisfying (6.2) and (6.4) respectively. Then, either of these a priori estimates
holds, uniformly in n
ν − 2Mo k∇un k2
def
≤ γg = k|x|f k2 + π(CπM12 + νM1 ),
(7.2)
ν − 4M1 k∇un k2
where C > 0 is a constant that depends on δ and R, introduced in the proof
of Proposition 6.1.
Proof. Since un ∈ Vn , it can be regarded as un ∈ V , by extending it to
be zero outside En . Likewise, the test function ϕ ∈ Vn is regarded as in V .
Insert ϕ = u in the weak formulation of (7.1), and transform and estimate
the various terms by using the assumptions (6.4) on f and (6.2) on b. In this
process we use the elementary calculation7
Z
dx
= π2 .
2 2
R3 (1 + |x| )
We have
ν
Z
En
∇un : ∇un dx ≤
Z
En
+
+
Z
un · (b · ∇)un dx
En
un · (un · ∇)b dx
En
f · un dx
Z
Z
∆b · un dx
+ ν
E
Z n
+
(b · ∇)b · un dx .
En
7
Combine (8.5) of Chapter 2 and Problem 2.3 of the Complements of the same
chapter.
7 Existence of Solutions to (6.1)
605
The various terms above are transformed and estimated as follows:
Z
f · un dx ≤ k|x|f k2 k|x|−1 un k2 ≤ 2k|x|f k2 k∇un k2 ;
En
Z
ν∆b · un dx ≤ νk∇bk2 k∇un k2 ≤ νM1 πk∇un k2 ;
En
Z
un · (b · ∇)un dx = 0 since b is solenoidal;
Z
ZEn
un · (un · ∇)b dx ≤ M1
|x|−2 |un |2 dx ≤ 4M1 k∇un k22 ;
En
En
Z
Z
un · (un · ∇)(b − a∞ + a∞ ) dx
un · (un · ∇)b dx =
En
ZEn
un · (un · ∇)(b − a∞ ) dx
≤
En
Z
≤
un · (un · ∇)(b − a∞ ) dx
R3
Z
(b − a∞ ) · (un · ∇)un dx
=
R3
Z
≤ Mo
|x|−1 |un ||∇un |dx ≤ 2Mo k∇un k22 ;
E
Z
Z
un · (b · ∇)b dx =
un · (b · ∇)b dx
En
En ∩supp ∇b
Z
=
b · (b · ∇)un dx
En ∩supp ∇b
2
k∇un k2,;En ∩supp ∇b
≤ kbk4;E
n ∩supp ∇b
≤ C(δ, R)k∇bk22 k∇un k2;En ∩supp ∇b
≤ C(δ, R)k∇bk22 k∇un k2
= C(δ, R)M12 π 2 k∇un k2 .
Combining these calculations proves (7.2).
The proposition provides an a priori estimate for k∇un k2 , independent of n
if either Mo or M1 is sufficiently small. In what follows, assume
max{2Mo; 4M1 } < ν
and set
ν − max{2Mo ; 4M1 } = α > 0.
(7.3)
For unbounded E, introduce the space
H = {completion of V in the norm k · kH = k∇ · k2 }.
This is a separable Hilbert space by the inner product h·, ·iH = h∇·, ∇·iH . By
construction V ⊂ H, as elements in H are not required to be in L2 (E; R3 ).
606
14 NAVIER–STOKES EQUATIONS
7.2 The Limiting Process
If (7.3) holds, then {un } is a sequence bounded in H, and hence weakly precompact in the same space. Every element u in the weak closure of {un }
is a weak solution of (4.5) in the following sense. First, u ∈ H∗ and hence
u ∈ H by the Riesz identification map. Next, having fixed ϕ ∈ V, let F be its
support and consider the sequence {un |F } of restrictions of un to F . Since F
is bounded, by the embedding inequalities (2.5)–(2.6), the norm k∇ · k2;F is
equivalent to the norm of W 1,2 (F ; R3 ). Therefore, there exists a constant C,
depending on F , such that
kun |F kW 1,2 (F ;R3 ) ≤ C
uniformly in n.
By the Rellich–Kondrachov compact embedding theorem (Theorem 2.2 of
Chapter 9), the embedding W 1,2 (F ; R3 ) ֒→ Lp (F ; R3 ) is compact for all 1 ≤
p < 6. Therefore, a subsequence {un′ |F } ⊂ {un |F } can be selected so that
{un′ } → u
{un′ } → u
weakly in W 1,2 (F ; R3 ), and
strongly in Lr (F ; R3 ).
(7.4)
Theorem 7.1. Let (6.2), (6.3), and (7.3) hold. Then, (4.5) admits a solution
u ∈ H satisfying
ν
Z
E
∇u : ∇ϕdx =
Z
E
{[u · (u · ∇) + u · (b · ∇) + b · (u · ∇)] ϕ + g · ϕ} dx
(7.5)
for all ϕ ∈ V.
Proof. Let u ∈ H be in the closure of {un } in the weak topology of H. Write
down (4.5) for un over En for ϕn ∈ Vn . Having fixed ϕ ∈ V, let F be its
support and let nF be so large that F ⊂⊂ Bn for all n ≥ nF . Then, for
such ϕ fixed, (4.5) will hold for all n ≥ nF . Letting n → ∞ along proper
subsequences depending of ϕ satisfying (7.4) establishes (7.5)
Remark 7.1 Notice that the indicated limiting process can be carried out
for a fixed ϕ of compact support and not for ϕ ∈ V . Thus, (7.5) holds only for
ϕ ∈ V and, in general, not for ϕ ∈ V . Once u in the weak closure of {un } has
been identified, the choice of subsequences for which (7.4) holds depends on
the selected testing function ϕ. However, the limiting identity (7.5) continues
to hold for all ϕ ∈ V. Also, for unbounded E, solutions are found in H and
in general not in V .
