Journal of Inequalities in Pure and
Applied Mathematics
W 2,2 ESTIMATES FOR SOLUTIONS TO NON-UNIFORMLY ELLIPTIC
PDE’S WITH MEASURABLE COEFFICIENTS
volume 6, issue 3, article 69,
2005.
ANDRÁS DOMOKOS
Department of Mathematics and Statistics
California State University Sacramento
6000 J Street, Sacramento
CA, 95819, USA
EMail: domokos@csus.edu
URL: http://webpages.csus.edu/d̃omokos
Received 14 February, 2005;
accepted 27 May, 2005.
Communicated by: A. Fiorenza
Abstract
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2000
Victoria University
ISSN (electronic): 1443-5756
036-05
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Abstract
We propose to extend Talenti’s estimates on the L2 norm of the second order
derivatives of the solutions of a uniformly elliptic PDE with measurable coefficients satisfying the Cordes condition to the non-uniformly elliptic case.
2000 Mathematics Subject Classification: 35J15, 35R05
Key words: Cordes conditions, Elliptic partial differential equations
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
The author would like to thank the suggestions of an anonymous referee that significantly improved the presentation of this paper.
András Domokos
Contents
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Main Result . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
References
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6
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1.
Introduction
The Cordes conditions first were used by H. O. Cordes [1] and later by G.
Talenti [5] to prove C α , C 1,α and W 2,2 estimates for the solutions of second
order linear and elliptic partial differential equations in non-divergence form
Au =
n
X
aij (x)Dij u,
i,j=1
where A = (aij ) ∈ L∞ (Ω, Rn×n ) is a symmetric matrix function. As an introductory remark about the Cordes condition we can say that by using the
normalization (see [5])
n
X
aii (x) = 1
András Domokos
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i=1
or strictly positive lower and upper bounds (see [1])
0<p≤
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
n
X
aii (x) ≤ P
i=1
we get a condition equivalent to the uniform ellipticity condition in R2 and
stronger than it in Rn , n ≥ 3. At the same time it seems to be the weakest
condition which implies that A is an isomorphism between the spaces W02,2 (Ω)
and L2 (Ω) and implicitly gives existence and uniqueness for boundary value
problems with measurable coefficients [4]. As an application it was used to
prove the second order differentiability of p−harmonic functions [3].
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If we assume that the Cordes condition is satisfied, then it is possible to give
an optimal upper bound of the L2 norm of the second order derivatives to the
solution u ∈ W02,2 (Ω) of the problem
Au = f, f ∈ L2 (Ω)
in terms of a constant times the L2 norm of f . An interesting method, that
connects linear algebra to PDE’s, has been developed in [5]. In this paper we
will extend this method to not necessarily uniformly elliptic problems and as
an application we will also show a change in Talenti’s constant. More exactly,
estimate (1.2) below holds in the case of operators with constant coefficients,
but needs a change to cover the general case.
Let us consider the bounded domain Ω ∈ Rn with a sufficiently regular
boundary and the Sobolev space
n
o
W 2,2 (Ω) = u ∈ L2 (Ω) : Dij u ∈ L2 (Ω) , ∀ i, j ∈ {1, . . . , n}
(u, v)W 2,2 =
u(x)v(x) +
Ω
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endowed with the inner-product
Z
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
n
X
II
I
!
Dij u(x) · Dij v(x) dx.
i,j=1
Let W02,2 (Ω) be the closure of C0∞ (Ω) in W 2,2 (Ω) and denote by D2 u the
matrix of the second order derivatives.
We state now Talenti’s result using our setting.
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Theorem 1.1 ([5]). Let us suppose that for a fixed 0 < ε < 1 and almost every
x ∈ Ω the following conditions hold:
(1.1)
n
X
aii (x) = 1 and
i=1
n X
i,j=1
2
aij (x) ≤
1
.
n−1+ε
Then, for all u ∈ W02,2 (Ω) we have
(1.2) ||D2 u||L2 (Ω)
√
p
n − 1 + ε √
n − 1 + ε + (1 − ε)(n − 1) ||Au||L2 (Ω) .
≤
ε
W 2,2 Estimates for Solutions to
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with Measurable Coefficients
András Domokos
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2.
