Elliptic partial differential equations with almost-real coefficients

120 Pages·2013·0.91 MB·English

by  Ariel Barton

MEMOIRS
of the
American Mathematical Society
Volume 223 • Number 1051 (fifth of 5 numbers) • May 2013
Elliptic Partial Differential Equations
with Almost-Real Coefficients
Ariel Barton
ISSN 0065-9266 (print) ISSN 1947-6221 (online)
American Mathematical Society
Number 1051
Elliptic Partial Differential Equations
with Almost-Real Coefficients
Ariel Barton
May2013 • Volume223 • Number1051(fifthof5numbers) • ISSN0065-9266
Library of Congress Cataloging-in-Publication Data
Barton,Ariel,1982-
Ellipticpartialdifferentialequationswithalmost-realcoefficients/ArielBarton.
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ISBN978-0-8218-8740-0(alk. paper)
1.Differentialequations,Elliptic. 2.Differentialequations,Partial. I.Title.
QA329.42.B37 2013
515(cid:2).3533–dc23 2012051365
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Contents
Chapter 1. Introduction 1
1.1. History 3
Chapter 2. Definitions and the Main Theorem 9
2.1. Geometric definitions 9
2.2. Definitions of function spaces 11
2.3. Layer potentials 13
2.4. The main theorem 14
2.5. Additional definitions 16
Chapter 3. Useful Theorems 21
3.1. Nontangential maximal functions 21
3.2. Bounds on solutions 23
3.3. Existence results 25
3.4. Preliminary uniqueness results 27
3.5. The Neumann and regularity problems in unusual domains 28
Chapter 4. The Fundamental Solution 33
4.1. A fundamental solution exists 33
4.2. Uniqueness of the fundamental solution 35
4.3. Symmetry of the fundamental solution 36
4.4. Conjugates to the fundamental solution 38
4.5. Caldero´n-Zygmund kernels 39
4.6. Analyticity 39
Chapter 5. Properties of Layer Potentials 43
5.1. Limits of layer potentials and the adjoint formulas 44
Chapter 6. Boundedness of Layer Potentials 49
6.1. Proof for a small Lipschitz constant: preliminary remarks 49
6.2. A B for the TB theorem 51
1
6.3. Weak boundedness of operators 54
6.4. The adjoint inequalities 56
6.5. Proof for a small Lipschitz constant: final remarks 63
6.6. Buildup to arbitrary special Lipschitz domains 64
6.7. Patching: special Lipschitz domains to bounded Lipschitz domains 67
Chapter 7. Invertibility of Layer Potentials and Other Properties 69
7.1. Nontangential maximal functions of layer potentials 69
7.2. Jump relations 73
7.3. Layer potentials on H1(∂V) 76
iii
iv CONTENTS
7.4. Invertibility of layer potentials on Lp(∂V) 79
Chapter 8. Uniqueness of Solutions 83
8.1. Counterexamples to uniqueness 83
8.2. Uniqueness results 84
Chapter 9. Boundary Data in H1(∂V) 89
9.1. Solutions with boundary data in H1 89
9.2. Invertibility of layer potentials on H1(∂V) 95
Chapter 10. Concluding Remarks 97
10.1. Converses 98
Bibliography 105
Abstract
In this monograph we investigate divergence-form elliptic partial differential
equations in two-dimensional Lipschitz domains whose coefficient matrices have
small (but possibly nonzero) imaginary parts and depend only on one of the two
coordinates.
We show that for such operators, the Dirichlet problem with boundary data
in Lq can be solved for q < ∞ large enough. We also show that the Neumann
and regularity problems with boundary data in Lp can be solved for p > 1 small
enough, and provide an endpoint result at p=1.
ReceivedbytheeditorNovember8,2010,and,inrevisedform,February7,2012.
ArticleelectronicallypublishedonOctober24,2012;S0065-9266(2012)00677-0.
2010 MathematicsSubjectClassification. Primary35J25;Secondary31A25.
Keywordsandphrases. Ellipticequations,complexcoefficients,Dirichletproblem,Neumann
problem,regularityproblem,Lipschitzdomains,Hardyspaces,layerpotentials.
Affiliationattimeofpublication: SchoolofMathematics,University ofMinnesota,Vincent
Hall,206ChurchSt. SE,Minneapolis,Minnesota55455-0488,email: [email protected].
(cid:2)c2012 American Mathematical Society
v

CHAPTER 1
Introduction
In this monograph, we consider solutions to boundary value problems for the
second-order divergence form partial differential equation
divA(X)∇u(X)=0.
The matrix of coefficients A is taken tobe measurable; we do not assume that A is
differentiable. Thus, thesolutionsulieintheSobolevspaceW1,2 offunctionswith
loc
one weak derivative, and the equation divA∇u=0 must be interpreted weakly. If
V is an open set, we say that divA∇u=0 in V if
ˆ
(1.1) ∇η·A∇u=0 for all η ∈C∞(V).
0
V
We always assume that the coefficient matrix A is bounded and elliptic, that
is, there exist some constants Λ>λ>0 such that
(1.2) λ|η|2 ≤Reη¯·A(X)η, |ξ·A(X)η|≤Λ|η||ξ|
for every X ∈Rn and every ξ, η ∈Cn. In this monograph, we will prove results in
thespecialcaseoftwodimensions(V ⊂R2),andofcoefficientsA(x,t)independent
ofoneofthetwocoordinates. Undertheseconditions,solutionsuarelocallyHo¨lder
continuous and their gradients are locally bounded.
We consider three boundary-value problems. If 1 < q < ∞, then we say
that the Dirichlet problem with boundary data in Lq(∂V), or (D)A, holds in the
q
domain V with constant C, if for every f ∈Lq(∂V), there exists a unique function
u∈W1,2(V) such that
loc ⎧
⎪⎨divA∇u=0 in V,
(D)A u=f on ∂V,
q ