A short story about the Monge-Ampère equation
Wojciech O»a«ski
14 October 2015
Contents
1 Introduction
1
1.1
Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2
Literature review . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
1.3
Structure of the note . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
2 Basic Concepts
5
2.1
Convex sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2
Supporting hyperplanes
7
2.3
Some facts about convex functions
. . . . . . . . . . . . . . . . . . .
9
2.4
Properties of the normal mapping . . . . . . . . . . . . . . . . . . . .
13
. . . . . . . . . . . . . . . . . . . . . . . . .
3 The Monge-Ampère measure
15
3.1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
15
3.2
Convergence properties . . . . . . . . . . . . . . . . . . . . . . . . . .
16
3.3
Inequalities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.4
Aleksandrov maximum principle . . . . . . . . . . . . . . . . . . . . .
26
3.5
Comparison principle . . . . . . . . . . . . . . . . . . . . . . . . . . .
26
4 Existence theorems
28
4.1
Generalised solutions of the MA equation . . . . . . . . . . . . . . . .
28
4.2
Homogeneous Dirichlet problem . . . . . . . . . . . . . . . . . . . . .
29
4.3
Inhomogeneous Dirichlet problem . . . . . . . . . . . . . . . . . . . .
32
4.4
At most two C solutions in 2D . . . . . . . . . . . . . . . . . . . . .
39
4.5
Similarities to the Laplace equation . . . . . . . . . . . . . . . . . . .
41
2
5 Bounds on the solution
43
5.1
Generalised Comparison Principle . . . . . . . . . . . . . . . . . . . .
43
5.2
Bounds on the solution . . . . . . . . . . . . . . . . . . . . . . . . . .
46
6 Conclusion
48
1
1
INTRODUCTION
Introduction
The Monge-Ampère equation detD2 φ = f is a fully nonlinear second order partial
dierential equation which arises in several areas, including the problem of prescribed
Gauss curvature in Riemannian geometry and optimal transport. The subject of this
note is the application of the Monge-Ampère equation in the theory of the twodimensional, incompressible Navier-Stokes equations. Namely, writing the velocity
u as u = (φy , −φx ) for some scalar function φ one can obtain a two-dimensional
Monge-Ampère equation for φ:
1
detD2 φ = ∆p.
2
(1.1)
The study of this equation is an attempt to answer an open problems for the 2D
Navier-Stokes equations, namely whether one can determine the velocity u of the
uid from the pressure p. It is not known how to solve the resulting Monge-Ampère
equation (1.1) as ∆p is not in general nonnegative and hence the equation is not elliptic. The purpose of the note is to study this equation with a possibly negative/signchanging right-hand side.
1.1
Motivation
Let us consider the 2D incompressible Navier-Stokes equations
ut + (u · ∇)u − ν∆u + ∇p = 0
div u = 0,
(1.2)
(1.3)
where ν > 0 is the kinematic viscosity. Following the calculation presented in Robinson [40], p. 6, we take the divergence of (1.2) and using the incompressibility condition
(1.3), we obtain
∇ · [(u · ∇)u] + ∆p = 0.
(1.4)
Note that the condition (1.3) is equivalent to ∂x u1 + ∂y u2 = 0. This means that u
can be described by a velocity potential, i.e. one can write
u = (u1 , u2 ) = (∂y φ, −∂x φ).
1
(1.5)
1.1
Motivation
1
INTRODUCTION
Substituting this into (1.4) one obtains
−∆p =
(φy )2
xx
+ (φx )2
yy
− 2(−φx φy )xy
= 2φxy φxy + 2φy φyxx + 2φx φxyy + 2φxy φxy − 2φxxy φy − 2φxx φyy
−2φxy φxy − 2φx φxyy = −2 φxx φyy − (φxy )2 .
Hence one obtains the equation
1
φxx φyy − (φxy )2 = ∆p.
2
(1.6)
Hence the function φ satises the Monge-Ampère equation, which was rst observed
by Larchevêque [29]. This equation is elliptic when the right-hand side is positive (recall that
second-order equation F (x, φ, ∇φ, D2 φ) = 0 is elliptic
n a nonlinear
o
∂F
if the matrix ∂r
is positive denite, where F = F (x, z, p, r), and substitute
ij
F (x, z, p, r) = det r, see Gilbarg & Trudinger [24], p. 441, for the details). We
note that the right hand side 21 ∆p is not, in general, positive. This means that the
general theory of the Monge-Ampère equation (see e.g. Gilbarg & Trudinger [24], pp.
467-482) will, in general, not apply in this case as the character of the equation may
possibly change inside the domain.
It is natural to ask for which pressure functions p there exists a solution φ of
(1.6). This is, however, a dicult question. Nevertheless, it turns out that that we
can eliminate some classes of functions for the pressure p. For this let us consider
u ∈ H01 (Ω). This translates, using the substitution (1.5), to boundary conditions for
φ of the form
on ∂Ω.
∇φ = 0
In particular this means that
∂φ
∂ν
(1.7)
= 0 on ∂Ω, where ν denotes the outward normal
vector at the boundary ∂Ω. It is straightforward to show that there is no solution
φ ∈ C 2 (Ω) to the problem

φ φ − (φ )2 = 1 ∆p
xx yy
xy
2
 ∂φ = 0
∂ν
in Ω,
on ∂Ω
(1.8)
if ∆p > 0 in Ω. In fact, we can prove this directly, without introducing any theory of
the Monge-Ampère equation. Writing
φxx φyy ≥ φxx φyy − φ2xy = det D2 φ > 0
2
1.1
Motivation
1
INTRODUCTION
we see that φxx , φyy must have the same sign in Ω (by a continuity argument), i.e.
either
φxx , φyy > 0 in Ω
or
φxx , φyy < 0 in Ω.
(1.9)
Hence, supposing that φxx , φyy > 0, we can use the divergence theorem to obtain
ˆ
ˆ
∆φ dxdy =
0<
Ω
∂Ω
∂φ
dS = 0,
∂ν
a contradiction (and similarly for φxx , φyy < 0).
If we weaken the condition ∆p > 0 to ∆p ≥ 0 in Ω then the nonexistence result
of solution of (1.8) no longer holds. In fact, one can come up with the following
counterexample (or rather an example of existence).
Example 1.1.
Let Ω = (−2, 2)2 ⊂ R2 and let φ(x, y) := v(x), where v : R → R is
any function such that v ∈ C02 ([−1, 1]). It is easy to see that the boundary condition
∂φ
∂ν
= 0 is satised everywhere on ∂Ω. Moreover
det D2 φ = det
v 00 0
0
!
0
=0
in Ω
(obviously, this example could be modied to the case of a domain Ω that is of smooth
regularity).
However, there are no results (of which the author is aware) of nonexistence of a
solution φ to the problem (1.8) if we consider the boundary condition (1.7) instead
of the condition ∂φ = 0.
∂ν ∂Ω
Let us now focus on the case of ∆p ≤ 0 in Ω. Considering constant boundary
data φ|∂Ω = C , which comes from the boundary condition (1.7) provided sucient
regularity of ∂Ω, one can also show a nonexistence result. Namely, there is no φ ∈
W 2,2 (Ω) (with Ω ⊂ R2 open, bounded, strictly convex) that is a solution of the
problem