Main Result
Consider the matrix valued mapping A : Ω → Mn (R), where A(x) = (aij (x))
with aij ∈ L∞ (Ω), and let
Au =
(2.1)
n
X
aij (x)Dij (u).
i,j=1
p
We use P
the notations ||a|| = a21 + · · · + a2n for a = (a1 , . . . , an ) ∈ Rn and
trace A = ni=1 aii for the trace of an n × n matrix A = (aij ). q
Also, we denote
Pn
Pn
2
by hA, Bi = i,j=1 aij bij the inner product and by ||A|| =
i,j=1 aij the
Euclidean norm in Rn×n .
Definition 2.1 (Cordes condition Kε ). We say that A satisfies the Cordes condition Kε if there exists ε ∈ (0, 1] such that
(2.2)
0 < ||A(x)||2 ≤
2
1
trace A(x) ,
n−1+ε
for almost every x ∈ Ω and
W 2,2 Estimates for Solutions to
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with Measurable Coefficients
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∈ L2loc (Ω).
trace A
Remark 1. We observe that inequality (2.2) implies that for
√
n
σ(x) =
trace A(x)
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we have
0<
(2.3)
2
1
1
2
≤
||A(x)||
≤
trace
A(x)
σ 2 (x)
n−1+ε
with σ(·) ∈ L2loc (Ω). Therefore without a strictly positive lower bound for
trace A(x), the Cordes condition Kε does not imply uniform ellipticity. As an
example we can mention
p xy y
2
A(x, y) = p xy
x
2
defined on
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
András Domokos
o
n
y
2
2
2
Ω = (x, y) ∈ R : x > 0, y > 0, 0 < x + y < 1, 1 < < 2 .
x
Contents
In this case inequality (2.2) looks like
x2 + xy + y 2 <
1
(x + y)2 .
1+ε
Considering the lines y = mx we see that
m
2
ε = inf
:1<m<2 =
2
m +m+1
7
and
√
σ(x) =
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2
.
x+y
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Remark 2. In the case when we want to have a strictly positive lower bound
for trace A we should use a Cordes condition Kε,γ that asks for the existence of
a number γ > 0 such that
2
1
1
1
≤ ||A(x)||2 ≤
trace A(x)
(2.4)
0< ≤ 2
γ
σ (x)
n−1+ε
for almost every xP
∈ Ω. In this way the normalized condition (1.1) corresponds
to the Kε,n , since ni=1 aii = 1 implies that γ = n.
We recall the following lemma from [5].
Lemma 2.1. Let a = (a1 , . . . , an ) ∈ Rn . Suppose that
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
András Domokos
(a1 + · · · + an )2 > (n − 1)||a||2 .
(2.5)
If for α > 1 and β > 0 the condition
1
1
1
2
2
(2.6) (a1 + · · · + an ) ≥ n − 1 +
||a|| +
n−1+
(α − 1),
α
β
α
holds, then we have
(2.7)
||k||2 + 2α
X
ki kj ≤ β(a1 k1 + · · · + an kn )2
i<j
for all k = (k1 , . . . , kn ) ∈ Rn .
The next lemma is the nonsymmetric version of the original one in Talenti’s
paper [5]. By nonsymmetric version we mean that we drop the assumption that
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A is symmetric. On the other hand, it is easy to see that Lemma 2.2 below will
not hold for arbitrary nonsymmetric matrices P , even in the case when A is
diagonal. For the completeness of our paper we include the proof, which can be
considered as a natural extension of the original one.
Lemma 2.2. Let A = (aij ) be an n × n real matrix. Suppose that
(trace A)2 > (n − 1)||A||2 .
(2.8)
If for α > 1 and β > 0 the condition
1
1
1
2
2
(2.9)
(trace A) ≥ n − 1 +
||A|| +
n−1+
(α − 1)
α
β
α
holds, then we have
(2.10)
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
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n X
pii pij
2
||P || + α
pij pjj
i,j=1
≤ β hA, P i2
for all real and symmetric n × n matrices P = (pij ).
Proof. Consider an arbitrary but fixed real and symmetric matrix P . It follows
that there exists a real orthogonal matrix C and a diagonal matrix


k1
0


..
D=

.
0
kn
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such that P = C −1 DC. Observe that
n 1 X pii pij
−
pij pjj
2
i,j=1
is the coefficient of λn−2 in the characteristic polynomial of P , therefore
n 1 X pii pij X
=
ki k j .
2 i,j=1 pij pjj i<j
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
Moreover,
n
X
(2.11)
p2ij
2
= trace(P ) =
i,j=1
n
X
ki2 .
András Domokos
i=1
Hence, inequality (2.10) can be rewritten as
|k|2 + 2α
(2.12)
X
k i kj ≤ β
i<j
Let B = CAC
(2.13)
−1
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n
X
!2
aij pij
i,j=1
. Then trace B = trace A and
hA, P i = trace(AP )
= trace(CAP C −1 )
= trace(CAC −1 CP C −1 )
= trace(BD)
n
X
=
bii ki .
i=1
.
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Also, because B and A are unitary equivalent, we have
n
n
n
X
X
X
2
2
bii ≤
bij =
a2ij .
i=1
ij
i,j=1
Therefore, b = (b11 , . . . , bnn ), α and β satisfy the condition (2.6) from Lemma
2.1, and hence
n
X
X
k12 + 2α
ki kj ≤ β(b11 k1 + · · · + bnn kn )2 = βhA, P i2 .
i=1
W 2,2 Estimates for Solutions to
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with Measurable Coefficients
i<j
Using (2.11) – (2.13) we get (2.10).
Theorem 2.3. Suppose that A satisfies the Cordes condition Kε . Then for all
u ∈ C0∞ (Ω) we have
p
1 √
2
n − 1 + ε + (1 − ε)(n − 1) ||σAu||L2 (Ω) .
(2.14) ||D u||L2 (Ω) ≤
ε
Proof. Fix x ∈ Ω such that (2.3) holds and consider an arbitrary α > 1/ε. Then
!2 n
X
1
aii (x) > n − 1 +
||A(x)||2 .
α
i=1
In order to choose β(x) > 0 such that
!2
n
X
(2.15)
aii (x)
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≥
1
n−1+
α
1
||A(x)||2 +
β(x)
1
n−1+
α
(α − 1),
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observe that condition Kε is equivalent to
!2 n
X
1
1
2
aii (x) ≥ n − 1 +
||A(x)|| + ε −
||A(x)||2 .
α
α
i=1
Therefore we have to ask β(x) to satisfy
1
1
1
2
||A(x)|| ≥
n−1+
(α − 1),
ε−
α
β(x)
α
and hence
(n − 1)α2 + (2 − n)α − 1
.
εα − 1
Considering the function f : (1/ε, +∞) → R defined by
(2.16)
β(x) ≥ σ 2 (x)
(n − 1)α2 + (2 − n)α − 1
,
εα − 1
we get that its minimum point is
p
n − 1 + (n − 1)(1 − ε)(n − 1 + ε)
α=
.
(n − 1)ε
f (α) =
Therefore, the minimum value of σ 2 (x) f (α), which is coincidentally the best
choice of β(x), is
p
2ε − εn + 2n − 2 + (n − 1)(1 − ε)(n − 1 + ε)
2
β(x) = σ (x)
ε2
2
2
p
σ (x) √
=
n
−
1
+
ε
+
(1
−
ε)(n
−
1)
.
ε2
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with Measurable Coefficients
András Domokos
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Applying Lemma 2.2 in the case of u ∈ C0∞ (Ω) and pij = Dij u(x), we get
(2.17)
Z X
n
X Z Dii u(x) Dij u(x) (Dij u(x)) dx + α
Dij u(x) Djj u(x) dx
Ω i,j=1
i6=j Ω
Z
≤
β(x)(Au(x))2 dx.
2
Ω
But, integrating by parts two times we get
Z
Z
(2.18)
Dii u(x)Djj u(x)dx =
Dij u(x)Dij u(x)dx,
Ω
Ω
and hence
(2.19)
W 2,2 Estimates for Solutions to
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with Measurable Coefficients
András Domokos
Z Dii u(x) Dij (x)
Ω Dij u(x) Djj u(x)
dx = 0.
Therefore, for all u ∈ C0∞ (Ω) we have
p
1 √
n − 1 + ε + (1 − ε)(n − 1) ||σAu||L2 (Ω) .
||D2 u||L2 (Ω) ≤
ε
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Theorem 2.3 clearly implies the following result.
Corollary 2.4. Suppose that A satisfies Cordes condition Kε,γ . Then for all
u ∈ W02,2 (Ω) we have
√ p
γ √
2
(2.20) ||D u||L2 (Ω) ≤
n − 1 + ε + (1 − ε)(n − 1) ||Au||L2 (Ω) .
ε
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Remark 3. In the case of trace A = 1 we get that
√ p
n √
2
n − 1 + ε + (1 − ε)(n − 1) ||Au||L2 (Ω) .
(2.21) ||D u||L2 (Ω) ≤
ε
If we compare estimate (1.2) with ours from (2.21) we realize that our constant
on the right hand side is larger. The interesting fact is that the two constants in
(1.2) and (2.21) coincide in the case when A = n1 I and ε = 1, and give (see
[2])
||D2 u||L2 (Ω) ≤ ||∆u||L2 (Ω) , for all u ∈ W02,2 (Ω) .
Looking at Talenti’s paper [5] we realize that the way in which the constant B
is chosen on page 303 leads to
(2.22)
||A(x)||2 ≥
1
.
n−1+ε
Comparing this inequality to (1.1) which gives
1
1
≤ ||A(x)||2 ≤
n
n−1+ε
and therefore
1
||A(x)||2 =
,
n−1+ε
we conclude that (2.22) (and hence (1.2)) holds for constant matrices A but may
fail for a nonconstant A(x) on a subset of Ω with positive Lebesgue measure.
Therefore, the estimate (2.21) is the right one for nonconstant matrix functions
A(x) satisfying (1.1).
W 2,2 Estimates for Solutions to
Non-Uniformly Elliptic PDE’S
with Measurable Coefficients
András Domokos
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Remark 4. Another interesting fact is found when applying our method to the
case of convex functions u. In this case we can further generalize the Cordes
condition in the following way: We say that A satisfies the condition Kε(x) if
1
∈ L2loc (Ω)
trace A
and there exists a measurable function ε : Ω → R such that 0 < ε(x) ≤ 1 for
a.e. x ∈ Ω and 1ε ∈ L2 (Ω), and the following inequalities hold:
2
2
trace A(x)
trace A(x)
1
(2.23)
0< 2
=
≤ ||A(x)||2 ≤
.
σ (x)
n
n − 1 + ε(x)
Inequality (2.17) in this case looks like
Z X
n
XZ
Dii u(x) Dij u(x) 2
dx
(Dij u(x)) dx +
α(x) D
u(x)
D
u(x)
ij
jj
Ω i,j=1
i6=j Ω
Z
≤
β(x)(Au(x))2 dx .
Ω
Observe that the convexity of u implies that D2 u(x) is positive definite, which
makes the determinants
Dii u(x) Dij u(x) Dij u(x) Djj u(x) positive. We conclude in this way that under the Cordes condition Kε(x) for all
convex functions u ∈ W 2,2 (Ω) we still have
p
1 √
2
||D u||L2 (Ω) ≤ n − 1 + ε + (1 − ε)(n − 1) σAu
2 .
ε
L (Ω)
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with Measurable Coefficients
András Domokos
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References
[1] H.O. CORDES, Zero order a priori estimates for solutions of elliptic differential equations, Proceedings of Symposia in Pure Mathematics, IV (1961),
157–166.
[2] D. GILBARG AND N.S. TRUDINGER, Elliptic Partial Differential Equations of Second Order, Springer-Verlag, 1983.
[3] J.J. MANFREDI AND A. WEITSMAN, On the Fatou Theorem for
p−Harmonic Functions, Comm. Partial Differential Equations 13(6)
(1988), 651–668
[4] C. PUCCI AND G. TALENTI, Elliptic (second-order) partial differential
equations with measurable coefficients and approximating integral equations, Adv. Math., 19 (1976), 48–105.
[5] G. TALENTI, Sopra una classe di equazioni ellitiche a coeficienti misurabili, Ann. Mat. Pura. Appl., 69 (1965), 285–304.
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