8 Time-Dependent Navier–Stokes Equations in Bounded
Domains
Continue to denote by E ⊂ R3 an open, bounded set with boundary ∂E of
class C 1 and satisfying the segment property. For 0 < T < ∞ let ET =
E × (0, T ), and introduce the spaces
8 Time-Dependent Navier–Stokes Equations in Bounded Domains
607
L2 (0, T ; V ) = {v(·, t) ∈ V for a.e. t ∈ (0, T ) with finite norm k∇vk2;ET } ;
(
)
v(·, t) ∈ V for a.e. t ∈ (0, T ) with finite norm
W =
;
2 = ess sup
2
2
kvkW
(0,T ) kv(·, t)k2;E + k∇vk2;ET
C ∞ (0, T ; V) = ϕ ∈ C ∞ (ET ; R3 ) with ϕ(·, t) ∈ V for all t ∈ (0, T ) .
For these spaces the operations of ∇ and div are meant weakly and with
respect to the space variables only. Functions ϕ ∈ C ∞ (0, T ; V) are divergencefree and of compact support in E, in the space variables, but are permitted
not to vanish for t = 0 or for t = T .
10
Lemma 8.1 Let v ∈ W . Then, v ∈ L 3 (ET ; R3 ) and
3
kvk 10
≤ γ 5 kvkW ,
3 ;ET
where γ is the constant of the embedding of V into L6 (E; R3 ).
Proof.
Z
0
T
Z
10
E
|v| 3 dxdt =
≤
≤
Z
Z
T
0
0
4
E
T
Z
Z
|v| 3 |v|2 dxdt
E
32 Z
13
|v|2 dx
|v|6 dx dt
E
ess sup kv(·, t)k2;E
2
(0,T )
10
3
34 Z
T
0
kv(·, t)k26;E dt
≤ γ kvkW
The last inequality follows from the embedding (2.4).
Consider a viscous fluid of Reynolds number ν −1 filling a rigid, still container
E and stirred by a forcing term f . Its time evolution over (0, T ) is modeled,
formally, by the system
vt − ν∆v + (v · ∇)v + ∇p = f
div v = 0;
v(·, t)
∂E
in ET ;
(8.1)
= 0;
v(·, 0) = vo
in E.
The homogeneous boundary condition for the velocity v, also called no-slip
condition, says that at the boundary, the fluid will have zero velocity with
respect to the same boundary.
Multiply the first of these, formally, by ϕ ∈ C ∞ (0, T ; V) and integrate by
parts over Et = E × (0, t) for t ∈ (0, T ]. Using that div v = 0 gives
608
14 NAVIER–STOKES EQUATIONS
Z
E
v(t) · ϕ(t)dx −
Z tZ
0
Z tZ
E
v · ϕτ dxdτ
ν∇v : ∇ϕ + (v · ∇)v · ϕ dxdτ
0
E
Z tZ
Z
vo · ϕ(0)dx +
=
f · ϕ dxdτ.
+
E
0
(8.2)
E
∞
Assuming momentarily that v ∈ C (0, T ; V) take ϕ = v and observe that
the nonlinear term gives, formally, zero contribution. Using Lemma 8.1 also
yields the formal energy inequality
2
ess supkv(t)k2;E
+ 2νk∇vk22;ET
(0,T )
≤ kvo k22;E + 2
≤
≤
2
kvo k2;E
kvo k22;E
Z TZ
0
E
f · v dxdt
(8.3)
+ 2kf k2;ET kvk2;ET
√
+ 2 T kf k2;ET ess sup kv(·, t)k2;E .
(0,T )
In what follows, the set of parameters {ν, T, |E|, kvok2;E , kf k2;ET } constitutes
the given data and we denote by γ a generic positive constant that can be
determined quantitatively, a priori only in these terms. With this notation,
by a standard application of the Cauchy–Schwarz inequality, (8.3) implies
(8.4)
kvkW ≤ γ kvo k2;E + kf k2;ET .
These formal remarks suggest that we define a weak solution to (8.1) as an
element of W satisfying (8.2) for all ϕ ∈ C ∞ (0, T ; V), and the energy estimate
(8.4). The membership v(·, t) ∈ V for a.e. t ∈ (0, T ) gives meaning, in the sense
of traces, to the homogeneous boundary data on ∂E. The same membership
ensures that div v = 0 weakly in ET . As for the initial data, observe that, for
solutions in this class, all integrals in (8.2) are well defined. As a consequence,
by Vitali’s absolute continuity of the integral, all integrals extended over Et
tend to zero as t → 0. Therefore,
Z
Z
vo · ϕ(0)dx for all ϕ ∈ C ∞ (0, T ; V).
lim
v(t) · ϕ(t)dx =
t→0
E
E
Thus, the initial datum vo is taken in the sense of such a weak continuity of
v(·, t) in L2 (E; R3 ). The same continuity also implies that div vo = 0 weakly
in E. The latter emerges then as a compatibility condition to be imposed on
the initial datum vo for a solution to exist.
Theorem 8.1 (Hopf [123]). Let f ∈ L2 (ET ; R3 ) and let vo ∈ L2 (E; R3 ) be
weakly divergence-free in E. Then there exists a weak solution to (8.1).
Remark 8.1 In the following we refer to such a solution as Hopf ’s solution.
9 The Galerkin Approximations
609
9 The Galerkin Approximations
Let e = (e1 , . . . , en , . . . ) be a complete system for V . Since V is dense in V ,
by sequential selection and Zorn’s lemma, the elements ej can be chosen in V.
Also, by sequential orthonormalization, although not necessarily orthonormal
with respect to the inner product h·, ·iV defined in (3.1), they can be chosen
to be orthonormal in L2 (E; R3 ), i.e., hei , ej iH = δij . Write a possible solution
in the form
v = vn + vr,n ,
where vn =
n
P
cj (t)ej
and vr,n =
j=1
P
cj (t)ej
(9.1)
j>n
for scalar functions (0, T ) ∋ t → cj (t). The remainder vr,n of the series satisfies
vr,n k2V = vr,n k22 + k∇vr,n k22 → 0
as n → ∞.
Since (e1 , . . . , en , . . . ) is complete in V , it is also complete in L2 (E; R3 ). Therefore, by the indicated orthonormalization in L2 (E; R3 ) and Parseval’s identity
kvk22;E =
P
j≥1
c2j .
Write v in (8.2) in the form (9.1) and observe that the terms involving vr,n
tend to zero as n → ∞. This suggests defining an
Pnapproximate solution to
(8.1) as a function vn ∈ C ∞ (0, T ; V), with vn = i=1 cn,i ei , satisfying (8.2)
for ϕ = ei , for all i = 1, . . . , n, i.e.,
Z
0
T
h
osym
nZ
n
P
c′n,i +
cn,j
ν∇ej : ∇ei dx
j=1
+
n
P
E
cn,j
j=1
nZ
E
ij
ei · (vn · ∇)ej dx
oskew
ij
dτ −
Z
(9.2)
E
i
f · ei dx dτ = 0.
sym
= {· · · }sym
define the entries of a n × n time
For fixed n ∈ N the terms Aij
ij
independent symmetric matrix Ansym , whereas the terms Askew
= {· · · }skew
ij
ij
skew
define the entries of a n × n skew symmetric matrix An
linearly dependent
on the time-dependent vector cn = (cn,1 , . . . , cn,n ). The last term defines a
vector fn = (f1 , . . . , fn ) dependent on t. Set also
Z
vo · ei dx,
cn (0) = co .
co = (co,1 , . . . , co,n ),
co,i =
E
Requiring the integrand over (0, T ) in (9.2) to vanish identically gives the
differential system in cn
c′n,i +
n
X
j=1
cn,j = fi
Asym
+ Askew
ij
ij
with cn,i (0) = co,i .
(9.3)
The unique solvability of this system hinges upon some a priori estimates,
which we derive next.
610
14 NAVIER–STOKES EQUATIONS
Proposition 9.1 Let cn = (cn,1 , . . . , cn,n ) be a solution to (9.3) and set vn =
P
n
i=1 cn,i ei . There is a constant γ depending only on the data and independent
of n and i, such that
kvn kW ≤ γ;
ess sup |cn,i (t)| ≤ γ;
(9.4)
(0,T )
√
|cn,i (t2 ) − cn,i (t1 )| ≤ γ(1 + k∇ei k∞;E ) t2 − t1
for all (t1 , t2 ) ⊂ (0, T ).
Proof. Multiply (9.3) by cn,i , add over i = 1, . . . , n, and observe that
ctn Askew
cn = 0, where ctn denotes the transpose of the vector cn . This gives
n
Z
n
n
n
P
P
1 d P
cn,i ei dx
c2n,i + ν
∇
cn,j ej : ∇
2 dt i=1
j=1
i=1
E
P
21 P
12
n
n
n
P
fi cn,i ≤
=
fi2
c2n,i
= kfn (t)k2;E kvn (t)k2;E .
i=1
i=1
i=1
Equivalently,
1 d
kvn (t)k22;E + νk∇vn (t)k22;E ≤ kfn (t)k2;E kvn (t)k2;E .
2 dt
To prove the first of (9.4), integrate this over (0, t) ⊂ (0, T ) to get
ess sup kvn (t)k22;E + 2νk∇vn k22;ET
(0,T )
√
≤ kvo k22;E + 2 T kfn k2;ET ess sup kvn (t)k2;E .
(0,T )
The proof is concluded by a standard application of Cauchy–Schwarz inequality in the last term. The second of (9.4) follows from this and Parseval’s
identity. To prove the last of (9.4), return to (9.3) and, for fixed i ∈ {1, . . . , n},
estimate
|c′n,i |
Z
Z
Z
≤ν
∇vn : ∇ei dx +
(vn · ∇)ei · vn dx +
|f | dx
E
E
E
Z
Z
Z
≤ν
|∇vn ||∇ei | dx +
|vn |2 |∇ei | dx +
|f | dx
E
E
E
Z
Z
Z
|vn |2 dx +
|f | dx
|∇vn | dx + k∇ei k∞;E
≤νk∇ei k∞;E
E
E
E
1
≤νk∇ei k∞;E k∇vn (t)k2;E |E| 2 + k∇ei k∞;E ess sup kvn (t)k22;E
(0,T )
1
2
+ |E| kf (t)k2;E
1
1
=k∇ei k∞;E ν|E| 2 k∇vn (t)k2;E + ess sup kvn (t)k22;E + |E| 2 kf (t)k2;E .
(0,T )
10 Selecting Subsequences Strongly Convergent in L2 (ET ; R3 )
611
Integrating over (t1 , t2 ) ⊂ (0, T ) and using the first of (9.4) gives
Z t2
Z t2
′
|c′n,i (t)|dt
cn,i (t)dt ≤
|cn,i (t2 ) − cn,i (t1 )| ≤
t1
≤νk∇ei k∞;E |E|
1
2
t1
t2
Z
t1
Z
E
12
|∇vn |2 dx dt
+ k∇ei k∞;E ess sup kvn (t)k22;E (t2 − t1 )
1
+ |E| 2
Z
(0,T )
t2
t
Z
E
12
|f |2 dx dt
1
√
≤ γ(ν, T, |E|, kvo k2 , kf k2;ET )(1 + k∇ei k∞ ) t2 − t1 .
The existence and uniqueness of the solutions to (9.3) can be established in the
small, for example, by a contraction fixed point argument. Then the solution
can be continued in the whole (0, T ), so long as it remains bounded. Such a
bound, independent of t, n, and i, is ensured by the second of (9.4).
10 Selecting Subsequences Strongly Convergent in
L2 (ET ; R3 )
It follows from Proposition 9.1 that for fixed j ∈ N the sequences {cn,j }∞
n=1 are
equibounded and equicontinuous, so that by the Ascoli–Arzelà theorem a subsequence {cnj ,j } ⊂ {cn,j }∞
n=1 can be selected converging to some cj uniformly
in (0, T ). By the Cantor diagonalization procedure a further subsequence can
be selected and relabelled with n, such that {cn,j } → cj uniformly in [0, T ].
However, it should be noted that, because of the last of (9.4), the rate of
convergence depends on the index j. Set formally
v=
∞
P
cj e j .
j=1
Proposition 10.1 For the same constant γ as in the first of (9.4) there holds
ess sup kv(·, t)k2;E + k∇vk2;ET ≤ γ.
(0,T )
Moreover, {vn (·, t)} → v(·, t) weakly in L2 (E; R3 ), uniformly in t ∈ (0, T ).
Proof. For a fixed positive integer k and all n
k
P
j=1
c2j (t) ≤
≤
k
P
j=1
k
P
j=1
c2j (t) −
k
P
j=1
c2n,j (t) +
k
P
j=1
c2n,j (t)
c2j (t) − c2n,j (t) + kvn (·, t)k22;E .
612
14 NAVIER–STOKES EQUATIONS
By the first of (9.4) the last term is bounded by a constant γ depending only
upon the data and independent of t and n. Letting n → ∞ the first term in
the right-hand side tends to zero by the uniform convergence of {cn,j } → cj
Pk
Pk
for j = 1, . . . , k. Thus, j=1 cj2 (t) ≤ γ. Since k is arbitrary, the series j=1 cj2
2
converges to kv(·, t)k2;E
and ess sup(0,T ) kv(·, t)k2;E ≤ γ. To prove the second
Pk
statement, fix k ∈ N and take first a function of the form ϕ = j=1 ϕj ej . For
such a function, by the othonormality of (e1 , . . . , en , . . . )
Z
E
(vn − v) · ϕ dx =
k
P
j=1
(cn,j − cj )ϕj → 0 as n → ∞
by the
of {cn,j } → cj for j = 1, . . . , k. For a general
P uniform convergence
3
),
having
fixed ε > 0, there exists kε , depending on ε
ϕ = ϕj ej ∈ L2 (E;
R
P
and ϕ, such that j>kε ϕj2 < ε. Then estimate
Z
E
[vn (t) − v(t)] · ϕ dx ≤
kε
P
j=1
|cn,j (t) − cj (t)| |ϕj |
+ ess sup kvn (t) − v(t)k2;E
(0,T )
P
j>kε
ϕj2
! 12
.
By the first of (9.4) a further subsequence out of {vn } can be selected and relabeled with n, such that {vn } → v′ and {∇vn } → ∇w weakly in L2 (ET ; R3 ).
By the uniqueness of the weak limit v′ = v and ∇w = ∇v. By the weak lower
semi-continuity of the norm and the first of (9.4)
k∇vk2;ET ≤ lim inf k∇vn k2;ET ≤ γ.
Proposition 10.2 {vn } → v strongly in L2 (ET ; R3 ).
The proof uses the following lemma.
Lemma 10.1 (Friedrichs [86]) For every ε > 0 there exist a positive integer Nε depending only on ε and |E|, and independent of vn , and Nε linearly
2
3
ε
independent functions {ψ ℓ }N
ℓ=1 ⊂ L (E; R ) such that
kvn −
vk22;ET
≤
Nε
P
ℓ=1
Z
0
T
Z
E
2
(vn − v) · ψ ℓ dx dt + εk∇(vn − v)k22;ET . (10.1)
Inequality (10.1) is a special case, applied to (vn − v) of a more general
Friedrichs’ Lemma, which we prove in Section 10c of the Complements.
Proof (of Proposition 10.2). Fix ε > 0 and determine Nε and the system
2
3
ε
{ψ ℓ }N
ℓ=1 ⊂ L (E; R ). Let now n → ∞ in (10.1). The first term goes to zero
because of the weak uniform convergence of (vn − v) in L2 (E; R3 ). The last
term is majorized by 2γ 2 ε, where γ is the constant in the first of (9.4).
11 The Limiting Process and Proof of Theorem 8.1
613
11 The Limiting Process and Proof of Theorem 8.1
Pk
Let ϕk = ℓ=1 ϕℓ eℓ for fixed k ∈ N. Multiply (9.3) by ϕi , add for i = 1, . . . , k
and integrate over (0, t) ⊂ (0, T ) to obtain for n ≥ k
Z tZ
Z
vn (t) · ϕk (t)dx −
vn · ϕk,τ dxdτ
E
0 E
Z tZ
+ν
∇vn : ∇ϕk dxdτ
0 E
Z tZ
+
(vn · ∇)vn · ϕk dxdτ
0 E
Z
=
E
vo · ϕk (0)dx +
Z tZ
0 E
f · ϕk dxdτ.
In turn, this is averaged in time over (t, t + h) ⊂ (0, T ), for a fixed h > 0,
sufficiently small so that 0 < t + h < T . Denoting by
Z t+h
Z
1 t+h
{· · · }dτ =
−
{· · · }dτ
h t
t
such averages gives
Z t+hZ
Z t+hZ τZ
−
vn (τ ) · ϕk (τ ) dxdτ − −
vn (s) · ϕk,s (s) dxdsdτ
t
E
t
0
E
Z t+hZ τZ
+ ν−
∇vn (s) : ∇ϕk (s) dxdsdτ
t
0
E
Z t+hZ τZ
+−
(vn (s) · ∇)vn (s) · ϕk (s) dxdsdτ
=
t
Z
E
0
E
Z t+hZ τZ
vo · ϕk (0) dx + −
f (s) · ϕk (s) dxdsdτ.
t
0
E
Let n → ∞ by keeping k fixed, to get
Z t+hZ
Z t+hZ τZ
−
v(τ ) · ϕk (τ ) dxdτ − −
v(s) · ϕk,s (s) dxdsdτ
t
E
t
0
E
Z t+hZ τZ
∇v(s) : ∇ϕk (s) dxdsdτ
+ ν−
t
0
E
Z t+hZ τZ
+−
(v(s) · ∇)v(s) · ϕk (s) dxdsdτ
t
=
Z
E
0
E
Z t+hZ τZ
vo · ϕk (0) dx + −
f (s) · ϕk (s) dxdsdτ.
t
0
E
The various limits are justified by the weak convergence {∇vn } → ∇v and
the strong convergence {vn } → v. In particular, such a strong convergence
614
14 NAVIER–STOKES EQUATIONS
permits one to pass to the limit in the nonlinear term. In Section 11c of the
Complements we discuss a counterexample to show that in general, having
weak convergence does not suffice to pass to the limit
P in such a term.
Next, take ϕ ∈ C ∞ (0, T ; V), write it as ϕ =
ϕj ej , and let ϕk be its
truncated series. Because of the predicated smoothness of ϕ
{ϕk }, {∇ϕk }, {ϕk,t } → ϕ, ∇ϕ, ϕt in L2 (ET ),
and also {ϕk } → ϕ in L5 (ET ; R3 ). Compute and estimate
ZZ
ZZ
lim sup
∇v : ∇ϕ dxdτ
∇v : ∇ϕk dxdτ −
k→∞
Et
Et
≤ k∇vk2;ET lim k∇(ϕk − ϕ)k2;ET = 0.
k→∞
The limits in all the other terms but the nonlinear one are treated similarly.
For the nonlinear term
ZZ
ZZ
(v · ∇)v · ϕk dxdτ
(v · ∇)v · ϕk dxdτ −
lim sup
k→∞
Et
Et
≤ k∇vk2;ET kvk 10
lim kϕk − ϕk5;ET .
3 ;ET
k→∞
Letting k → ∞ yields, for all ϕ ∈ C ∞ (0, T ; V)
Z t+hZ τZ
Z t+hZ
−
v(τ ) · ϕ(τ ) dxdτ − −
v(s) · ϕs (s) dxdsdτ
t
t
E
0
E
Z t+hZ τZ
+ ν−
∇v(s) : ∇ϕ(s) dxdsdτ
t
0
E
Z t+hZ τZ
+−
(v(s) · ∇)v(s) · ϕ(s) dxdsdτ
t
=
Z
E
0
E
Z t+hZ τZ
vo · ϕ(0) dx + −
f (s) · ϕ(s) dxdsdτ.
t
0
E
Finally, let h → 0 and notice that
Z t+h Z
Z
v(t) · ϕ(t) dx
lim −
v(τ ) · ϕ(τ ) dxdτ =
h→0 t
E
E
for a.e. t ∈ (0, T ),
since, for integrable functions in (0, T ), a.e. t is a Lebesgue point. Thus, the
function v thus constructed satisfies the definition (8.2) of a weak solution. It
should be stressed that the testing functions ϕ cannot, in general, be taken out
of C 1 (0, T ; V ) as the limiting process for k → ∞ requires a further smoothness,
guaranteed in general by taking ϕ ∈ C ∞ (0, T ; V).
12 Higher Integrability and Some Consequences
615
12 Higher Integrability and Some Consequences
The Hopf solution has a limited degree of regularity due to the nonlinear term
(v · ∇)v · ϕ. The weak formulation (8.2) holds for all ϕ ∈ C ∞ (0, T ; V) ⊂ W ,
whereas the solution v is required to be in W . If in (8.2) one could take
ϕ = v, then, since div v = 0, the nonlinear term would vanish and further
regularity could be inferred on v. Optimal local and global regularity of the
Hopf solutions is unknown and it is currently a major topic of investigation. To
underscore this point, here we indicate some consequences of assuming higher
integrability on v and on the various terms of (8.1), including the pressure
term ∇p.
5
Lemma 12.1 Let v be a Hopf solution of (8.1). Then, (v·∇)v ∈ L 4 (ET ; R3 ),
and
k(v · ∇)vk 54 ;ET ≤ kvk 10
k∇vk2;ET .
3 ;ET
Proof. Let q, q ′ > 1 be Hölder conjugate and for p > 1 to be chosen, compute
and estimate
ZZ
1′
ZZ
q1 Z Z
′
q
p
pq
|v|pq dxdt
. (12.1)
|(v · ∇)v| dxdt ≤
|∇v| dxdt
ET
ET
ET
Choose pq = 2 and pq ′ =
10
3 ,
which yields p =
5
4.
5
4
(ET ; R3 ) and set
Assume momentarily that ∇p ∈ Lloc
5
4
Φ = f − ∇p − (v · ∇)v ∈ Lloc
(ET ; R3 ).
Then the weak formulation (8.2) yields8
vt − ν∆v = Φ
weakly in ET for all ϕ ∈ Co∞ (ET ; R3 ).
(12.2)
5
4
(ET ; R3 ).
This is a linear parabolic system with the forcing term Φ ∈ Lloc
Then, by classical parabolic theory [67], the weak derivatives vxi xj and vt
5
4
are in Lloc
(ET ; R3 ). The argument can be repeated to yield further regularity
on v. Therefore, assuming a moderate degree of integrability of ∇p yields a
considerably higher regularity on v.
In § 20, we get back to the regularity of the pressure for Hopf solutions.
12.1 The Lp,q (ET ; RN ) Spaces
For p, q > 1 let


 Lebesgue measurable functions f : ET → RN with 
1
.
Lp,q (ET ; RN ) =
 finite norm kf kp,q;E = R T kf (·, t)kq dt q 
T
p;E
0
8
See 12.1. of the Complements.
616
14 NAVIER–STOKES EQUATIONS
In the scalar case, we have already introduced these spaces in Section 1 of
Chapter 11. In what follows we let p > N and q > 2 be linked by
N
2
+ = 1.
p
q
(12.3)
Condition (12.3) is usually known as the Ladyzhenskaya–Prodi–Serrin condition.
Recall also the following special case of the Gagliardo–Nirenberg embedding inequality9
2
N
2p
p
q
.
kvkr;E ≤ γ(N, p)k∇vk2;E
kvk2;E
,
where r =
p−2
Lemma 12.2 There exists a constant γ(N, p) depending only on N and p,
such that for any triple (u, v, w) with u ∈ Lp,q (ET ; RN ), v ∈ W , and w ∈ W ,
there holds
Z TZ
(v · ∇)w · u dxdt ≤ γkukp,q;ET kvkW k∇wk2;ET ;
0
Z TZ
0
E
E
(12.4)
(w · ∇)w · u dxdt
≤γ
Z
T
0
ku(·, t)kqp;E kw(·, t)k22;E dt
1q
1+ N
k∇wk2;ETp .
Proof. By Hölder’s inequality with conjugate exponents
1
1 1 1
1 1
+ = ,
i.e.,
+ + = 1,
r
p
2
r
p 2
also using the indicated special case of Gagliardo–Nirenberg inequality we
have
Z
(v · ∇)w · u dx ≤ kvkr;E k∇wk2;E kukp;E
E
2
N
q
p
≤ γkvk2;E
k∇vk2;E
k∇wk2;E kukp;E .
Next, integrate over (0, T ) and use Hölder’s inequality with conjugate exponents
N
1 1
+ + = 1,
2p q
2
Z TZ
Z T
q1
(v · ∇)w · u dxdt ≤ γ
kv(·, t)k22;E ku(·, t)kqp;E dt
0
E
×
Z
0
T
0
k∇v(·, t)k22;E dt
≤ γ ess sup kv(·, t)k2;E
(0,T )
N Z
2p
2q
0
See DiBenedetto [50], Chap. 10, Theorem 1.1.
N
p
k∇w(·, t)k22;E dt
12
k∇vk2;ET k∇wk2;ET kukp,q;ET
≤ γkvkW k∇wk2;ET kukp,q;ET .
9
T
13 Energy Identity with Higher Integrability
617
This proves the first of (12.4). The proof of the second is the same by interchanging the roles of v and w.
12.2 The Case N = 2
Lemma 12.3 Let N = 2. Then, for all v ∈ W (12.3) holds with p = q = 4,
and
1
kvk4;ET ≤ π − 4 kvkW .
Proof. The Gagliardo–Nirenberg multiplicative inequality for u ∈ Wo1,p (E)
reads10
kukp∗;E ≤ γ(N, p)k∇ukp;E ,
where
p∗ =
Np
N −p
and
1≤p<N
for a constant γ(N, p) depending only on N and p. When p = 1 the optimal
N1
, where ωN is the measure of the unit sphere
constant is γ(N, 1) = N1 ωNN
in RN . Apply the inequality for N = 2, with u = |v|2 and p = 1 to get
Z
Z
Z
12
1 1
1
|v|4 dx
≤ √
|v| ∇v dx
∇|v|2 dx ≤ √
2 π E
π E
E
Z
Z
12
1
1 |∇v|2 dx
|v|2 dx 2
≤ √
π E
E
Z
Z
12
21 1 2
√
≤
|∇v|2 dx .
ess sup
|v| dx
π (0,T ) E
E
From this
Z
1
|v(·, t)| dx ≤ kvk2W
π
E
4
Integrating over (0, T ) yields
kvk44;ET ≤
Z
E
|∇v(·, t)|2 dx.
1
kvk2W k∇vk22;ET .
π
Corollary 12.1 Any Hopf solution to (8.1) for N = 2 satisfies (12.3) for
p = q = 4.
13 Energy Identity for the Homogeneous Boundary
Value Problem with Higher Integrability
We get back to (8.1) with f = 0 to which we refer as the homogeneous problem
and label it as (8.1)o . A weak solution is meant in the sense of (8.2)o , with
f = 0, for all ϕ ∈ C ∞ (0, T ; V). Although a weak solution has been constructed
by Hopf’s procedure we assume here that one is given and meant weakly.
10
See DiBenedetto [50], Chap. 10, Corollary 1.1.
618
14 NAVIER–STOKES EQUATIONS
Proposition 13.1 (Prodi [207]) Let v be a weak solution to (8.1)o . Moreover, assume that v ∈ Lp,q (ET ; RN ) with p > N and q > 2 satisfying (12.3).
Then,
kv(·, t)k22;E + 2νk∇vk22;ET = kvo k22;E
for a.e. t ∈ (0, T ).
(13.1)
Proof. The proof consists in taking formally ϕ = v in (8.2)o . The assumption
(12.3) makes this possible by a series of approximations. First, since v ∈
L2 (0, T ; V ) there exists a sequence {vk } ⊂ C ∞ (0, T ; V) such that {vk } → v
in L2 (0, T ; V ). Next, let J(·) be the Friedrichs’ mollifying kernel in R and
denote by Jε (·) its rescaling by a parameter ε ∈ (0, 1),
2 (
1 τ exp τ 2τ−1
for |τ | < 1,
J(τ ) = C
Jε (τ ) = J
,
ε
ε
0
for |τ | ≥ 1,
where C > 0 is a constant that normalizes the kernel J. Notice that
J(−t) = J(t),
J ′ (−t) = −J ′ (t).
Then, for a.e. t ∈ (0, T ] fixed, set
vε,k (τ ) =
Z
0
t
Jε (τ − s)vk (s) ds;
vε (τ ) =
Z
t
0
Jε (τ − s)v(s) ds.
(13.2)
One verifies that vk,ε ∈ C ∞ (0, T ; V) and therefore, it is an admissible test
function in the weak formulation (8.2)o . Such a choice gives
Z
E
v(t) · vε,k (t) dx −
+
=
Z tZ
0
E
Z tZ
Z0
E
E
v · vε,k;τ dxdτ
ν∇v : ∇vε,k + (v · ∇)v · vε,k dxdτ
vo · vε,k (0) dx.
Letting k → ∞ now gives
Z
Z tZ
v(t) · vε (t) dx −
v · vε;τ dxdτ
E
0
E
Z tZ
ν∇v : ∇vε + (v · ∇)v · vε dxdτ
+
Z0 E
vo · vε (0) dx.
=
(13.3)
E
The various limits, but the first one and the one regarding the nonlinear term,
are justified since {vε,k } → vε in L2 (0, T ; V ).
13 Energy Identity with Higher Integrability
619
The limit of the first term is justified, for fixed ε > 0 since {vk } → v in
L2 (ET ; RN ) and the definition of vε . Indeed,
Z
v(t) · [vε,k (t) − vε (t)] dx
E
Z t
Z
≤
Jε (t − s)|vk (s) − v(s)| dsdx
|v(t)|
E
0
Z t
Z
Jε (t − s)
=
|v(t)||vk (s) − v(s) dx ds
0
E
Z t
≤
Jε (t − s)kv(t)k2;E kvk (s) − v(s)k2;E ds
0
≤ ess sup kv(t)k2;E
(0,T )
≤ kvkW
Z
Z
Jε (t − s)kvk (s) − v(s)k2;E ds
0
Jε2 (t) dt
R
T
21
kvk − vk2;ET .
The last term tends to zero as k → ∞ since {vk } → v in L2 (ET ; RN ). As for
the nonlinear term, compute and estimate
Z tZ
Z tZ
(v · ∇)v · (vε,k − vε ) dxdτ =
(v · ∇)(vε,k − vε ) · v dxdτ
0
E
0
E
≤ γkvkW kvkp,q;ET k∇(vε,k − vε )k2;ET ,
by virtue of Lemma 12.2. This is indeed the role of the assumption (12.3) and
the ensuing Lemma. The last term tends to zero as k → ∞ since {vε,k } → vε
in L2 (0, T ; V ).
Next, we let ε → 0 in (13.3). For the first term we have
Z
E
Z
Z
t
v(t)
Jε (t − s)v(s) dsdx
E
0
Z Z t
=
Jε (η)v(t − η) · v(t) dηdx
E 0
Z Z t
=
Jε (η)|v(t)|2 dηdx
E 0
Z Z t
+
Jε (η)v(t) · [v(t − η) − v(t)] dηdx.
v(t) · vε (t)dx =
E
0
Since Jε is even and it has been normalized, as ε → 0,
Z
Z Z t
1
|v(t)|2 dx.
Jε (η)|v(t)|2 dηdx →
2 E
E 0
On the other hand,
620
14 NAVIER–STOKES EQUATIONS
Z
t
Jε (η)
0
Z
E
v(t)·[v(t − η) − v(t)] dxdη
≤
Z
t
Jε (η)
0
Z
E
v(t) · [v(t − η) − v(t)] dx dη
and the integral tends to zero as |η| < ε → 0 by the weak continuity of
t → v(t) in L2 (E). A similar result holds for the right-hand side of (13.3).
The second term is identically zero in ε. Indeed, after interchanging the order
of integration, it can be written as
Z Z tZ t
Jε′ (τ − s)v(s) · v(τ ) dsdτ dx.
E
0
0
Now the integral in (· · · ), for a.e. fixed x ∈ E, is a double integral extended over the rectangle of vertices {(0, 0), (t, 0), (t, t), (0, t)}, which, in turn
is the union of two disjoint, equal triangles of vertices {(0, 0), (t, 0), (t, t)} and
{(0, 0), (t, t), (0, t)}. Now the argument v(s)·v(τ ) is even with respect to these
triangles, whereas Jε′ (τ − s) is odd.
Next,
Z Z t
Z tZ
Z
|∇v| Jε (τ − s)|∇[v(s) − v(τ )]| dsdτ dx
∇v : ∇(vε − v) dxdτ ≤
0
E
E
0
R
and this tends to zero as ε → 0. Finally, for the nonlinear term compute and
estimate, with the aid of Lemma 12.2,
Z t
Z tZ
Jε (τ − s)[v(s) − v(τ )] dxdsdτ
(v · ∇)v ·
0 E
0
Z Z t Z t
2
12
≤ γkvkW kvkp,q;ET
Jε (τ − s)[∇v(s) − ∇v(τ )]ds dτ dx ,
E
0
0
which tends to zero as ε → 0 by the property of the mollifiers. Observe that
the limit of the nonlinear term
Z tZ
Z tZ
lim
(v · ∇)v · vε dxdτ =
(v · ∇)v · v dxdτ = 0
ε→0
0
E
0
E
gives zero contribution since div v = 0. Collecting these calculations proves
(13.1).
Remark 13.1 For N = 2 condition (12.3) is redundant, as already stated in
Lemma 12.3.
14 Stability and Uniqueness for the Homogeneous
Boundary Value Problem with Higher Integrability
Proposition 14.1 [177] Let v and u be two weak solutions of (8.1) with
f = 0, originating from initial data vo and uo in L2 (E; RN ), meant in the
14 Stability and Uniqueness with Higher Integrability
621
sense of (8.2)o , for all ϕ ∈ C ∞ (0, T ; V). Moreover, assume that at least one v
or u, say, for example, u is in Lp,q (ET ; RN ) with p > N and q > 2 satisfying
(12.3). Assume finally that they both satisfy the energy estimates
kv(·, t)k22;E + 2νk∇vk22;Et ≤ kvo k22;E
for a.e. t ∈ (0, T ).
2
ku(·, t)k22;E + 2νk∇uk22;ET ≤ kuo k2;E
(14.1)
Then, there exist a constant γ depending only upon N and ν such that setting
w = v − u there holds
Z t
q
2
2
ku(·, τ )kp;E dτ
kw(·, t)k2;E ≤ kwo k2;E exp γ
0
for a.e. t ∈ (0, T ).
Remark 14.1 If both v and u are in Lp,q (ET ; RN ) with p > N and q > 2
satisfying (12.3) then by Proposition 13.1, the energy estimates (14.1) are
satisfied. The Proposition is a statement of stability and uniqueness. If N = 2,
v and u are both in L4 (ET ; R2 ) and therefore, weak solutions are unique.
Proof. Let v and u be two weak solutions to (8.1) originating from initial
data vo and uo in L2 (E), meant in the sense of (8.2)o , with f = 0, for all
ϕ ∈ C ∞ (0, T ; V). In the weak formulation of v take the testing function
uε,k defined as in (13.2) and in the weak formulation of u take the testing
function vε,k . Then let k → ∞ by the same arguments as in the proof of
Proposition 13.1, and add the resulting identities getting
Z
[v(t) · uε (t) + vε (t) · u(t)] dx
E
Z Z t Z t
−
Jε′ (τ − s)[v(τ ) · u(s) + v(s) · u(τ )]dsdτ dx
E
0
0
Z tZ
∇v : ∇uε + ∇vε : ∇u dxdτ
+ν
0
+
=
Z tZ
Z0
E
E
E
(v · ∇)v · uε + (u · ∇)u · vε dxdτ
[vo · uε (0) + vε (0) · uo ] dx.
Arguing as in the proof of Proposition 13.1, the second integral is identically
zero in ε since the argument [v(τ )u(s) + v(s)u(τ )] is even with respect to the
two triangles of vertices {(0, 0), (t, 0), (t, t)} and {(0, 0), (t, t), (0, t)} and Jε′ is
odd with respect to the same triangles. We may now let ε → by the same
arguments and get
622
14 NAVIER–STOKES EQUATIONS
Z
v(·, t) · u(·, t) dx + 2ν
E
+
=
Z tZ
Z tZ
Z0
E
E
0
E
∇v : ∇u dxdτ
(v · ∇)v · u + (u · ∇)u · v dxdτ
(14.2)
vo · uo dx.
Next observe that since weak solutions are divergence-free
Z tZ
Z tZ
(v · ∇)v · w dxdτ
(v · ∇)v · u dxdτ = −
0 E
0 E
Z tZ
Z tZ
(u · ∇)u · w dxdτ,
(u · ∇)u · v dxdτ =
0
0
E
E
where we have set w = v − u. Using again that w is divergence-free, the sum
of these terms equals
Z tZ
Z tZ
(w · ∇)w · u dxdτ.
(v · ∇)v · u + (u · ∇)u · v dxdτ = −
0
E
0
E
Adding the energy inequalities (14.1) and subtracting (14.2) multiplied by 2
gives
Z tZ
2
2
2
kw(t)k2;E + 2νk∇wk2;Et ≤ kwo k2;E +
(w · ∇)w · u dxdτ .
0
E
The right-hand side is estimated by the second of (12.4) of Lemma 12.2, and
N
Young’s inequality with conjugate exponents q1 and 21 + 2p
, and gives
Z tZ
0
E
(w · ∇)w · u dxdτ ≤ γ
Z
t
0
ku(τ )kqp;E kw(τ )k22;E dτ + 2νk∇wk22;Et ,
for a constant γ depending only upon N and ν. Combining these estimates
gives
Z t
2
2
kw(t)k2;E ≤ kwo k2;E + γ
ku(τ )kqp;E kw(τ )k22;E dτ.
0
The proof is concluded by an application of Gronwall’s inequality.
15 Local Regularity of Solutions with