φ φ − (φ )2 = 1 ∆p
xx yy
xy
2
φ = C
in Ω,
on ∂Ω
(1.10)
with ∆p ≤ 0 in Ω, ∆p 6≡ 0 (see Corollary 5.3). This statement is by no means
an elementary fact. As we will see, it requires investigation of the theory of the
Monge-Ampère equation. We will nd out that the non-existence of the solution of
the problem (1.10) can be proved essentially using a generalized comparison principle
3
1.2
Literature review
1
INTRODUCTION
(see Theorem 5.1), which implies that any possible solution of this problem is bounded
below by the unique convex solution of the respective Dirichlet boundary value problem and, similarly, bounded above by the unique concave solution (see Theorem 5.2).
Since in the case of ∆p ≤ 0 both the convex and the concave solution will turn out
to be constant functions, the nonexistence result will follow, see Corollary 5.3. This
is the main result presented in this note.
1.2
Literature review
The theory of the Monge-Ampère equation goes back to the work of A. V. Pogorelov
[35], [36], [37] and A. D. Aleksandrov (see e.g. Kutateladze [28]), which focused
mostly on geometry of convex surfaces and on the problem of nding a closed convex surface in Rn with prescribed Gaussian curvature. A more recent exposition,
with the viewpoint of partial dierential equations, can be found in Bakelman [5],
Gutiérrez [25], Chapter 8 of Aubin [4] and Chapter 17 of Gilbarg & Trudinger [24].
Some important results regarding the Monge-Ampère equation include the notion of
generalised solution (see Pogorelov [35], [36]), the notion of viscosity solution and its
equivalence to the generalised solution (see Crandall, Ishii & Lions [16] and Caarelli
[10], pp. 137-139), Aleksandrov maximum principle (see Aleksandrov [2]), classication of Monge-Ampère equations (see Lychagin, Rubtsov & Chekalov [31]) and, most
notably, Caarelli's regularity results, see Caaralli [11] (see also Caarelli [10], [12],
[13], [14]). The Monge-Ampère equation has wide applications in geometry, particularly in the problem of prescribed Gaussian curvature (see Kutateladze [28], Pogorelov
[36], [37], Urbas [42] and Oliker [32], [33]) and in optimal control, particularly in the
mass transfer problems (see Evans [21], Evans & Gangbo [22], Dacorogna & Moser
[18] and Benamou & Brenier [6]).
1.3
Structure of the note
The structure of the note is as follows: Section 2 presents the concepts that are
fundamental in the theory of Monge-Ampère equation. It includes a description of the
properties of convex sets, convex functions and the concept of supporting hyperplanes,
which is a generalisation of the concept of tangent hyperplanes. This leads to the idea
of the normal mapping and consequently to the concept of Monge-Ampère measures,
which is the main subject of the next section, Section 3. In that section, through
the discussion of the convergence properties of Monge-Ampère measures, we learn
the importance of uniform convergence on compact sets of convex functions (this
4
2
BASIC CONCEPTS
convergence is sucient for the weak convergence of the corresponding Monge-Ampère
measures). We further present the inequalities between Monge-Ampère measures of
two convex functions, Aleksandrov maximum principle and comparison principle of
convex functions. All these results will turn out essential in Section 4 where we show
the existence of generalised solutions to homogeneous and inhomogeneous Dirichlet
problems with continuous boundary data. In the same section we also prove the
existence of at most two C 2 solutions of a 2D "strictly inhomogeneous" Dirichlet
problem and we reect on the similarities between Monge-Ampère equation and the
Laplace equation. In Section 5 we prove a generalised comparison principle, which
is a comparison principle for not necessarily convex functions. We then use this
comparison principle to show that any solution of the Monge-Ampère equation, with
not necessarily positive right-hand side, is bounded below and above by, respectively,
the unique concave solution and the unique convex solution of a certain corresponding
Dirichlet problem. Section 6 summarises the main results of the note and outlines
future perspectives.
The results presented in the note are, in most part, borrowed from Gutiérrez [25].
We also include a number of the author's own contributions, which include the main
result of this note, Section 5.2. Moreover, the presented proofs are, unless stated
otherwise, constructed independently.
2
Basic Concepts
The main aim of this section is to understand the essential properties of convex
functions and normal mapping, which we will use in the presentation of the MongeAmpère theory later.
2.1
Convex sets
Let Ω ⊂ Rn be an open set and φ : Ω → R. In most of the analysis of the Monge-
convex (i.e. if x, y ∈ Ω then λx + (1 − λ)y ∈ Ω
for all λ ∈ [0, 1]) or strictly convex (i.e. if x, y ∈ Ω then λx + (1 − λ)y ∈ Ω for all
Ampère equation we will consider Ω
λ ∈ (0, 1)).
One of the geometric importance of convexity is that we have supporting hy-
perplanes at each point on the boundary. In other words, we have the following
elementary results.
Lemma 2.1. Let n ≥ 2, Ω ⊂ Rn be an open, nonempty convex set and let x0 ∈ ∂Ω.
5
2.1
Convex sets
2
BASIC CONCEPTS
(a) (Separation lemma) There exists an ane hyperplane κ, dened by an equation
a · x + α = 0, where a ∈ Rn \ {0}, α ∈ R, such that
a · x0 + α = 0
and
Ω ⊂ {a · x + α > 0}.
(b) If Ω is strictly convex then the supporting hyperplane κ is such that for each
r > 0 there exists η > 0 such that
{x ∈ Ω : a · x + α ≤ η} ⊂ B(x0 , r)
(see Fig. 1)
(b)
(a)
a·x+α=0
Ω
r
Ω
x0
x0
a·x+α=η
Figure 1
Clearly property (b) does not hold for general convex sets (take for instance Ω :=
{(x1 , . . . , xn ) ∈ Rn : x1 > 0}).
Proof.
(a) This fact follows from the Geometric Hahn-Banach Theorem, see Leunberger
[30], p. 133, for the proof and Ekeland & Témam [19], p. 5, for interesting
applications. We note that this fact is true also in any innite-dimensional
topological vector space.
(b) Suppose otherwise that for all η > 0 we have {x ∈ Ω : a·x+α ≤ η} 6⊂ B(x0 , r).
This means that there exist sequences {ηk }k≥1 ⊂ R,{xk }k≥1 ⊂ Rn such that
k→∞
ηk −→ 0+ and xk ∈ Ω,
a · xk + α ≤ ηk ,
(2.1)
and |xk − x0 | ≥ r for all k . Moreover we can take {xk }k≥1 to be such that
|xk − x0 | = r for all k . In fact, for a given xk we can dene xk :=
6
r
x
|xk −x0 | k
+
2.2
Supporting hyperplanes
1−
r
|xk −x0 |
2
BASIC CONCEPTS
x0 . Then xk ∈ Ω by convexity of Ω,
r
r
(a · xk + α) + 1 −
(a · x0 + α) ≤ ηk
a · xk + α =
{z
}
|xk − x0 | |
|xk − x0 | | {z }
≤ηk
=0≤ηk
and
r
r
r
|xk − x0 | = xk + 1 −
x0 − x0 =
|xk − x0 | = r.
|xk − x0 |
|xk − x0 |
|xk − x0 |
Hence we can indeed assume that {xk }k≥1 satises |xk − x0 | = r for all k . As
j→∞
∂B(x0 , r) ∩ Ω is compact we can extract a subsequence xkj such that xkj −→ y
for some y ∈ ∂B(x0 , r) ∩ Ω. In particular y 6= x0 . Taking the limit kj → ∞
in (2.1) we obtain a · y + α ≤ 0. Hence, by strict convexity of Ω we obtain
1
(y
2
+ x0 ) ∈ Ω and thus, by (a)
1
1
1
0 < a · (y + x0 ) + α = (a · y + α) + (a · x0 + α) ≤ 0,
2
2
2
a contradiction.
2.2
Supporting hyperplanes
Denition 2.2.
the point x0 is
Given x0 ∈ Ω, a
supporting hyperplane to the function φ at
an ane function l(x) = φ(x0 ) + p · (x − x0 ), p ∈ Rn , such that
φ(x) ≥ l(x) for all x ∈ Ω (see Fig. 2).
supporting hyperplanes
φ
Figure 2: Supporting hyperplanes (note this is a 1D sketch of an n-dimensional
situation).
We note that, even though the supporting hyperplane is dened for any function φ,
its usual application is for φ convex. That is because any convex function φ : Ω → R
has at least one supporting hyperplane at each point x ∈ Ω (see Lemma 2.7 (a) ). Most
of the theory of the Monge-Ampère equation presented in this report is concerned
7
2.2
Supporting hyperplanes
2
BASIC CONCEPTS
with supporting hyperplanes and convex functions. However, considering hyperplanes
supporting a function from above, i.e. ane functions l(x) := φ(x0 ) + p · (x − x0 ) such
that φ(x) ≤ l(x) for all x ∈ Ω, we can obtain similar results for concave functions.
From now on, we will be concerned only with supporting hyperplanes from below (and
hence with convex functions) and we will keep in mind that the analysis of supporting
hyperplanes from above (which corresponds to concave functions) is analogous.
We also note that for a convex function φ, a sucient condition for l(x) = φ(x0 ) +
p · (x − x0 ) to be a supporting hyperplane to u at point x0 ∈ Ω is that l(x) supports
φ over any open neighbourhood U of x0 .
Lemma 2.3. If φ : Ω → R is convex and φ(x) ≥ φ(x0 ) + p · (x − x0 ) for all x ∈ U ,
where U ⊂ Ω is open and x0 ∈ U , then φ(x) ≥ φ(x0 ) + p · (x − x0 ) for all x ∈ Ω.
Proof. Suppose otherwise that there exists x1 ∈ Ω such that φ(x1 ) < φ(x0 ) + p · (x1 −
x0 ). As U is open there exists λ ∈ (0, 1) such that xλ := λx1 + (1 − λ)x0 ∈ U and we
get
φ(xλ ) ≤ λφ(x1 ) + (1 − λ)φ(x0 ) < λ(φ(x0 ) + p · (x1 − x0 )) + (1 − λ)φ(x0 )
= φ(x0 ) + p · (xλ − x0 ) ≤ φ(xλ ),
a contradiction.
Denition 2.4.
The
normal mapping of
φ, or
subdierential of φ,
is the set-
valued function ∂φ : Ω → P(R ) dened by
n
∂φ(x0 ) := {p ∈ Rn : φ(x) ≥ φ(x0 ) + p · (x − x0 ) ∀x ∈ Ω}.
Given E ⊂ Ω we dene ∂φ(E) = ∪x∈E ∂φ(x).
We will need the following lemma.
Lemma 2.5. If γ > 0, then ∂(γφ)(E) = γ(∂φ(E))
Proof.
p ∈ ∂(γφ)(E) ⇔ ∃x0 ∈ E ∀x ∈ Ω γφ(x) ≥ γφ(x0 ) + p · (x − x0 )
1
1
⇔ ∃x0 ∈ E ∀x ∈ Ω φ(x) ≥ φ(x0 ) + p · (x − x0 ) ⇔ p ∈ ∂φ(E)
γ
γ
8
2.3
Some facts about convex functions
2.3
2
BASIC CONCEPTS
Some facts about convex functions
There are several properties of convex functions that make them the main subject of
the analysis related to the Monge-Ampère equation. First of all, we have the following
bound on the slope of the supporting hyperplane to a convex function.
Lemma 2.6 (Bound on |p|). Let Ω ⊂ Rn be a bounded convex open set and φ a convex
function in Ω such that φ ≤ M on ∂Ω for some M ∈ R. If x ∈ Ω and p ∈ ∂φ(x),
then
|p| ≤
M − φ(x)
dist(x, ∂Ω)
(see Fig. 3).
dist(x, ∂Ω)
M − φ(x)
the slope of
this line
is greater or equal than the slope of
this supporting hyperplane
φ
φ(x)
Figure 3: Bound on |p| (note this is a 1D sketch of an n-dimensional situation).
Proof (we simplify the proof from Gutiérrez [25], p. 48). The claim follows trivially
if p = 0. If p 6= 0 we have φ(y) ≥ φ(x) + p · (y − x) for all y ∈ Ω and, by continuity of
p
φ, also for all y ∈ Ω. In particular, if y0 := x + dist(x, ∂Ω) |p|
∈ ∂Ω then M ≥ φ(y0 ) ≥
φ(x) + p · (y0 − x) = φ(x) + dist(x, ∂Ω)|p|.
We have the following properties of convex functions.
Lemma 2.7. Let φ : Ω → R be convex. Then
(a) φ has a supporting hyperplane at every point x ∈ Ω,
(b) φ is continuous,
(c) for K ⊂ Ω compact, ∂φ(K) is compact,
(d) for K ⊂ Ω compact, φ is uniformly Lipschitz in K ,
(e) φ is dierentiable a.e. in Ω.
9
2.3
Some facts about convex functions
2
BASIC CONCEPTS
Proof.
(a) If φ is convex then A := {(x, z) ∈ Ω × R : φ(x) < z} is convex. Indeed, if
(x1 , z1 ), (x2 , z2 ) ∈ A then φ(x1 ) < z1 , φ(x2 ) < z2 and hence, for all λ ∈ [0, 1],
φ(λx1 + (1 − λ)x2 ) ≤ λφ(x1 ) + (1 − λ)φ(x2 ) < λz1 + (1 − λ)z2 ,
which means that λ(x1 , z1 ) + (1 − λ)(x2 , z2 ) ∈ A.
Now let x0 ∈ Ω. From the separation lemma (Lemma 2.1, (a) ) we know that
there exists a hyperplane κ ⊂ Rn+1 that separates A and (x0 , φ(x0 )). In other
words, if a · x + αz + β = 0, a ∈ Rn , α, β ∈ R with (a, α) 6= 0 is the equation of
κ, then
a · x + αz + β > 0 ∀(x, z) ∈ A,
a · x0 + αφ(x0 ) + β = 0.
(2.2)
We will show that α > 0. Taking any z0 greater than φ(x0 ) we have (x0 , z0 ) ∈ A
and hence
a · x0 + αz0 + β > 0 = a · x0 + αφ(x0 ) + β.
Therefore α(φ(x0 ) − z0 ) < 0 and consequently α > 0. Hence
1
β
z > − a · x − =: l(x) ∀(x, z) ∈ A.
α
α
Hence, as (x, φ(x)) ∈ A for each x ∈ Ω, we obtain, by continuity, φ(x) ≥ l(x)
for all x ∈ Ω. Noting that φ(x0 ) = l(x0 ) we deduce that l(·) is a supporting
hyperplane of φ at x0 .
(b) Let x0 ∈ Ω, we will show that φ is continuous at x0 . Without loss of generality
we can assume that x0 = 0 and that Q := [−1, 1]n ⊂ Ω (the general case follows
by translation and scaling). For x ∈ Q we have
x1 +1
2
∈ [−1, 1] and, by convexity,
x1 + 1
1 − x1
φ(x) = φ
(−1, x2 , . . . , xn ) +
(1, x2 , . . . , xn )
2
2
x1 + 1
1 − x1
≤
φ(−1, x2 , . . . , xn ) +
φ(1, x2 , . . . , xn )
2
2
x1 + 1
1 − x1
≤
sup φ(y) +
sup φ(y) = sup φ(y).
2 y∈∂Q
2 y∈∂Q
y∈∂Q
Hence φ is bounded by its values on ∂Q. Repeating this reasoning for each
±1
of the sides of ∂Q (i.e. considering a function ψm
: [−1, 1]n−1 , dened by
10
2.3
Some facts about convex functions
2
BASIC CONCEPTS
±1
(y1 , . . . , yn−1 ) := φ(y1 , . . . , ym−1 , ±1, ym , . . . , yn−1 ) for m = 1, . . . , n) it folψm
lows that φ is bounded by its values at the corners of Q, i.e.
sup φ(y) ≤
y∈Q
max
s:{1,...,n}→{0,1}
φ
n
X
!
(−1)s(i) ei
=: C,
i=1
where ei denotes the unit vector in the i-th direction. Note that C < ∞ as we
take the maximum over a nite number of points.
For y ∈ ∂Q and λ ∈ [0, 1] write
φ(λy) ≤ (1 − λ)φ(0) + λφ(y) = φ(0) + λ(φ(y) − φ(0)),
which is equivalent to
φ(λy) − φ(0) ≤ λ(φ(0) − φ(−y)).
(2.3)
Also, for all t ∈ 0, 12 we have
t
t
y + t(−y) ≤ (1 − t)φ
y + t φ(−y),
φ(0) = φ (1 − t)
1−t
1−t
which, after dividing by 1 − t and setting λ :=
t
1−t
∈ [0, 1], gives
(1 + λ)φ(0) ≤ φ(λy) + λφ(−y)
or, equivalently,
−λ(φ(−y) − φ(0)) ≤ φ(λy) − φ(0).
This, together with (2.3) gives as the following lower and upper bounds on the
dierence φ(λy) − φ(0):
−λ(φ(−y) − φ(0)) ≤ φ(λy) − φ(0) ≤ λ(φ(0) − φ(−y)).
(2.4)
Now for λ ∈ [0, 1] let Qλ := {λy : y ∈ Q}. For z ∈ ∂Qλ (2.4) implies that
|φ(z) − φ(0)| ≤ λ sup φ(y) − φ(0) ≤ λ(C − φ(0)).
y∈∂Q
On the other hand, for any z ∈ Qλ , there exists λ1 ≤ λ such that z ∈ ∂Qλ1 and
11
2.3
Some facts about convex functions
2
BASIC CONCEPTS
hence
|φ(z) − φ(0)| ≤ λ1 (C − φ(0)) ≤ λ(C − φ(0))
∀z ∈ Qλ
Taking λ → 0+ gives continuity of φ at 0.
(c) (we modify the proof from Gutiérrez [25], p. 2) Let {pk }k≥1 ⊂ ∂φ(K) be any
sequence. For each k there exists xk ∈ K such that pk ∈ ∂φ(xk ), that is
for all x ∈ Ω.
φ(x) ≥ φ(xk ) + pk · (x − xk )
Since K is compact, there exists a subsequence, which we will also denote
k→∞
xk , such that xk −→ x0 for some x0 ∈ K . Since the set Kδ := {x ∈ Ω :
dist(x, K) ≤ δ} is compact and contained in Ω for suciently small δ and since
xk ∈ IntKδ for all k , we have, by Lemma 2.6,
1
|pk | ≤
δ
max φ − min φ ,
Kδ
K
where (maxKδ φ − minK φ) < ∞ by continuity of φ (see (b) ). Hence the
sequence {pk }k≥1 is bounded. Consequently, there exists a convergent subm→∞
sequence {pkm }m≥1 such that pkm −→ p0 for some p0 ∈ Rn with |p0 | ≤
1
δ
(maxKδ φ − minK φ). We will show that p0 ∈ ∂φ(K). We note that
φ(x) ≥ φ(xkm ) + pkm · (x − xkm ) ∀x ∈ Ω, ∀m.
Hence, taking the limit m → ∞ and using continuity of φ we get φ(x) ≥
φ(x0 ) + p0 · (x − x0 ) for all x ∈ Ω. Hence p0 ∈ ∂φ(x0 ) ⊂ ∂φ(K).
(d) (we paraphrase the proof from Gutiérrez [25], p. 3) From (a) φ has a supporting
hyperplane at any x ∈ Ω, hence, for each x ∈ K we have px ∈ ∂φ(x) such that
φ(y) ≥ φ(x) + p · (y − x) for all y ∈ Ω. Let C := supp∈∂φ(K) |p|. From (c) we
have C < ∞ and therefore we may write
φ(y) − φ(x) ≥ −|px ||y − x| ≥ −C|y − x| ∀x, y ∈ K.
This and reversing the roles of x and y gives |φ(y) − φ(x)| ≥ C|y − x| for all
x, y ∈ K .
(e) (following Lemma 1.1.7 in Gutiérrez [25]) The function φ is locally Lipschitz
(by (d) ) and hence it is dierentiable a.e. in Ω by Rademacher's Theorem (see
12
2.4
Properties of the normal mapping
2
BASIC CONCEPTS
Evans & Gariepy [23], pp. 81-84).
Corollary 2.8. If
compact
φ ∈ C(Ω) is convex and supΩ |φ| ≤ M then for every K ⊂ Ω
|φ(x) − φ(y)| ≤
2M
|x − y|
dist(K, ∂Ω)
∀x, y ∈ K.
Proof. Follows directly from Lemma 2.6 and the proof of point (d) of Lemma 2.7.
2.4
Properties of the normal mapping
The Legendre transform is a useful tool in studying the normal mapping.
Denition 2.9. The Legendre transform of the function φ : Ω → R is the function
φ∗ : Rn → R dened by
φ∗ (p) := sup(x · p − φ(x))
x∈Ω
(see Fig. 4).
x·p
φ(x)
φ∗ (p)
Rn
0
Figure 4: Interpretation of the Legendre transform (note this is a 1D sketch of an
n-dimensional situation).
We have the following two simple facts.
Fact 2.10. Let φ∗ be the Legendre transform of φ. Then
(a) φ∗ is convex in Rn ,
(b) if Ω is bounded and φ is bounded in Ω, then φ∗ is nite (at each p ∈ Rn ).
Proof.
(a) Let p1 , p2 ∈ Rn and λ ∈ [0, 1]. We have, for all x ∈ Ω,
x · (λp1 + (1 − λ)p2 ) − φ(x) = λ(x · p1 − φ(x)) + (1 − λ)(x · p2 − φ(x))
≤ λφ∗ (p1 ) + (1 − λ)φ∗ (p2 ).
Taking supx∈Ω nishes the proof.
13
2.4
Properties of the normal mapping
2
BASIC CONCEPTS
(b) We have φ∗ (p) ≤ supx∈Ω |x| |p| + supx∈Ω |φ(x)| ≤ C1 |p| + C2 for each p ∈ Rn .
We can use Legendre transform to prove the following, result, which is important in
studying convergence properties of Monge-Ampère measures (see Lemma 3.6 (b)) and
in the proof of existence of solution of the homogeneous Dirichlet problem (Theorem
4.2)
Lemma 2.11. If Ω is open and φ is a continuous function in Ω, then the set of points
in Rn that belong to the image by the normal mapping of more than one point of Ω
has Lebesgue measure zero. That is, the set
S := {p ∈ Rn : ∃x, y ∈ Ω, x 6= y and p ∈ ∂φ(x) ∩ ∂φ(y)}
has measure zero. This also means that the set of supporting hyperplanes that touch
the graph of φ at more than one point has measure zero.
Proof (we paraphrase the proof from Gutiérrez [25], p. 4). We may assume that
Ω is bounded because otherwise we can write Ω =
S
k≥1
Ωk , where Ωk is open, Ωk
is compact and Ωk ⊂ Ωk+1 for all k (one can take for instance Ωk := Ω ∩ B(0, k)).
If p ∈ S , then there exist x, y ∈ Ω, x 6= y with φ(z) ≥ φ(x) + p · (z − x), φ(z) ≥
φ(y) + p · (z − y) for all z ∈ Ω. Since {Ωk }k≥1 is an increasing family of sets, x, y ∈ Ωk0
for some k0 ≥ 1 and obviously the previous inequalities hold true for z ∈ Ωk0 . That
is, if
n
o
Sk0 := p ∈ Rn : ∃x, y ∈ Ω, x 6= y and p ∈ ∂ φ|Ωk (x) ∩ ∂ φ|Ωk (y)
0
then we have p ∈ Sk0 . This means that S ⊂
S
k
0
Sk and then we show that each Sk
has measure zero. Hence we may assume boundedness of Ω.
Let φ∗ be the Legendre transform of φ. By Fact 2.10 φ∗ is convex and hence differentiable almost everywhere (see Lemma 2.7 (e) ). We will show that if p ∈ S
then u∗ is not dierentiable at p, which implies |S| = 0. Indeed, if p ∈ S , then
p ∈ ∂φ(x1 ) ∩ ∂φ(x2 ) with x1 6= x2 . This means that φ(z) ≥ φ(xi ) + p · (z − xi ) for all
z ∈ Ω, i = 1, 2 or, equivalently,
φ∗ (p) = xi · p − φ(xi ),
i = 1, 2.
On the other hand φ∗ (z) ≥ xi · z − φ(xi ) for all z ∈ Ω and so φ∗ (z) ≥ φ∗ (p) + xi · (z − p)
for all z and i = 1, 2. Hence, if φ∗ were dierentiable at p we would have ∇φ∗ (p) = xi ,
i = 1, 2.
14
3
3
THE MONGE-AMPÈRE MEASURE
The Monge-Ampère measure
3.1
Introduction
We dene the Monge-Ampère measure to be the Lebesgue measure of the normal
mapping. We have the following lemma.
Lemma 3.1. For φ ∈ C(Ω) let
S := {E ⊂ Ω : ∂φ(E) is Lebesgue measurable}
Then S is a Borel σ-algebra and the set function M φ : S → R dened by
M φ(E) := |∂φ(E)|,
where R := R ∪ {∞}, is a measure, that is nite on compact sets. This measure is
called the Monge-Ampère measure associated with the function φ.
The proof of the above lemma is rather technical and hence it is omitted. It can
be found in Gutiérrez [25], p. 5.
The reason for the name of this measure is clear from the following lemma.
Lemma 3.2. If
φ ∈ C 2 (Ω) then M φ is absolutely continuous with respect to the
Lebesgue measure and
ˆ
for all Borel sets E ⊂ Ω.
det D2 φ dx
M φ(E) =
E
The proof of this lemma, which uses Sard's Theorem, can be found in Gutiérrez
[25], p. 4.
Example 3.3 (Examples of Monge-Ampère measures).
Let x0 ∈ Ω.
(a) If φ ∈ C 2 (Ω) then, by Lemma 3.2,
M φ(E) = |∇φ(E)|
for all Borel subsets E ⊂ Ω.
In particular, if φ(x) := q · x + α, where q ∈ Rn , α ∈ R, then M φ = 0.
15
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
(b) If φ(x) := δ|x − x0 | then
∂φ(x) =

B(0, δ)
δ
x−x0
|x−x0 |
Hence for each Borel subset E ⊂ Rn

B(0, δ)
M φ(E) =
|∇φ(E)|
for x = x0 ,
for x 6= x0 .
if {x0 } ⊂ E,
if {x0 } 6⊂ E.
(c) If φ(x) := δ|x − x0 |2 then, D2 φ = 2δI and so by Lemma 3.2,
ˆ
M φ(E) =
E
det D2 φ dx = (2δ)n |E|.
(3.1)
(d) If γ > 0, then M (γφ) = γ n M φ for every φ convex (cf. Lemma 2.5).
3.2
Convergence properties
We want to prove certain convergence properties of Monge-Ampère measures.
Denition 3.4.
Let µ, µj be Borel measures dened on a σ -algebra of Borel subsets
of an open set Ω ⊂ Rn and let µ(Ω), µj (Ω) ≤ M for some M > 0. We say that µj
converge weakly to µ (denoted µj *µ) if
ˆ
Ω
j→∞
f (x)dµj (x) −→
ˆ
Ω
f (x)dµ(x) ∀f ∈ C0 (Ω).
We will show that uniform convergence on compact subsets of convex functions is
sucient for the weak convergence of the respective Monge-Ampère measures, which
will later turn out to be essential in showing the existence of solutions to an inhomogeneous Dirichlet problem of Monge-Ampère equation (Section 4.3). The convergence
properties can also be used in proving inequalities regarding Monge-Ampère measures
(Section 3.3). Namely, they enable one to use approximation arguments in showing
the superadditivity of Monge-Ampère measure in the case of convex functions that
are not C 2 (see Lemma 3.10).
In the statement of convergence lemma (Lemma 3.6), which we borrow from
Gutiérrez [25], pp. 6-7, we will apply the formulation of the Fatou lemma which diers
´
from its usual occurence in analysis (i.e. if fj ∈ L1 , fj ≥ 0 and lim inf j→∞ fj < ∞
16
3.2
then
Convergence properties
´
3
lim inf j→∞ fj ≤ lim inf j→∞
´
THE MONGE-AMPÈRE MEASURE
fj (see Alt [3], p. 96)) and which is not explained
in detail in Gutiérrez [25]. For this reason we rst introduce this formulation. Recall
the denitions
lim sup Ak :=
k→∞
\ [
Am ,
lim inf Ak :=
k→∞
k≥0 m≥k
[ \
Am .
k≥0 m≥k
innitely many
Ak and lim inf k→∞ Ak consists of elements that belong to all but nitely many Ak .
In other words, lim supk→∞ Ak consists of elements that belong to
Obviously, lim inf k→∞ Ak ⊂ lim supk→∞ Ak . We have the following result.
Lemma 3.5 (Fatou lemma). Let |·| be a measure dened on a σ-algebra Σ. If Ak ∈ Σ
then lim inf k→∞ Ak ∈ Σ and
| lim inf Ak | ≤ lim inf |Ak |.
k→∞
k→∞
(3.2)
In particular if A ⊂ lim inf k→∞ Ak then |A| ≤ lim inf k→∞ |Ak |.
Similarly, lim supk→∞ Ak ∈ Σ,
lim sup |Ak | ≤ | lim sup Ak |
k→∞
k→∞
and if lim supk→∞ Ak ⊂ A then lim supk→∞ |Ak | ≤ A.
Proof (inspired by the proof of the "usual" Fatou lemma in Alt [3], pp. 96-97).
We have lim inf k→∞ Ak =
S
k≥0
T
m≥k
Am , hence lim inf k→∞ Ak ∈ Σ as a countable
intersection and sum of sets Am ∈ Σ. Let
Gk :=
\
Am .
m≥k
Then lim inf k→∞ Ak =
S
k
Gk , Gk is an increasing family of sets (i.e. Gk ⊂ Gk+1 ) and
Gk ⊂ Aj for all j ≥ k . Hence also |Gk | ≤ |Aj | for all j ≥ k and taking lim inf j→∞ we
get
|Gk | ≤ lim inf |Aj |
j→∞
∀k
(3.3)
Letting k → ∞ and using the monotonicity Gk ⊂ Gk+1 we get
[ (3.3)
| lim inf Aj | = Gk = lim |Gk | ≤ lim inf |Aj |.
j→∞
j→∞
k→∞
k
S
T
For the lim supk→∞ Ak replace "lim inf ", " ", " ", "⊂", "≤", "increasing" with
17
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
T
S
"lim sup", " ", " ", "⊃", "≥", "decreasing" respectively.
We can now state the convergence lemma (which is a reformulation of Lemma
1.2.2 from Gutièrrez [25]).
Lemma 3.6
(Convergence lemma).
Let φk ∈ C(Ω) be convex functions such that
φk −→ φ uniformly on compact subsets of Ω. Then
k→∞
(a) If K ⊂ Ω is compact, then
lim sup ∂φk (K) ⊂ ∂φ(K),
k→∞
and by the Fatou lemma
lim sup M φk (K) ≤ M φ(K).
(3.4)
k→∞
(b) If U ⊂ Ω is open and {Vk }k≥1 is a family of open sets such that Vk+1 ⊂ Vk and
U ⊂ Vk ⊂ Ω for each k , then
∂φ(U ) ⊂ lim inf ∂φk (Vk ),
k→∞
where the inequality holds for almost every point of the set on the left-hand side,
and by the Fatou lemma
M φ(U ) ≤ lim inf M φk (Vk ).
k→∞
(3.5)
Proof (we modify the proof of Lemma 1.2.2 from Gutièrrez [25])
(a) If p ∈ lim supk→∞ ∂φk (K), then for each n there exist kn and xkn ∈ K such
that p ∈ ∂φkn (xkn ). By selecting a subsequence xj of xkn we may assume that
xj → x0 ∈ K . On the other hand,
φj (x) ≥ φj (xj ) + p · (x − xj ),
∀x ∈ Ω,
and by letting j → ∞ we get, by the uniform convergence of φj on compact
sets,
φ(x) ≥ φ(x0 ) + p · (x − x0 ),
that is p ∈ ∂φ(x0 ) ⊂ ∂φ(K).
18
∀x ∈ Ω,
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
(b) Let S := {p : p ∈ ∂φ(x1 ) ∩ ∂φ(x2 ) for some x1 , x2 ∈ Ω, x1 6= x2 }. By Lemma
2.11, |S| = 0. Consider ∂φ(U ) \ S . If p ∈ ∂φ(U ) \ S , then there exists a unique
x0 ∈ U such that p ∈ ∂φ(x0 ). Hence
(3.6)
φ(x) > φ(x0 ) + p · (x − x0 ) ∀x ∈ Ω, x 6= x0 .
Otherwise, if φ(x1 ) = φ(x0 ) + p · (x1 − x0 ) for some x1 ∈ Ω, x1 6= x0 then
φ(x) ≥ φ(x0 ) + p · (x − x0 ) = φ(x1 ) + p · (x − x1 ) for all x ∈ Ω. Letting λ ∈ (0, 1)
be such that xλ := λx0 + (1 − λ)x1 ∈ U (such a λ exists because U is open).
Since p ∈ ∂φ(x0 ) and since φ is convex, we have for all x ∈ Ω
φ(x) = λφ(x) + (1 − λ)φ(x)
≥ λ(φ(x0 ) + p · (x − x0 )) + (1 − λ)(φ(x1 ) + p · (x − x1 ))
= λφ(x0 ) + (1 − λ)φ(x1 ) + p · (x − xλ ) ≥ φ(xλ ) + p · (x − xλ ),
that is, p ∈ ∂φ(xλ ), which contradicts the uniqueness of x0 .
Now, since V1 ⊂ Ω is compact and φk → φ uniformly on V1 , we have from (3.6)
that for any δ > 0 there exists kδ such that φk (x) ≥ φk (x0 ) + p · (x − x0 ) − δ for
all x ∈ V1 and k ≥ kδ . Fix any δ > 0 and hence also kδ . Let
δk := min{φk (x) − φk (x0 ) − p · (x − x0 ) + δ}.
x∈Vk
(3.7)
This minimum is attained at some xk ∈ Vk , i.e.
δk = φk (xk ) − φk (x0 ) − p · (xk − x0 ) + δ.
(3.8)
Hence we have
(3.7)
(3.8)
φk (x) ≥ φk (x0 ) + p · (x − x0 ) − δ + δk = φk (xk ) + p · (x − xk ) ∀x ∈ Vk (3.9)
and, in particular,
φk (x) ≥ φk (xk ) + p · (x − xk ) ∀x ∈ U
(3.10)
We will show that p is the slope of a supporting hyperplane to φk at the point xk
for all but nitely many k . Note that, by Lemma 2.3 (for an ane function to
support a convex function at a given point over Ω it is sucient to support it over
19
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
an open neighbourhood of this point), we only need to show that xk ∈ Vk for all
but nitely many k . Suppose otherwise that xk ∈ ∂Vk for innitely many k , i.e.
suppose that there exists a subsequence {xkm }m≥1 ⊂ V1 such that xkm ∈ ∂Vkm
for all m ≥ 1. As V1 is compact, there exists a subsequence, which we will also
m→∞
denote {xkm }m≥1 , such that xkm −→ y for some y ∈ V1 . Moreover y 6= x0
as |xkm − x0 | ≥ dist(x0 , ∂Vkm ) ≥ dist(x0 , ∂U ) > 0 for all m. By the uniform
m→∞
convergence of φk to a continuous function φ we get φkm (xkm ) −→ φ(y) (indeed,
one can write |φkm (xkm ) − φ(y)| ≤ |φkm (xkm ) − φ(xkm )| + |φ(xkm ) − φ(y)| ≤
m→∞
||φkm − φ||L∞ (V1 ) + |φ(xkm ) − φ(y)| −→ 0). Hence, taking limm→∞ in (3.10) and
letting x := x0 we get
(3.6)
φ(x0 ) ≥ φ(y) + p · (x0 − y) > φ(x0 ),
a contradiction. Therefore indeed xk ∈ Vk for all but nitely many k and
consequently p ∈ ∂φk (xk ) ⊂ ∂φk (Vk ) for all but nitely many k . This means
that p ∈ lim inf k→∞ ∂φk (Vk ).
One of the consequences of this lemma is the weak convergence of Monge-Ampère
measures. Namely, we have the following result (formulated without proof in Gutièrrez [25], p. 8).
Lemma 3.7. If φk are convex functions in Ω such that φk → φ uniformly on compact
subsets of Ω and M φ(Ω), M φk (Ω) ≤ L for some constant L > 0, then the associated
Monge-Ampère measures M φk tend to M φ weakly, that is
ˆ
Ω
ˆ
f (x)dM φk (x) →
f (x)dM φ(x)
Ω
for every f continuous with compact support in Ω (cf. Denition 3.4).
Proof (inspired by a suggestion of one of the supervisors). We will assume that f ≥ 0
(the general case follows by considering the positive and negative parts f + , f − and
writing f = f + − f − ).
Fix > 0 and let F := maxΩ f and Ω0 := {x ∈ Ω : f (x) > 0} ≡ Int(supp f ).
Since f is integrable with respect to M φ, there exist N > 0, ai ∈ (0, L], Ui ⊂ Ω0 Borel
P
sets, i = 1, . . . , N , such that Ui ∩ Uj = ∅ for i 6= j , N
i=1 ai χUi ≤ f (where χE denotes
the characteristic function of a borel set E ) and
N
X
i=1
ˆ
ai M φ(Ui ) ≤
Ω0
f dM φ ≤
20
N
X
i=1
ai M φ(Ui ) + .
(3.11)
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
Moreover, we will show that, by continuity of f , we can take Ui to be open. For this
reason let
T := {τ ∈ R : M φ({x ∈ Ω0 : f = τ }) = 0}.
S
Note that (−∞, 0] ∪ (F, +∞) ⊂ T . We have (0, F ] \ T = n≥1 Tn , where
Tn := {τ ∈ (0, F ] : M φ({x ∈ Ω0 : f = τ }) ≥ 1/n}.
P
P
It is clear that #Tn ≤ Ln (otherwise L <
τ ∈Tn 1/n ≤
τ ∈Tn M φ({f = τ }) =
S
0
M φ( τ ∈Tn {f = τ }) ≤ M φ(Ω ) ≤ L, a contradiction). Consequently, [0, F ] \ T is
at most countable. We will use this fact to see that T is dense in (0, F ]. In fact,
if T is not dense in (0, F ], then there exists an open interval (a, b) ⊂ (0, F ] such
that (a, b) ∩ T = ∅, i.e. (a, b) ⊂ (0, F ] \ T , which is a contradiction since (a, b) is
uncountable. Hence T is indeed dense in (0, F ].
Therefore, we can nd N ≥ 1 and a sequence {τi }i=0,...,N ⊂ T such that 0 =
for all i = 1, . . . , N (for instance
1
⊂ T such that τ0 := 0, τN := F 1 + 4N
and
τ0 < τ1 < . . . < τN −1 < F < τN and τi − τi−1 ≤
take any N ≥ 4F L/ and {τi }i=0,...,N
|τi − i M
|≤
N
F
4N
for i = 1, . . . , N − 1).
L
We now dene
Ui := {x ∈ Ω0 : τi−1 < f (x) < τi }
for i = 1, . . . , N.
We note that the sets Ui , i = 1, . . . , N are open (by continuity of f ) and mutually
P
disjoint and that N
i=1 τi−1 χUi ≤ f . We also have
!
M φ Ω0 \
[
Ui
=
N
X
i=1
i=1,...,N
M φ({f = τi }) = 0
(3.12)
and therefore
ˆ
Ω0
f dM φ −
N
X
i=1
τi−1 M φ(Ui ) =
N ˆ
X
Ui
i=1
(f − τi−1 ) dM φ ≤
N ˆ
X
i=1
Ui
(τi − τi−1 ) dM φ
N
X
≤
M φ(Ui ) ≤ M φ(Ω0 ) ≤ ,
L i=1
L
which means that (3.11) holds (with ai := τi−1 for all i = 1, . . . , N ), as claimed.
For each i let {Vik }k≥1 be a family of open sets such that Vik ⊂ Ω, Ui ⊂ Vik ,
Vik+1 ⊂ Vik and M φk (Vik ) ≤ M φk (Ui ) +
21
2i τN
for all k ≥ 1 (such a family exists
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
by compactness of supp f and by the regularity of Borel measures M φk , k ≥ 1, see
Parthasarathy [34], p. 27). Note that, by point (b) of the above lemma, we have
M φ(Ui ) ≤ lim inf k→∞ M φk (Vik ) for each i = 1, . . . , N and hence
ˆ
ˆ
f dM φ =
Ω0
Ω
N
X
f dM φ ≤
N
X
≤ lim inf
k→∞
k→∞
ˆ
≤ lim inf
τi−1 M φ(Ui ) + ≤
τi−1 M φk (Ui ) +
i=1
N
X
= lim inf
i=1
i=1
N
X
i
2 τN
+ ≤ lim inf
N
X
k→∞
τi−1 M φk (Ui ) + 2
i=1
!!
ˆ
k→∞
Ui
+ 2
i=1,...,N
(3.13)
f dM φk + 2.
Ω
´
1, . . . , N , let
[
Ω0 \
τi−1 M φk (Ui ) + 0 · M φk
Ω0
We will now show that
k→∞
i=1
f dM φk + 2 = lim inf
k→∞
τi−1 lim inf M φk (Vik ) + f dM φ ≥ lim supk→∞
Ω
´
Ω
f dM φk − . For each i =
Wi := {x ∈ Ω0 : τi−1 ≤ f (x) ≤ τi }.
S
Note that each Wi , i = 1, . . . , N , is a compact subset of Ω and that Ω0 = i=2,...,N (Wi \
P
0
Wi−1 ) ∪ W1 . We have f ≤ N
i=1 τi χWi and, since M φ({x ∈ Ω : f = τi }) = 0 for all
i = 1, . . . , N we have
N
X
i=1
ˆ
τi M φ(Wi ) −
f dM φ =
N ˆ
X
Ω
i=1
≤
L
Wi
N
X
N ˆ
X
(τi − f ) dM φ ≤
M φ(Wi ) =
i=1
i=1
Wi
(τi − τi−1 ) dM φ
M φ(Ω0 ) ≤ .
L
Now, using point (a) of the above lemma, we can write
ˆ
Ω
f dM φ ≥
N
X
i=1
τi M φ(Wi ) − ≥
≥ lim sup
k→∞
N
X
i=1
N
X
i=1
≥ lim sup
k→∞
ˆ
Ω0
k→∞
τi M φk (Wi ) − ≥ lim sup τ1 M φk (W1 ) +
k→∞
τi lim sup M φk (Wi ) − N
X
i=2
!
τi M φk (Wi \ Wi−1 )
ˆ
f dM φk − = lim sup
k→∞
22
Ω
f dM φk − .
−
3.2
Convergence properties
3
THE MONGE-AMPÈRE MEASURE
From here and from (3.13), taking the limit → 0+ , we get limk→∞
´
f dM φ.
Ω
´
Ω
f dM φk =
Another proof of this lemma, which uses a roundabout approach of a certain space
of currents, can be found in Rauch & Taylor [39], pp. 353-354.
We also state other consequences of Lemma 3.6.
Corollary 3.8. If
φk , ψk ∈ C(Ω) are convex in Ω such that φk → φ, ψk → ψ
uniformly on compact subsets of Ω and
M φk ≤ M ψk
in Ω ∀ k
then
(3.14)
in Ω.
Mφ ≤ Mψ
(3.15)
Proof. It is enough to prove (3.15) for K ⊂ Ω with K compact (by regularity of Borel
measures, see Parthasarathy [34], p. 27). Let Um := {x ∈ Ω : dist(x, K) < δ/m} for
T
δ > 0 such that U1 ⊂ Ω. We have K = m≥1 Um . For a xed m consider a family of
k
k+1
k
k
k
and Um ⊂ Um
⊂ Um
⊂ Ω, Um
for all k (take for
}k≥1 such that Um
open sets {Um
k
:= {x ∈ Ω : dist(x, Um ) < 1/k}). Then, by Lemma 3.6,
instance Um
(3.5)
k
k)
M φ(K) ≤ M φ(Um ) ≤ lim inf M φk (Um
) ≤ lim inf M φk (Um
k→∞
k→∞
(3.14)
(3.4)
k ) ≤ lim sup M ψ (U k ) ≤ M ψ(U ).
≤ lim sup M φk (Um
k
m
m
k→∞
k→∞
Therefore, taking m → ∞, we obtain
!
M φ(U ) ≤ lim M ψ(Um ) = M ψ
m→∞
\
Um
= M ψ(K),
m≥1
which nishes the proof.
Corollary 3.9. Similarly, if
φk , ψk , wk ∈ C(Ω) are convex in Ω such that φk → φ,
ψk → ψ , ϕk → w uniformly on compact subsets of Ω and M φk ≤ M ψk + M ϕk in Ω
then
Mφ ≤ Mψ + Mϕ
Proof. Similar to the proof of Corollary 3.8.
23
in Ω.
(3.16)
3.3
3.3
Inequalities
3
THE MONGE-AMPÈRE MEASURE
Inequalities
The last Corollary can be applied in showing the superadditivity of Monge-Ampère
measure for C(Ω) functions.
Lemma 3.10
(Superadditivity of Monge-Ampère measure).
convex with Ω ⊂ R convex. Then
n
Let φ, ψ ∈ C(Ω) be
M (φ + ψ)(E) ≥ M φ(E) + M ψ(E) ∀ Borel sets E ⊂ Ω.
(3.17)
Proof. First suppose that ψ, φ ∈ C 2 (Ω). Then (3.17) is equivalent to
det (D2 φ + D2 ψ) ≥ det D2 φ + det D2 ψ
in Ω.
(3.18)
As φ, ψ are convex functions, each of D2 φ, D2 ψ , D2 (φ + ψ) is a positive denite
matrix in Ω.
A, B ∈ R
n×n
Therefore (3.18) follows as det (A + B) ≥ det A + det B for any
positive denite (in fact a stronger inequality holds: (det (A + B))1/n ≥
(det A)1/n + (det B)1/n , see Horn & Johnson [27], pp. 510511, or Bhatia [9], p. 114,
for further generalisations).
Now let ψ, φ ∈ C(Ω) and x K ⊂ Ω compact.
Let > 0 be such that <
1
dist(K, ∂Ω),
2
Ω := {x ∈ Ω : dist (x, ∂Ω) > }
(note that Ω is convex) and let η ∈ C0∞ (B(0, )) be a nonnegative function
such that
−1 x 2
´
η dx = 1 (for instance take a standard mollier η := C exp
−1
,
B(0,) ´
where C > 0 is chosen such that B(0,) η dx = 1). Let
ˆ
ψ := (ψ ∗ η )(x) ≡
B(0,)
ψ(x − y)η (y)dy
∀ x ∈ Ω
be a mollication of a function ψ and similarly dene φ := φ ∗ η . We have ψ , φ ∈
24
3.3
Inequalities
3
THE MONGE-AMPÈRE MEASURE
C ∞ (Ω ) (as η ∈ C0∞ (B(0, ))) and ψ , φ are convex. Indeed
ˆ
ψ (λx1 + (1 − λ)x2 ) =
B(0,)
ˆ
=
B(0,)
ˆ
≤ λ
ψ(λx1 + (1 − λ)x2 − y)η (y)dy
ψ(λ(x1 − y) + (1 − λ)(x2 − y))η (y)dy
ˆ
ψ(x1 − y)η (y)dy + (1 − λ)
ψ(x2 − y)η (y)dy
B(0,)
B(0,)
= λψ (x1 ) + (1 − λ)ψ (x2 ) ∀λ ∈ [0, 1], x1 , x2 ∈ Ω .
From the above argument we have that
M (φ + ψ )(K) ≥ M φ (K) + M ψ (K)
→0+
(3.19)
→0+
Noting that ψ −→ ψ and φ −→ φ uniformly on compact subsets of Ω we may use
Corollary 3.9 to take the limit → 0+ in (3.19) and get M (φ + ψ)(K) ≥ M φ(K) +
M ψ(K).
One of the important implications of this lemma is that adding an ane function
to a convex function does not change its Monge-Ampère measure.
Corollary 3.11. Let ψ ∈ C(Ω) be convex and let q ∈ Rn , α ∈ R. Then the function
φ(x) := ψ(x) + l(x), where l(x) := q · x + α satises M φ(E) = M ψ(E) for all Borel
sets E ⊂ Ω.
Proof. Recall that M l = 0 by Example 3.3 (a). Hence
M ψ = M ψ + M l ≤ M (ψ + l) = M φ = M φ + M (−l) ≤ M (φ − l) = M ψ.
Another important implication of Lemma 3.10 is that adding a quadratic functional to a convex function strictly increases its Monge-Ampère measure. As we will
see later, this trick makes it straightforward to prove a comparison principle for the
solutions of the Monge-Ampère equation (see Theorem 3.15). In other words, we have
the following result.
Corollary 3.12. Let ψ ∈ C(Ω) be convex and let x0 ∈ Ω. For δ > 0 let
ψδ (x) := ψ(x) + δ|x − x0 |2 .
Then
M ψδ (E) ≥ M ψ(E) + (2δ)n |E|
25
∀ Borel set E ⊂ Ω,
(3.20)
3.4
Aleksandrov maximum principle
3
THE MONGE-AMPÈRE MEASURE
where |E| denotes the Lebesgue measure of a Borel set E .
Proof.
(3.17)
(3.1)
M ψδ (E) ≥ M ψ(E) + M (δ| · −x0 |2 )(E) = M ψ(E) + (2δ)n |E|
3.4
Aleksandrov maximum principle
Here we state the Aleksandrov maximum principle, which is a classical result in the
theory of the Monge-Ampère equation and will be useful in constructing the solution
to the inhomogeneous Dirichlet problem for the Monge-Ampère equation (i.e. problem
(4.6)).
Lemma 3.13 (Aleksandrov maximum principle). If Ω ⊂ Rn is a bounded, open and
convex set with diameter D, then for all φ ∈ C(Ω) convex with φ = 0 on ∂Ω
|φ(x)|n ≤ C Dn−1 dist(x, ∂Ω)|∂φ(Ω)|
∀x ∈ Ω,
where C is a constant depending only on the dimension n.
The proof of this lemma can be found in Gutiérrez [25], p. 12.
3.5
Comparison principle
Let Ω ⊂ Rn be a bounded open set.
Lemma 3.14. Let φ, ψ ∈ C(Ω). If φ = ψ on ∂Ω and ψ ≥ φ in Ω, then
∂ψ(Ω) ⊂ ∂φ(Ω)
(see Fig. 5).
ψ
a
φ
Figure 5: (based on Fig. 1.2. from Gutiérrez [25], p. 11)
26
3.5
Comparison principle
3
THE MONGE-AMPÈRE MEASURE
Proof (we simplify the proof from Gutiérrez [25], pp. 10-11). If p ∈ ∂ψ(Ω) then there
exists x0 ∈ Ω such that
ψ(x) ≥ ψ(x0 ) + p · (x − x0 )
∀x ∈ Ω.
(3.21)
Let
a := sup(ψ(x0 ) + p · (x − x0 ) − φ(x))
x∈Ω
(see Fig. 5). Since ψ(x0 ) ≥ φ(x0 ), we have a ≥ 0. Moreover, since Ω is bounded and
φ is continuous, there exists x1 ∈ Ω such that a = ψ(x0 ) + p · (x1 − x0 ) − φ(x1 ) and so
φ(x) ≥ ψ(x0 ) + p · (x − x0 ) − a = φ(x1 ) + p · (x − x1 ) ∀x ∈ Ω,
where the inequality holds by denition of a. Clearly, φ(x1 )+p·(x−x1 ) is a supporting
hyperplane to the function φ at x1 . This means that p ∈ ∂φ(Ω) if x1 ∈ Ω. Otherwise,
if x1 ∈ ∂Ω, then ψ(x1 ) = φ(x1 ) and by taking x := x1 in (3.21), we have
φ(x1 ) = ψ(x1 ) ≥ ψ(x0 ) + p · (x1 − x0 ) = φ(x1 ) + a.
Hence a = 0 and therefore ψ(x0 ) = φ(x0 ). Also φ(x) ≥ φ(x0 ) + p · (x − x0 ) for all
x ∈ Ω and hence p ∈ ∂φ(x0 ) ⊂ ∂φ(Ω).
We can now state the comparison principle for convex (concave) functions.
Theorem 3.15
that
(Comparison principle).
Let φ, ψ ∈ C(Ω) be convex functions such
on Ω.
Mφ ≤ Mψ
(3.22)
Then
min(φ − ψ)(x) = min (φ − v)(x).
x∈∂Ω
x∈Ω
(3.23)
Proof (from Gutiérrez [25], pp. 16-17). Suppose (3.23) is not true and we have
a := min(φ − ψ)(x) < min (φ − ψ)(x) =: b.
x∈∂Ω
x∈Ω
Then there exists x0 ∈ Ω such that a = φ(x0 ) − ψ(x0 ). Let δ > 0 be such that
δ (diam Ω)2 <
b−a
2
and let
ϕ(x) = ψ(x) + δ|x − x0 |2 +
27
b+a
.
2
4
EXISTENCE THEOREMS
Consider the set G := {x ∈ Ω : φ(x) < ϕ(x)}. We have x0 ∈ G. Also, G∩∂Ω = ∅. In
order to see it, we note that for every x ∈ ∂Ω we have ψ(x) ≤ φ(x) − b (by denition
of b) and hence
ϕ(x) ≤ φ(x) + δ|x − x0 |2 −
b−a
b−a
≤ φ(x) + δ(diam Ω)2 −
< φ(x),
2
2
which means that ϕ(x) < φ(x) for x ∈ ∂Ω. Hence G ∩ ∂Ω = ∅ and consequently
∂G = {x ∈ Ω : ϕ(x) = φ(x)}. By Lemma 3.14 we obtain ∂ϕ(G) ⊂ ∂φ(G). Noting
that ∂ϕ = ∂(ψ + δ| · −x0 |2 ) (adding a constant does not change the normal mapping)
and using Corollary 3.12 we obtain
M φ(G) ≥ M ϕ(G) = |∂(ψ + δ| · −x0 |2 )(G)| = M ψ(G) + (2δ 2 )n |G|,
which contradicts (3.22).
4
Existence theorems
First, let us introduce the generalised notion of solution to the Monge-Ampère equation.
4.1
Generalised solutions of the Monge-Ampère equation
Let Ω ⊂ Rn be open and convex.
Denition 4.1. Let ν be a Borel measure dened in Ω. The convex function φ ∈
C(Ω) is a generalised solution of the Monge-Ampère equation
det D2 φ = ν
(4.1)
if the Monge-Ampère measure M φ associated with φ equals ν .
If φ ∈ C 2 (Ω) and the Borel measure ν is absolutely continuous with respect to
the Lebesgue measure with density f ∈ C(Ω), f ≥ 0 then φ is a generalised solution
of (4.1) i det D2 φ = f in Ω.
28
4.2
Homogeneous Dirichlet problem
4.2
4
EXISTENCE THEOREMS
Homogeneous Dirichlet problem
Let Ω ⊂ Rn be bounded and strictly convex and let us consider a homogeneous
Dirichlet problem for the Monge-Ampère equation

det D2 φ = 0
in Ω,
φ = g
on ∂Ω,
(4.2)
where g : ∂Ω → R is continuous. Let
F := {a(x) : a is an ane function and a ≤ g on ∂Ω}.
We will show that problem (4.2) has a unique convex generalized solution (see Denition 4.1). What is more, the solution φ is given by
φ(x) := sup a(x).
(4.3)
a∈F
Let us give a geometric interpretation of this characterisation of the solution φ. Consider a set G := {(x, z) ∈ Rn × R : x ∈ ∂Ω and z = g(x)}, i.e. a plot of g in Rn × R.
Then one can think of φ as of the "lower boundary" of the set conv G (where conv
denote the convex hull of a set), i.e. φ(x) = minz: (x,z)∈G z . Also note that F is the
set of hyperplanes supporting G from below (i.e. below with respect to the xn+1
direction).
Similarly one can show that there exists a unique concave solution to (4.2) (in a
generalized sense) and the concave solution is the "upper boundary" of conv G.
We make these comments precise in the following theorem.
Theorem 4.2. Let
Ω ⊂ Rn be bounded and strictly convex, and g : ∂Ω → R be
a continuous function. There exists a unique convex function φ ∈ C(Ω) that is a
generalized solution of the problem (4.2).
Similarly, there exists a unique concave C(Ω) solution to (4.2).
Proof (we paraphrase and simplify the proof from Gutiérrez [25], pp. 17-19). First,
we note that uniqueness of solution follows directly from the comparison principle
(Theorem 3.15). For the existence we will show that
(a) F 6= ∅ and φ is convex with φ(x) ≤ g(x) for x ∈ ∂Ω,
(b) φ = g on ∂Ω,
29
4.2
Homogeneous Dirichlet problem
4
EXISTENCE THEOREMS
(c) φ ∈ C(Ω),
(d) ∂φ(Ω) ⊂ {p ∈ Rn : there exists x, y ∈ Ω, x 6= y and p ∈ ∂φ(x) ∩ ∂φ(y)}.
Then, by Lemma 2.11, we will have M φ(Ω) = |∂φ(Ω)| = 0, which will nish the
proof. We have
(a) As g is continuous we have a(x) := miny∈∂Ω g(y) ∈ F . Hence F 6= ∅ and φ is
well dened. We have, for x, y ∈ Ω, λ ∈ [0, 1],
φ(λx + (1 − λ)y) = sup a(λx + (1 − λ)y) ≤ λ sup a(x) + (1 − λ) sup a(y)
{z
}
a∈F |
a∈F
a∈F
=λa(x)+(1−λ)a(y)
= λφ(x) + (1 − λ)φ(y),
i.e. φ is convex. Also, φ(x) ≤ g(x) for x ∈ ∂Ω as each a(x) ∈ F has this
property.
(b) Let ξ ∈ ∂Ω. By continuity of g , given > 0 there exists δ > 0 such that
|g(x) − g(ξ)| ≤ for |x − ξ| < δ , x ∈ ∂Ω. Let l(x) = 0 be the equation of the
supporting hyperplane to Ω at the point ξ (see Lemma 2.1 (a) ). Since Ω is
strictly convex, there exists η > 0 such that S = {x ∈ Ω : l(x) ≤ η} ⊂ B(ξ, δ)
(see Lemma 2.7 (b) ). Let
M := min{g(x) : x ∈ ∂Ω, l(x) ≥ η}
and consider
a (x) := g(ξ) − − A l(x),
where
A := max
(4.4)
g(ξ) − − M
,0 .
η
We have l(ξ) = 0 and hence a (ξ) = g(ξ)−. We will show that a ∈ F . Indeed,
if x ∈ ∂Ω ∩ S , then x ∈ ∂Ω ∩ B(ξ, δ) and g(x) ≥ g(ξ) − ≥ g(ξ) − − A l(x) =
a (x). If x ∈ ∂Ω ∩ S c , then l(x) > η and by the denition of M and the choice
of A we have
g(x) ≥ M = a (x) + M − g(ξ) + + A l(x) ≥ a (x) + M − g(ξ) + + A η ≥ a (x).
Therefore a ∈ F for each > 0. In particular φ(ξ) ≥ a (ξ) = g(ξ) − for every
> 0. Taking → 0+ we get φ(ξ) ≥ g(ξ).
30
4.2
Homogeneous Dirichlet problem
4
EXISTENCE THEOREMS
(c) Since φ is convex in Ω, φ is continuous in Ω (see Lemma 2.7 (b) ). Now let
ξ ∈ ∂Ω and let {xn }n≥1 ⊂ Ω be a sequence converging to ξ . We will show
that φ(xn ) → g(ξ). Let a be dened by (4.4). As a ∈ F for each > 0,
we have φ(x) ≥ a (x) for all x ∈ Ω. In particular φ(xn ) ≥ a (xn ) and so
lim inf φ(xn ) ≥ lim inf a (xn ) = lim inf(g(ξ) − − A l(xn )) = g(ξ) − for all
> 0. Hence lim inf φ(xn ) ≥ g(ξ). We now prove that lim sup φ(xn ) ≤ g(ξ).
Let h be the solution of the problem

∆h = 0
in Ω,
h ∈ C(Ω) with h = g
on ∂Ω
(such an h exists as the boundary ∂Ω of Ω is suciently regular (i.e. as a convex
set it has a barrier at each point), see e.g. Theorem 2.14 in Gilbarg & Trudinger
[24]). On the other hand we have ∆a = 0 in Ω (trivially) and a ≤ g on ∂Ω for any
a ∈ F . Then a − h is also harmonic and, as a|∂Ω ≤ g = h|∂Ω , by the maximum
principle of harmonic functions we obtain a ≤ h in Ω. Taking supremum over
a ∈ F we obtain φ(x) ≤ h(x) for x ∈ Ω. In particular, φ(xn ) ≤ h(xn ) and
therefore lim sup φ(xn ) ≤ lim sup h(xn ) = g(ξ).
(d) If p ∈ ∂φ(Ω) then there exists x0 ∈ Ω such that
φ(x) ≥ φ(x0 ) + p · (x − x0 ) =: a(x)
∀x ∈ Ω.
(4.5)
Since φ = g on ∂Ω, we have g(x) ≥ a(x) for all x ∈ ∂Ω. There exists ξ ∈ ∂Ω such
that g(ξ) = a(ξ). Otherwise, there exists some > 0 such that g(x) ≥ a(x) + for all x ∈ ∂Ω and then, by denition of φ, φ(x) ≥ a(x) + for all x ∈ Ω. In
particular φ(x0 ) ≥ a(x0 ) + = φ(x0 ) + , a contradiction. Since Ω is convex,
the open segment I joining x0 and ξ is contained in Ω. We will show that a(x)
is a supporting hyperplane to φ at any z ∈ I . By (4.5) we only need to show
that φ(z) = a(z) for all z ∈ I . We have φ(x0 ) = a(x0 ), φ(ξ) = g(ξ) = a(ξ) and
if z = tx0 + (1 − t)ξ , t ∈ [0, 1] is any point in I , we have, by convexity of u,
φ(z) ≤ tφ(x0 ) + (1 − t)φ(ξ) = ta(x0 ) + (1 − t)a(ξ) = a(z) ≤ φ(z),
i.e. φ(z) = a(z) for all z ∈ I , as needed. This means that p ∈ ∂φ(z) for all
z ∈ I , which nishes the proof.
31
4.3
Inhomogeneous Dirichlet problem
4.3
4
EXISTENCE THEOREMS
Inhomogeneous Dirichlet problem
Let Ω ⊂ Rn be an open, bounded and a strictly convex set, µ a Borel measure in
Ω with µ(Ω) < ∞, and g ∈ C(∂Ω). We will focus on the inhomogeneous Dirichlet
problem

det D2 φ = µ
in Ω,
φ = g
on ∂Ω.
(4.6)
Let
F(µ, g) := {v ∈ C(Ω) : v is convex, M v ≥ µ, v = g on ∂Ω}.
We rst note that if ψ ∈ F(µ, g), then ψ ≤ φ0 , where φ0 ∈ C(Ω) is the convex
solution to the problem (4.2). Indeed, we have 0 = M φ0 ≤ µ ≤ M ψ in Ω and by
the comparison principle (Theorem 3.15) we have that ψ ≤ φ0 in Ω. Therefore, in
particular, all functions in F(µ, g) are uniformly bounded above.
We will show that if µ is a nite combination of delta masses, i.e. if µ =
PN
i=1
ai δx i
for N ≥ 1, and xi ∈ Ω, ai > 0 for i = 1, . . . , N , then the solution Φ to the problem
(4.6) is given by
Φ(x) :=
sup ψ(x).
(4.7)
ψ∈F (µ,g)
In general, i.e. for any Borel measure µ, we are going to use an approximation argument to claim existence of a solution. For this reason we need the following two
lemmas.
Lemma 4.3. Let
µ be a Borel measure on Ω with µ(Ω) < ∞. Then there exists a
sequence {µj }j≥1 of measures such that
µj =
Nj
X
aji δxj ,
i
i=1
supj µj (Ω) =: A < ∞ and µj * µ (see Denition 3.4 for the denition of the weak
j→∞
convergence).
We omit the proof.
Lemma 4.4. Let Ω ⊂ Rn be an open, bounded and strictly convex domain, µj , µ be
Borel measures in Ω, φj ∈ C(Ω) be convex and g ∈ C(∂Ω) such that
1. φj = g on ∂Ω,
2. M φj = µj in Ω,
32
4.3
Inhomogeneous Dirichlet problem
4
EXISTENCE THEOREMS
3. µj * µ in Ω, and
4. µj (Ω) ≤ A for all j .
Then {φj } contains a subsequence {φjk }, such that φjk k→∞
−→ φ uniformly on compact
subsets of Ω, where φ ∈ C(Ω) is convex and M φ = µ in Ω, φ = g on ∂Ω.
Proof (we paraphrase the proof from Gutiérrez [25], pp. 20-21). We have φj ∈
F(µj , g) and therefore φj are uniformly bounded above by φ0 (see the explanation
above the previous lemma). We will show that φj are also uniformly bounded below
in Ω. Similarly as in (4.4), let ξ ∈ ∂Ω, > 0 and a (x) := g(ξ) − − A l(x), where
l(x) = 0 is the equation of a supporting hyperplane of Ω at ξ and constant A is
dened as in (4.4). Recall that a (x) ≤ g(x) for x ∈ ∂Ω, l(ξ) = 0, l(x) > 0 for
x ∈ Ω and A ≥ 0. Set ψj (x) := φj (x) − a (x). Then the ψj s are convex in Ω and
ψj (x) = g(x) − a (x) ≥ 0 for x ∈ ∂Ω. If ψj (x) ≥ 0 for all x ∈ Ω then φj is bounded
below in Ω (by boundedness of Ω). If at some point x ∈ Ω we have ψj (x) < 0,
then by the Aleksandrov maximum principle (Lemma 3.13) applied to ψj on the set
G := {x ∈ Ω : ψj (x) ≤ 0} (G is convex by convexity of ψ ; indeed, if x1 , x2 ∈ G then
ψ(λx1 + (1 − λ)x2 ) ≤ λψ(x1 ) + (1 − λ)ψ(x2 ) ≤ 0 for all λ ∈ [0, 1]), we obtain
(−ψj (x))n ≤ C dist(x, ∂Ω)diam(G)n−1 M ψj (Ω) ≤ C dist(x, ∂Ω)diam(Ω)n−1 A,
and consequently ψj (x) ≥ −C dist(x, ∂Ω)1/n , that is
φj (x) ≥ g(ξ) − − A l(x) − C dist(x, ∂Ω)1/n ,
(4.8)
which proves that φj are uniformly bounded below in Ω.
Since φj ≤ φ0 for all j and dist(x, ∂Ω) ≤ |x − ξ| we get from (4.8) that
φ0 (x) ≥ φj (x) ≥ g(ξ) − − A l(x) − C|x − ξ|1/n ,
(4.9)
and φj (x) → g(ξ) as x → ξ uniformly in j .
Therefore, by Corollary 2.8, we get that φj are locally uniformly Lipschitz in Ω
and hence, by Arzèla-Ascoli theorem, there exists a subsequence {φjk }, and a convex
function φ in Ω such that φj → φ uniformly on compact subsets of Ω. We also have
from (4.9) that φ ∈ C(Ω) and that φ = g on ∂Ω. The weak convergence M φjk → M φ
follows from from Lemma 3.7 and hence M φ = µ by the uniqueness of weak limits of
measures.
33
4.3
Inhomogeneous Dirichlet problem
4
EXISTENCE THEOREMS
We can now state the main theorem of this section.
Theorem 4.5. If Ω ⊂ Rn is open, bounded and strictly convex, µ is a Borel measure
in Ω with µ(Ω) < +∞ and g ∈ C(∂Ω), then there exists a unique convex generalized
solution Φ to the problem (4.6).
We note that if µ is absolutely continuous with respect to the Lebesgue measure
with density function f , where f ∈ C(Ω) is positive, then the Caarelli's regularity
results give φ ∈ W 2,p (Ω) for all p ∈ (0, ∞). What is more, if f ∈ C α (Ω) for some
α > 0 then u ∈ C 2,α (Ω). See Theorem 1 and Theorem 2 in Caarelli [11].
Proof (of Theorem 4.5; we paraphrase and simplify the proof from Gutiérrez [25], pp.
21-24). The uniqueness follows by the comparison principle (Theorem 3.15).
By Lemma 4.3 there exists a sequence of measures {µj } converging weakly to µ
such that each µj is a nite combination of delta masses with positive coecients and
µj (Ω) ≤ A for all j . If we solve the Dirichlet problem for each µj with boundary data
g , then the theorem follows by Lemma 4.4. Therefore we assume from now on that
µ=
N
X
ai δxi ,
i=1
xi ∈ Ω, ai > 0.
We will show that the solution Φ is given by (4.7). We will show the following claims.
(a) F(µ, g) 6= 0 (so that Φ is well dened),
(b) If φ, ψ ∈ F(µ, g) then max(φ, ψ) ∈ F(µ, g),
(c) Φ ∈ F(µ, g),
(d) M Φ ≤ µ.
Clearly, these four claims prove the theorem.
(a) By Example 3.3 (b), M (|x − xi |) = ωn δxi , with ωn := |B(0, 1)|. Let
f (x) :=
N
1 X
1/n
ωn i=1
1/n
ai |x − xi |
and φ be a solution to the Dirichlet problem M φ = 0 in Ω with φ = g − f on
∂Ω (see Theorem 4.2). We claim that ψ := φ + f ∈ F(µ, g). Indeed, it is clear
34
4.3
Inhomogeneous Dirichlet problem
4
EXISTENCE THEOREMS
that ψ ∈ C(Ω), ψ is convex and ψ = g on ∂Ω. Using superadditivity of the
Monge-Ampère measure (Lemma 3.10), we have
M ψ = M (φ + f ) ≥ M φ + M f ≥
N
1 X
1/n
ωn
i=1
1/n
M (ai |x
− xi |) =
N
X
ai δxi = µ.
i=1
Hence indeed ψ ∈ F(µ, g) and therefore F(µ, g) 6= ∅. Consequently Φ given by
(4.7) is well dened.
(b) Let Ψ := max(φ, ψ), Ω1 := {x ∈ Ω : φ(x) ≥ ψ(x)}, Ω2 := {x ∈ Ω : φ(x) <
ψ(x)}. If E ⊂ Ω1 , then ∂Ψ(E) ⊃ ∂φ(E). Indeed, if p ∈ ∂φ(x0 ) for some
x0 ∈ E , then φ(x) ≥ φ(x0 ) + p · (x − x0 ) for all x ∈ Ω and, as Ψ(x0 ) = φ(x0 )
and Ψ ≥ φ in Ω, we also have Ψ(x) ≥ Ψ(x0 ) + p · (x − x0 ) for all x ∈ Ω, i.e.
p ∈ ∂Ψ(x0 ) ⊂ ∂Ψ(E). Similarly, if E ⊂ Ω2 , then ∂Ψ(E) ⊃ ∂ψ(E). For any
E ⊂ Ω we decompose E = (E ∩ Ω1 ) ∪ (E ∩ Ω2 ) and write
M Ψ(E) = M Ψ(E ∩ Ω1 ) + M Ψ(E ∩ Ω2 )
≥ M φ(E ∩ Ω1 ) + M ψ(E ∩ Ω2 )
≥ µ(E ∩ Ω1 ) + µ(E ∩ Ω2 ) = µ(E).
(c) We rst show that for each xed y ∈ Ω there exists a uniformly bounded
sequence ψm ∈ F(µ, g) converging uniformly on compact subsets of Ω to a
function ψ ∈ F(µ, g) such that ψ(y) = Φ(y). Indeed, by (a), let ψ0 ∈ F(µ, g).
For any φ ∈ F(µ, g) we have φ ≤ φ0 with φ0 being a solution of the homogeneous
Dirichlet problem (cf. explanation before Lemma 4.3). By denition of Φ there
exists a sequence ψm ⊂ F(µ, g) such that ψm (y) → Φ(y) as m → ∞. Let
ψm := max(ψ0 , ψm ). By (b), ψm ∈ F(µ, g) and therefore ψm (y) ≤ ψm (y) ≤ Φ(y)
and so ψm (y) → Φ(y) as m → ∞. Because ψ0 ≤ ψm ≤ φ0 in Ω, ψm is uniformly
bounded. Since each ψm is convex in Ω, it follows from Corollary 2.8 that
given K ⊂ Ω compact, ψm are uniformly Lipschitz in K with constant C =
C(K, supΩ |ψ0 |, supΩ |φ0 |). Therefore ψm are equicontinuous on K and bounded
in Ω. By Arzèla-Ascoli theorem there exists a subsequence ψmj converging
uniformly on compact subsets of Ω to a convex function ψ , which satises ψ(y) =
Φ(y). By Corollary 3.8 we have that M ψ ≥ µ and hence w ∈ F(µ, g). In
particular ψ ≤ Φ in Ω.
We now show that M Φ ≥ µ in Ω. It is enough to prove that M Φ({xi }) ≥ ai
35
4.3
Inhomogeneous Dirichlet problem
4
EXISTENCE THEOREMS
for i = 1, . . . , N . We prove for i = 1 (other cases are similar). By the above
analysis, there exists a uniformly bounded sequence ψm ∈ F(µ, g) such that
m→∞
ψm −→ ψ ∈ F(µ, g) uniformly on compacts of Ω with ψ(x1 ) = Φ(x1 ). We
have M ψ({x1 }) ≥ a1 . If p ∈ ∂ψ(x1 ), then ψ(x) ≥ ψ(x1 ) + p · (x − x1 ) in Ω
and hence Φ(x) ≥ ψ(x) ≥ ψ(x1 ) + p · (x − x1 ) = Φ(x1 ) + p · (x − x1 ), that is
p ∈ ∂Φ(x1 ). Therefore ∂Φ({x1 }) ⊃ ∂ψ({x1 }) and consequently M Φ({x1 }) ≥
M ψ({x1 }) ≥ a1 .
(d) We rst prove that the measure M Φ is concentrated on the set {x1 , . . . , xN }. Let
x0 ∈ Ω with x0 6= xi for all i = 1, . . . , N , and choose r > 0 so that |xi − x0 | > r
for i = 1, . . . , N and B(x0 , r) ⊂ Ω. Let ψ be the convex solution of the problem
M ψ = 0 in B(x0 , r) with ψ = Φ on ∂B(x0 , r) (see Theorem 4.2) and let φ be
the "lifting of Φ", which is dened by
φ(x) :=

Φ(x)
for |x − x0 | ≥ r, x ∈ Ω,
for |x − x0 | ≤ r.
ψ(x)
We claim that φ ∈ F(µ, g). It is clear that φ ∈ C(Ω) and φ|∂Ω = g . Moreover, φ is convex. Indeed, convexity in B(x0 , r) and in Ω \ B(x0 , r) follows by
denition. By (c) we have that M Φ ≥ µ ≥ 0 = M ψ in B(x0 , r) and hence
by the comparison principle (Theorem 3.15), Φ ≤ ψ = φ in B(x0 , r). To show
convexity of φ in Ω, let x ∈ B(x0 , r) and y ∈ Ω \ B(x0 , r) and let ν ∈ (0, 1)
be such that z := νx + (1 − ν)y ∈ ∂B(x0 , r). Let λ ∈ [0, 1]. If λ ≥ ν then
λx + (1 − λ)y in B(x0 , r) and take λ :=
λ−ν
1−ν
∈ [0, 1]. Then λ + (1 − λ)ν = λ,
(1 − λ)(1 − ν) = 1 − λ and, using convexity of φ in B(x0 , r) and convexity of Φ
in Ω, we obtain
φ(λx + (1 − λ)y) = φ(λx + (1 − λ)(νx + (1 − ν)y))
≤ λφ(x) + (1 − λ)φ(νx + (1 − ν)y)
= λφ(x) + (1 − λ)Φ(νx + (1 − ν)y)
≤ λφ(x) + (1 − λ)(ν Φ(x) +(1 − ν) Φ(y))
| {z }
|{z}
≤φ(x)
=φ(y)
≤ λφ(x) + (1 − λ)(νφ(x) + (1 − ν)φ(y))
= λφ(x) + (1 − λ)φ(y).
A similar calculation could be carried out for λ < ν (in this case one should
36
4.3
Inhomogeneous Dirichlet problem
take λ :=
λ
ν
4
EXISTENCE THEOREMS
∈ [0, 1) and one does not need to use convexity of φ ∈ B(x0 , r)).
Hence φ is convex in Ω. We will now verify that M φ ≥ µ in Ω. Noting that for
F ⊂ Ω \ B(x0 , r) we have ∂φ(F ) ⊃ ∂Φ(F ) (by the same argument as in showing
∂Φ({x1 }) ⊃ ∂ψ({x1 }) in the end of the proof of (d) ), we write, for any Borel
set E ⊂ Ω,
M φ(E) = M φ(E ∩ B(x0 , r)) + M φ(E \ B(x0 , r))
≥ M ψ(E ∩ B(x0 , r)) + M Φ(E \ B(x0 , r)) = 0 + M Φ(E \ B(x0 , r))
≥ µ(E \ B(x0 , r)) = µ(E ∩ {x1 , . . . , xN }) = µ(E).
Therefore indeed φ ∈ F(µ, g).
Hence φ ≤ Φ and, since φ = ψ ≥ Φ in B(x0 , r), we get ψ = Φ in B(x0 , r).
This means that M Φ = M ψ = 0 in B(x0 , r). Choosing any B(x0 , r) such that
B(x0 , r) ∩ {x1 , . . . , xN } = ∅ we get that M Φ(E) = 0 for any Borel set E ⊂ Ω
such that E ∩ {x1 , . . . , xN } = ∅. Therefore M Φ is concentrated on the set
{x1 , . . . , xN }, that is
MΦ =
N
X
λi ai δxi ,
i=1
with λi ≥ 1, i = 1, . . . , N .
We claim that λi = 1 for each i = 1, . . . , N . Suppose by contradiction that
λi > 1 for some i. Without loss of generality, we may assume that i = 1
and x1 = 0 (otherwise apply renumbering of {xn } and translate Φ). We have
|∂Φ({0})| = λa > 0. The set ∂Φ({0}) is convex. Indeed, if Φ(0) + p1 · x ≤ Φ(x)
and Φ(0) + p2 · x ≤ Φ(x) for all x ∈ Ω, then Φ(0) + (λp1 + (1 − λ)p2 ) · x =
λ(Φ(0) + p1 · x) + (1 − λ)(Φ(0) + p2 · x) ≤ Φ(x). Hence, as the set ∂Φ({0})
is convex and has positive measure, it contains an open ball, i.e. there exists
a ball B(p0 , 2) ⊂ ∂Φ({0}). This means that Φ(x) ≥ Φ(0) + p · x for all
p ∈ B(p0 , 2) and for all x ∈ Ω. Let Ψ(x) = Φ(x) − p0 · x (see Fig. 6). Then
Ψ(x) ≥ Ψ(0) + (p − p0 ) · x for all x ∈ Ω and p ∈ B(p0 , 2). Given x ∈ Ω take
x
∈ B(p0 , 2) and so
p = p0 + |x|
Ψ(x) ≥ Ψ(0) + |x|
∀x ∈ Ω.
Let α be a constant such that Ψ(0)−α is negative and Ψ(0)−α > − mini=2,...,N |xi |,
and dene Ψ(x) := Ψ(x) − α (see Fig. 6). Note that the choice of α implies that
37
4.3
Inhomogeneous Dirichlet problem
4
EXISTENCE THEOREMS
xi 6∈ {Ψ < 0} for all i = 2, . . . , N . We have that Ψ(0) is negative and small,
and Ψ(x) ≥ Ψ(0) + |x| for all x ∈ Ω. If r = − Ψ(0)
, then Ψ(x) ≥ Ψ(0) + |x| ≥ 0
for all |x| ≥ r. Let
ϕ(x) :=

Ψ(x)
λ−1/n Ψ(x)
if Ψ(x) ≥ 0,
if Ψ(x) < 0,
see Fig. 6. We will show that ϕ ∈ F(µ, g), where g are the boundary values of
Ψ(x) ≡ Φ(x) − p0 · x − α. Notice that since λ > 1, we have λ−1/n Ψ(x) > Ψ(x)
on the set {Ψ(x) < 0}. Therefore, similarly to the proof of convexity of φ,
Φ(x)
Ψ(x)
Ψ(x)
ϕ(x)
0
Figure 6: The functions Φ(x), Ψ(x), Ψ(x), ϕ(x) (note this is a 1D sketch of an
n-dimensional situation).
the function ϕ is convex in Ω. Also, on the set {Ψ(x) < 0}, we have M ϕ =
M (λ−1/n Ψ) =
1
MΨ
λ
=
1
MΦ
λ
= a1 δ0 = µ, where we have used Example 3.3
(d) and Corollary 3.11. On the other hand ϕ = Ψ on the set {Ψ(x) ≥ 0}, so
M ϕ = M Ψ = M Φ ≥ µ on the same set. Consequently, M ϕ ≥ µ in Ω. Hence
indeed ϕ ∈ F(µ, g).
We note that, by Corollary 3.11, the function v 0 (x) := v(x) − p0 · x − α belongs
to F(µ, g) if and only if v ∈ F(µ, g). Hence, by denition of Φ,
Ψ(x) = Φ(x) − p0 · x − α =
sup {v(x) − p0 · x − α} =
v∈F (µ,g)
sup
v 0 (x).
v 0 ∈F (µ,g)
Since ϕ ∈ F(µ, g), we get that ϕ(x) ≤ Ψ(x) for all x ∈ Ω. In particular,
ϕ(0) ≤ Ψ(0) or, equivalently, λ−1/n Ψ(0) ≤ Ψ(0). As Ψ(0) < 0 we obtain
λ−1/n ≥ 1
a contradiction since λ > 1.
38
4.4
At most two
4.4
C2
solutions in 2D
At most two
C2
4
EXISTENCE THEOREMS
solutions in 2D
We will now show that in dimension n = 2 we have at most 2 solutions to the
inhomogeneous Dirichlet problem (4.6) with measure µ given by a density f with
respect to the Lebesgue measure, where f ∈ C(Ω) is strictly positive. In other words
we have the following theorem.
Theorem 4.6. Let
Ω ⊂ R2 be an open set. Let f : Ω → R be continuous and
everywhere positive and let g : ∂Ω → R be continuous. Then the problem

det D2 φ = f
in Ω,
φ = g
on ∂Ω
(4.10)
has at most two C 2 (Ω) ∩ C(Ω) solutions.
We note that if f ∈ C α (Ω) for some α > 0 then the problem (4.10) has
exactly
two solutions (in a generalised sense), which are of regularity C 2,α (Ω) ∩ C(Ω) (see the
remark after the statement of Theorem 4.5).
The proof of Theorem 4.6 is adapted from Courant & Hilbert [15], pp. 324-325,
where a more general equation is considered. The proof we will present is a simplication of the analysis of this more general equation. Before the proof, we rst need
to recall a uniqueness theorem for elliptic dierential equations. Let Ω be a bounded
and connected domain in Rn . Let the dierential operator
Lψ ≡ aij (x)Dij ψ + bi (x)Di ψ + c(x)ψ
(4.11)
be elliptic in Ω, i.e. let the matrix {aij (x)}i,j=1,...,n be positive denite for each x ∈ Ω
(i.e. aij (x)ξi ξj > 0 ∀x ∈ Ω, ∀ξ ∈ Rn ) and let aij , bi , c, for i, j ∈ {1, . . . , n}, be
continuous. Let us also assume that
|bi (x)|
≤C
λ(x)
∀x ∈ Ω, ∀i = 1, . . . , n,
(4.12)
where λ(x) > 0 is the smallest eigenvalue of the matrix {aij (x)}i,j=1,...,n (λ > 0 since
all eigenvalues of a positive denite matrix are positive) and C > 0. We have the
following maximum principle.
Lemma 4.7 (Weak maximum principle). Suppose ψ ∈ C 2 (Ω)∩C(Ω) satises Lψ ≥ 0
39
4.4
At most two
C2
solutions in 2D
4
EXISTENCE THEOREMS
in Ω with c ≤ 0. Then ψ attains on ∂Ω its nonnegative maximum in Ω, i.e.
sup ψ ≤ sup ψ + .
∂Ω
Ω
Proof. See Gilbarg & Trudinger [24], p. 33.
Corollary 4.8. There exists at most one solution φ ∈ C 2 (Ω) ∩ C(Ω) to the problem

Lφ = f
in Ω,
φ = g
on ∂Ω,
(4.13)
where f ∈ C(Ω), g ∈ C(∂Ω) and L is as in Lemma 4.7.
Proof. If φ, ψ are two solutions to (4.13) then ϕ := φ − ψ satises (4.13) with f ≡ 0,
g ≡ 0 and lemma 4.7 implies supΩ ϕ ≤ sup∂Ω ϕ+ = 0. Analogously, taking −ϕ gives
supΩ (−ϕ) ≤ 0. Hence 0 ≤ ϕ ≤ 0 and consequently φ = ψ .
We can now prove Theorem 4.6.
Proof (of Theorem 4.6; adapted from Courant & Hilbert [15], pp. 324-325). If φ ∈
C 2 (Ω) ∩ C(Ω) is a solution to (4.10), then
φxx φyy − φ2xy > 0
(4.14)
in Ω. In particular neither φxx nor φyy can vanish at any point x ∈ Ω. Therefore it
remains to show that there exists at most one solution to (4.10) for which
φxx > 0 (and then also φyy > 0)
(4.15)
and at most one solution to (4.10) for which
φxx < 0 (and then also φyy < 0)
(4.16)
holds in Ω (cf. (1.9)). In what follows we will consider the case (4.15) (the case (4.16)
follows in a similar way). Suppose there exist two solutions φ, ψ ∈ C 2 (Ω) ∩ C(Ω) of
the problem (4.10) satisfying (4.15). Then, letting ϕ := φ − ψ we get
0 = det D2 (ψ + ϕ) − det D2 ψ = (ϕxx ϕyy − ϕ2xy ) + ψxx ϕyy + ψyy ϕxx − 2ψxy ϕxy ,
0 = det D2 (φ) − det D2 (φ − ϕ) = −(ϕxx ϕyy − ϕ2xy ) + φxx ϕyy + φyy ϕxx − 2φxy ϕxy .
40
4.5
Similarities to the Laplace equation
4
EXISTENCE THEOREMS
Adding the two equalities we obtain L(ψ+φ) ϕ = 0, where we dene
Lw ϕ := wyy ϕxx − 2wxy ϕxy + wxx ϕyy
(4.17)
for w ∈ C 2 (Ω) ∩ C(Ω). Hence ϕ is a solution to the problem

L
(ψ+φ) ϕ = 0
ϕ = 0
in Ω,
on ∂Ω.
(4.18)
We note that L(ψ+φ) is a dierential operator of the form (4.11). Indeed, we have
L(ψ+φ) = Lψ + Lφ and each of Lψ , Lφ is of the form (4.11) because of (4.14) and (4.15)
(note that (4.14) and (4.15) together are equivalent to the positive deniteness of the
matrix {φij }i,j=x,y ). We also note that L(ψ+φ) satises condition (4.12) (trivially).
Therefore, noting that ϕ ≡ 0 is a solution of (4.18), we may use Corollary 4.8 to
conclude that it is the only solution of this problem. Consequently φ ≡ ψ .
We note that assumption f > 0 in Theorem 4.6 is crucial in a sense that it implies
convexity or concavity of the solution φ, i.e. it enables us to use the trick (4.15-4.16).
4.5
Similarities to the Laplace equation
This is a good moment to reect on the similarities between the Monge-Ampère
equation and the Laplace equation.
Firstly, both the Laplace operator ∆φ of a function φ : Rn ⊃ Ω → R and the
determinant of the Hessian matrix det D2 φ are invariant under orthogonal transformations of variables. That is, if y := [y1 , . . . , yn ]T and x := [x1 , . . . , xn ]T are such that
y = Qx for some orthogonal Q ∈ Rn×n (i.e. Q−1 = QT ), then letting φ(y) := φ(x) we
have ∆φ(y) = ∆φ(x) and det D2 φ(y) = det D2 φ(x). This property is a consequence
of the fact that the trace and the determinant are two of the n scalar invariants of
orthogonal transformations of a n × n symmetric matrix (hence in 2D these are the
only such invariants), which can be proved using the characteristic polynomial (see
Horn & Johnson [27], pp. 49-51).
Secondly, one can compare the comparison principle, Theorem 3.15, to the maximum principle for subharmonic functions (and minimum principle for superharmonic
functions), that is: if v ∈ C 2 (Ω) (for Ω bounded) satises ∆v ≥ 0 (i.e. v is subharmonic) then maxΩ v = max∂Ω v (see e.g. Gilbarg & Trudinger [24], p. 15). One
could try to compare convex solutions of Monge-Ampère equation to superharmonic
41
4.5
Similarities to the Laplace equation
4
EXISTENCE THEOREMS
functions (and concave solutions to subharmonic functions), but one needs to be
careful, because "superharmonicity" and "subharmonicity" relate to the inequality
in the partial dierential equation (i.e. in the Laplace equation), while convexity and
concavity of solutions of the Monge-Ampère equation correspond to the two dierent
"modes" of solution of the same equation. It is interesting to observe how these two
modes appear as the only two solutions in the Dirichlet problem in Theorem 4.6 in
the section above.
Furthermore, the fact that the convex solution φ to the homogeneous Dirichlet
problem for the Monge-Ampère equation is given by (4.3) is reminiscent of the Perron's method for solving the Dirichlet problem for Laplace equation. Namely, for
Ω ⊂ Rn open, bounded and with C 2 boundary and for g ∈ C(∂Ω), let
S := {v ∈ C(Ω) : v is subharmonic and v ≤ g on ∂Ω}
(we note that one can also dene subharmonic (superharmonic) functions in the space
C(Ω)). Then the solution u to the problem ∆u = 0 in Ω, u = g on ∂Ω is given by
u(x) = sup v(x),
v∈S
see Han & Lin [26], pp. 125-130, for the details.
Finally, let us look at the part (d) of the proof of Theorem 4.5 (the existence of
solution to inhomogeneous Dirichlet problem for the Monge-Ampère equation). The
"lifting procedure" we used there (i.e. we lifted a convex function Φ with M Φ ≥ µ
on a ball B(x0 , r) using the convex solution ψ of the problem M ψ = 0 in B(x0 , r),
ψ = Φ on ∂B(x0 , r)) is analogous to the lifting procedure used in the Perron's method
(where one lifts a subharmonic function v on a ball B(x0 , r) using the unique solution
w of the problem ∆w = 0 in B(x0 , r), w = v on ∂B(x0 , r)), see Han & Lin [26], pp.
125-130, for the details.
We note that these similarities together with the application of the Monge-Ampère
equation in the 2D Navier-Stokes equation, which we described in the motivation
section (Section 1.1), were the author's inspiration to undertake this topic.
42
5
5
BOUNDS ON THE SOLUTION
Bounds on the solution to the Monge-Ampère equation
Here we state the main results of this note. We extend the approach presented in
[39] into the case of the inhomogeneous Dirichlet problem. In this section we will be
concerned with the case of even n only. We comment on the case of odd n at the end
of this section.
5.1
Generalised Comparison Principle
Let [ν]+ := max(ν, 0) denote the positive part of a Borel measure ν .
Theorem 5.1 (Generalised comparison principle). Let Ω be a bounded, strictly convex
set in Rn . Let ψ ∈ C(Ω) be a convex function in Ω and let φ ∈ W 2,n (Ω) be such that
M ψ ≥ [M φ]+
on Ω.
(5.1)
Then
min(φ − ψ)(x) = min (φ − ψ)(x).
x∈∂Ω
x∈Ω
(5.2)
Similarly, if ψ is concave in Ω and φ ∈ W 2,n (Ω) is such that M ψ ≥ [M φ]+ on Ω,
then
max(φ − ψ)(x) = max(φ − ψ)(x).
x∈∂Ω
x∈Ω
(5.3)
If φ, ψ ∈ C 2 (Ω) then (5.1) is equivalent to
+
det D2 ψ ≥ det D2 φ
(5.4)
Proof (of Theorem 5.1; we modify the proof from Rauch & Taylor [39], pp. 362-364).
Let ψ be convex (the case of ψ concave proceeds similarly).
1. First suppose that φ ∈ C 2 (Ω).
Suppose that there exists x0 ∈ Ω such that
(φ − ψ)(x0 ) = min(φ − ψ)(x) < min (φ − ψ)(x)
x∈∂Ω
x∈Ω
and consider the function ψ (x) := ψ(x) + |x − x0 |2 for > 0. It is clear that if
is suciently small then the function φ − ψ still does not attain its minimum
on ∂Ω (see Fig. 7). Fix such .
43
5.1
Generalised Comparison Principle
φ − ψ
5
BOUNDS ON THE SOLUTION
φ−ψ
x0
x
Figure 7: The points x0 and x (note this is a 1D sketch of a 2D situation).
Hence there exists x ∈ Ω such that
(φ − ψ )(x ) = min(φ − ψ )(x) < min (φ − ψ )(x).
x∈∂Ω
x∈Ω
(5.5)
We note that x can be chosen such that the Hessian matrix D2 φ(x ) is not
positive denite. Indeed, letting Ω1 := {x ∈ Ω : D2 φ > 0} we can see that φ
is convex on Ω1 . Therefore, using Corollary 3.12, M ψ ≥ M ψ ≥ M φ on Ω1 .
From Theorem 3.15, it follows that the minimum of φ − ψ over Ω1 is attained
on ∂Ω1 . Hence, as (φ − ψ )(x ) is the minimum of φ − ψ over Ω ⊃ Ω1 , we can
choose x ∈ Ω \ Ω1 , i.e. we can assume that D2 φ(x ) is not positive denite.
As any symmetric matrix is positive denite i all its eigenvalues are positive
(see e.g. Theorem 7.2.1 on p. 438 of Horn & Johnson [27]), we conclude that
D2 φ(x) has at least one nonpositive eigenvalue for each x ∈ Ω \ Ω1 . Thus let
λ ≤ 0 be an eigenvalue of D2 φ(x ) and let α ∈ Rn , |α | = 1, be the respective
eigenvector. Then, by the Taylor expansion in the α direction, we can write,
for small values of t ∈ R,
φ(x + tα ) − φ(x ) = l(t) + λt2 + o(t2 ),
(5.6)
where l1 (t) ≡ l1 t, l1 ∈ Rn , is a linear functional on R and o(·) : R → R is any
y→0
function such that o(y)/y −→ 0. As ψ is convex, it has a supporting hyperplane
at x (see Lemma 2.7 (a) ). Hence we have
ψ (x + tα) − ψ (x ) = ψ(x + tα) − ψ(x ) + (|x + tα − x0 |2 − |x − x0 |2 )
≥ l2 (t) + (|x + tα − x0 |2 − |x − x0 |2 )
= l3 (t) + t2 ,
where l2 (t) ≡ l2 t, l3 (t) ≡ l3 t, l2 , l3 ∈ Rn , are linear functions. Combining this
44
5.1
Generalised Comparison Principle
5
BOUNDS ON THE SOLUTION
inequality with (5.6), we get
(5.5)
(φ − ψ )(x1 )
≤
(φ − ψ )(x1 + tα)
(φ − ψ )(x1 ) + l(t) + (λ − )t2 + o(t2 )
≤
for small values of t. This means that the quadratic polynomial l(t) + (λ − )t2
attains its minimum at t = 0. Hence l(·) ≡ 0 and λ − ≥ 0, which contradicts
λ ≤ 0 < . Therefore the minimum principle (5.2) holds for φ ∈ C 2 (Ω).
2. Now let φ ∈ W 2,n (Ω).
Note that for φ ∈ W 2,n (Ω), we have det D2 φ ∈ L1 (Ω) by Hölder's inequality.
We are going to use an approximation argument, i.e. according to Lemma 3.7
we need a sequence φj ∈ C(Ω) converging to φ uniformly on compact sets. Since
Ω is only taken to be convex, ∂Ω is not in general C 2 (so we cannot use the
standard extension theorem (see e.g. Theorem 4.26 in Adams [1])) and we have
to justify that u can be extended to a W 2,n (Rn ) function of compact support.
Indeed, by Theorem 5 on p. 181 and Example 2 on p. 189 of Stein [41] we
get a continuous extension of φ to Rn with ||φ||W 2,n (Rn ) ≤ C||φ||W 2,n (Ω) and we
can then consider φξ , where ξ ∈ C0∞ (B(0, 2R)) is such that ξ ≡ 1 on B(0, R)
and R > 0 is chosen such that Ω ⊂ B(0, R). It is clear that φξ is the required
extension.
Hence, by the approximation property of Sobolev spaces (see e.g. Theorem 3.18
in Adams [1]), there exist a sequence of functions φj ∈ C ∞ (Rn ) such that each
j→∞
φj vanishes for |x| > 2R and φj −→ φ in W 2,n (Ω). Hence also, by Sobolev
embedding W 2,n (B(0, 2R)) ,→ C(B(0, 2R)) (see e.g. Evans [20], pp. 284-285),
j→∞
we have φj −→ φ in C(Ω).
Let µj := [M φj ]+ and µ := [M φ]+ , so that µj * µ by Lemma 3.7. Let ψj be
the unique convex function in C(Ω) such that

M ψ = µ
j
j
ψ = φ
j
j
in Ω,
on ∂Ω.
The existence and uniqueness of ψj is given by Theorem 4.5. As φj ∈ C 2 (Ω) we
45
5.2
Bounds on the solution
5
BOUNDS ON THE SOLUTION
have (from 1.) that
ψj ≤ φj
in Ω.
(5.7)
Furthermore, since φj → φ in C(∂Ω) and M ψj * µ, the functions ψj converge
in C(Ω) to the unique convex function ψ in C(Ω) such that ψ = φ on ∂Ω and
M ψ = µ in Ω (again by Lemma 3.7). Taking the limit j → ∞ in (5.7) we get
ψ ≤ φ in Ω. Moreover, since M ψ ≥ [M φ]+ = µ = M ψ and both ψ , ψ are
convex, we can use the usual comparison principle, Theorem 3.15, to obtain
min ψ − ψ ≥ min ψ − ψ .
∂Ω
Ω
(5.8)
Hence, for each x ∈ Ω we have
φ(x) − ψ(x) + ψ(x) − ψ(x) ≥ ψ(x) − ψ(x)
|
{z
}
≥0
≥ min ψ(y) − ψ(y) = min (φ(y) − ψ(y)) .
φ(x) − ψ(x) =
y∈∂Ω
y∈∂Ω
For a concave ψ use the same reasoning with φ, ψ and ψ replaced by −φ, −ψ and
−ψ respectively.
5.2
Bounds on the solution
The following are the author's own results.
Theorem 5.2. Let
Ω be a bounded, strictly convex set in Rn . Suppose that φ ∈
W 2,n (Ω) is a solution to

det D2 φ = f
φ = g
where f ∈ L1 (Ω). Then
Φconv ≤ φ ≤ Φconc
in Ω,
on ∂Ω,
in Ω,
(5.9)
(5.10)
where Φconv ∈ C(Ω) is the unique convex solution (Φconc ∈ C(Ω) is the unique concave
solution) of the problem

det D2 Φ = f +
in Ω,
Φ = g
on ∂Ω.
46
(5.11)
5.2
Bounds on the solution
5
BOUNDS ON THE SOLUTION
Proof. Clearly M Φconv ≥ M φ and M Φconc ≥ M φ. Hence, by Theorem 5.1, we get
min(φ − Φconv )(x) = min (φ − Φconv )(x) = 0,
x∈∂Ω
x∈Ω
max(φ − Φconc )(x) = max(φ − Φconc )(x) = 0
x∈∂Ω
x∈Ω
and the claim follows.
This theorem is a rather surprising result, which says that no matter how negative
det D2 φ is inside the domain Ω, the solution φ (if exists) will still be bounded above
and below by the two solutions: convex and concave, of the Dirichlet problem (5.11).
One of the applications of this result, used in the theory of thin shells (see Rauch &
Taylor [39], pp. 352, 364), corresponds to the case f ≡ 0. In short, it enables one to
prove that, in some parameter regimes, the potential energy of possible shell states
has a negative minimum (see Remark 2.2 in Rabinowitz [38]). The argument is based
on an observation that the only 2D surface with everywhere zero Gaussian curvature
and homogeneous boundary data is a zero surface (see Remark 2.2 in Rabinowitz
[38], Theorem 4.1 in Berger [8] and Lemma 3.2 in Berger [7]), i.e. a surface of a zero
function. Another consequence of Theorem 5.2, and the main result of this note,
is the following non-existence result for the Monge-Ampère equation with constant
boundary data and non-positive right-hand side.
Corollary 5.3. Let C ∈ R and f ∈ L1 (Ω) be a nonpositive function such that f 6≡ 0.
Then the problem

det D2 φ = f
in Ω,
φ = C
on ∂Ω
(5.12)
has no W 2,n (Ω) solution.
Proof. Suppose there exists a W 2,n (Ω) solution φ of the problem (5.12). The constant
function Φ ≡ C satises det D2 Φ = 0 = f + with Φ|∂Ω = C . Therefore, by Theorem 5.2, C ≤ φ ≤ C , i.e. φ ≡ C . Hence 0 ≡ det D2 φ ≡ f 6≡ 0, a contradiction.
We note that, since the regularity results for the Monge-Ampère equation (see the
comment after Theorem 4.5) are concerned with a nonnegative right-hand side of the
Monge-Ampère equation only, this corollary does not state nonexistence of less regular
solutions. What is more, it is not clear what kind of less regular notion of solution
would apply here (note that the concept of generalised solutions also corresponds to
nonnegative right-hand side only).
47
6
CONCLUSION
Finally, for odd space dimension n, the results of this section no longer hold. This
is, fundamentally, because for any matrix A ∈ Rn×n with odd n we have det (−A) =
−det A. Consequently, the proof of Theorem 5.1 would be no longer valid as the
negative deniteness of D2 φ would imply M φ < 0 and hence we wouldn't be able
to use the comparison principle (Theorem 3.15), as we did in part 1. of the proof
of Theorem 5.1. Instead, in the case of odd n, one could prove a modication of
Theorem 5.1, in which, in the case of concave ψ , the positive part [M φ]+ of the
measure M φ is replaced by the negative part [M φ]− (one would also have to be careful
with dening a negative Monge-Ampère measure; for instance M φ for a concave φ
could be dened as a Monge-Ampère measure of the convex −φ). This would lead to
the analogue of Theorem 5.2 of the following form.
Theorem 5.4. Let
Ω be a bounded, strictly convex set in Rn with n odd. Suppose
that φ ∈ W 2,n (Ω) is a solution to
where f ∈ L1 (Ω). Then

det D2 φ = f
in Ω,
φ = g
on ∂Ω,
Φconv ≤ φ ≤ Φconc
(5.13)
in Ω,
(5.14)
where Φconv ∈ C(Ω), Φconv is convex, Φconc ∈ C(Ω), Φconc is concave, are solutions of
the problems

det D2 Φ
Φ
conv
conv
=f
=g
+
in Ω,
on ∂Ω
and

det D2 Φ
conc
Φ
conv
=g
= −f −
in Ω,
on ∂Ω,
respectively.
However, we note that no nonexistence result of the form of Corollary 5.3 would
follow from the above theorem.
6
Conclusion
In summary, we have that if ∆p > 0 in Ω or ∆p ≤ 0 in Ω (for Ω ⊂ R2 strictly
convex) then there is no divergence-free u ∈ H01 (Ω) such that (1.4) holds. Therefore,
for any velocity eld u satisfying the 2D Navier-Stokes equations, the pressure p
cannot be superharmonic in Ω nor strictly subharmonic in Ω at any moment of time.
48
6
CONCLUSION
Nevertheless, although this fact gives us insight about the pressure p, it is still not
clear whether we can determine the velocity eld u from the pressure p.
We note that the theory developed in this note is, in the most part, disjoint with
the analysis of Navier-Stokes equations. It would be interesting to learn which of the
above results could be obtained from the Navier-Stokes theory (i.e. without using the
substitution (1.5)). In fact, using substitution (1.5), one can calculate that det D2 φ =
det ∇u, which translates the Monge-Ampère equation for φ into the Jacobian equation
for u. We note, however, that the Jacobian equation is, generally speaking, often
considered with boundary conditions of the form u|∂Ω = id (as the Jacobian equation
is usually studied within the context of dieomorphisms of a given domain) and a
strictly positive right-hand side (see e.g. Dacorogna & Kneuss [17] to understand why
a sign-changing Jacobian determinant is a problem; see also Dacorogna & Moser [18]
for an interesting dynamical systems approach to solving a Jacobian equation). An
attempt to generalise the presented results to the case of Jacobian equation det ∇u =
1
∆p
2
for the velocity eld u could possibly give us a valuable insight into the properties
of the velocity in Navier-Stokes equations.
There are also several directions of a further study of Monge-Ampère equation in
the presented context.
1. Generalised solutions to (5.12). One could try to construct a generalised notion
of solutions of the Monge-Ampère equation with a negative right-hand side
(note that Denition 4.1 corresponds only to a positive right-hand side) and
hence investigate the existence of solutions to (5.12) that are not W 2,2 .
2. Nonconstant boundary data in (5.12). Theorem 5.2 suggests that the amplitude
of "oscillations" of f inside Ω is bounded by the "positivity" of det D2 φ. Hence
one could "hope" for the nonexistence result of the following form. If osc∂Ω g :=
sup∂Ω g − inf ∂Ω g ≤ then there exists δ > 0 such that the problem

det D2 φ = f
in Ω,
φ = g
on ∂Ω,
has no solution when f ≤ −δ . Note that Corollary 5.3 corresponds to the case
= δ = 0.
3. Complex Monge-Ampère equation. One could investigate the theory of complex
Monge-Ampère theory (e.g. Aubin [4]) and possibly understand the geometrical
picture behind the presented results, particularly behind Corollary 5.3.
49
References
[1] R. A. Adams.
Sobolev Spaces. Pure and Applied Mathematics. Academic Press,
Inc., New York, 1975.
[2] A. D. Aleksandrov. Majorization of solutions of second-order linear equations.
Amer. Math. Soc. Transl., 2(68):120143, 1968.
Lineare Functionalanalysis, 6., überarbeitete Auage (in German).
[3] H. W. Alt.
Springer-Verlag, Berlin Heidelberg, 2012.
Nonlinear analysis on manifolds. Monge-Ampère equations, volume
252 of Grundlehren der Mathematischen Wissenschaften. Springer-Verlag, New
[4] T. Aubin.
York, 1982.
[5] I. J. Bakelman.
Convex analysis and nonlinear geometric elliptic equations.
Springer-Verlag, Berlin, 1994.
[6] J.-D. Benamou and Y. Brenier. A computational uid mechanics solution to the
Monge-Kantorovich mass transfer problem.
Numer. Math., 84(3):375393, 2000.
[7] M. S. Berger. On von Kármán's equations and the buckling of a thin elastic
plate. I. The clamped plate.
Comm. Pure Appl. Math., 20:687719, 1967.
[8] M. S. Berger. On the Existence of Equilibrium States of Thin Elastic Shells (I).
Indiana Univ. Math. J., 20(7):591602, 1971.
[9] R. Bhatia.
Positive Denite Matrices. Princeton Series in Applied Mathematics.
Princeton University Press, 2007.
[10] L. A. Caarelli. Interior a priori estimates for solutions of fully nonlinear equations.
Ann. of Math. (2), 130(1):189213, 1989.
[11] L. A. Caarelli. Interior W 2,p estimates for solutions of the Monge-Ampère
equation.
Ann. of Math. (2), 131(1):135150, 1990.
[12] L. A. Caarelli. A localization property of viscosity solutions to the MongeAmpère equation and their strict convexity.
Ann. of Math. (2), 131(1):129134,
1990.
[13] L. A. Caarelli. Some regularity properties of solutions of Monge Ampère equation.
Comm. Pure Appl. Math., 44(8-9):965969, 1991.
REFERENCES
REFERENCES
[14] L. A. Caarelli. A note on the degeneracy of convex solutions to Monge Ampère
equation.
Comm. Partial Dierential Equations, 18(7-8):12131217, 1993.
[15] R. Courant and D. Hilbert.
Methods of Mathematical Physics, Volume 2. Wiley-
VCH, 1962.
[16] M. G. Crandall, H. Ishii, and P.-L. Lions. User's guide to viscosity solutions
of second order partial dierential equations.
Bull. Amer. Math. Soc. (N.S.),
27(1):167, 1992.
[17] B. Dacorogna and O. Kneuss. Multiple Jacobian equations.
Appl. Anal., 13(5):17791787, 2014.
Commun. Pure
[18] B. Dacorogna and J. Moser. On a partial dierential equation involving the
Jacobian determinant.
Ann. Inst. H. Poincaré Anal. Non Linéaire, 7(1):126,
1990.
[19] I. Ekeland and R. Témam.
Convex Analysis and Variational Problems. Classics
in Applied Mathematics. Society for Industrial and Applied Mathematics, 1999.
[20] L. C. Evans. Partial dierential equations, Second Edition, volume 19 of Graduate
Studies in Mathematics. American Mathematical Society, Providence, RI, 1998.
[21] L. C. Evans. Partial dierential equations and Monge-Kantorovich mass transfer.
pages 65126, 1999.
[22] L. C. Evans and W. Gangbo. Dierential equations methods for the MongeKantorovich mass transfer problem.
Mem. Amer. Math. Soc., 137(653):viii+66,
1999.
[23] L. C. Evans and R. F. Gariepy.
Measure theory and ne properties of functions.
Studies in Advanced Mathematics. CRC Press, Boca Raton, FL, 1992.
[24] D. Gilbarg and N. S. Trudinger.
Elliptic Partial Dierential Equations of Second
Order. Classics in Mathematics. Springer, 2001.
The Monge-Ampère Equation, volume 44 of Progress in Nonlinear Dierential Equations and Their Applications. Birkhäuser, 2001.
[25] C. E. Gutiérrez.
Elliptic partial dierential equations, volume 1 of Courant
Lecture Notes in Mathematics. Courant Institute of Mathematical Sciences, New
[26] Q. Han and F. Lin.
York; American Mathematical Society, Providence, RI, second edition, 2011.
51
REFERENCES
REFERENCES
[27] R. A. Horn and C. R. Johnson.
Matrix analysis, Second Edition. Cambridge
University Press, 2013.
A. D. Alexandrov selected works. Part II. Intrinsic geometry
[28] S. S. Kutateladze.
of convex surfaces. Chapman & Hall/CRC, 2006. Translated from Russian by
S. Vakhrameyev.
[29] M. Larchevêque. Équation de Monge-Ampère et écoulements incompressibles
bidimensionnels.
C. R. Acad. Sci. Paris Sér. II Méc. Phys. Chim. Sci. Univers
Sci. Terre, 311(1):3336, 1990.
[30] D. G. Luenberger.
Optimization by Vector Space Methods. John Wiley & Sons,
Inc., 1969.
[31] V. V. Lychagin, V. N. Rubtsov, and I. V. Chekalov. A classication of MongeAmpère equations.
Ann. Sci. École Norm. Sup. (4), 26(3):281308, 1993.
[32] V. I. Oliker. The problem of embedding S n into Rn+1 with prescribed Gauss
curvature and its solution by variational methods.
Trans. Amer. Math. Soc.,
295(1):291303, 1986.
[33] V. I. Oliker. A variational solution of the A. D. Aleksandrov problem of existence
of a convex polytope with prescribed Gauss curvature. 69:8190, 2005.
[34] K. R. Parthasarathy.
Probability measures on metric spaces. Probability and
Mathematical Statistics, No. 3. Academic Press, Inc., New York-London, 1967.
[35] A. V. Pogorelov.
Monge-Ampère equations of elliptic type. P. Noordho, Ltd.,
Groningen, 1964.
[36] A. V. Pogorelov.
Extrinsic geometry of convex surfaces. American Mathematical
Society, Providence, R.I., 1973.
[37] A. V. Pogorelov.
The Minkowski multidimensional problem. V. H. Winston &
Sons, Washington, D.C.; Halsted Press [John Wiley & Sons], New York-TorontoLondon, 1978.
[38] P. H. Rabinowitz. A Note on Topological Degree for Potential Operators.
Math. Anal. Appl., (2):483492, 1975.
J.
[39] J. Rauch and B. A. Taylor. The Dirichlet problem for the multidimensional
Monge-Ampère equation.
Rocky Mountain J. Math., 7(2):345364, 1977.
52
[40] J. C. Robinson. Attractors and nite-dimensional behaviour in the 2D NavierStokes equations.
[41] E. M. Stein.
ISRN Math. Anal., 2013.
Singular Integrals and Dierentiability Properties of Functions.
Princeton Mathematical Series. Princeton University Press, 1970.
[42] J. Urbas. Global continuity estimates for two dimensional graphs of prescribed
Gauss curvature.
Manuscripta Math., 115(2):179193, 2004.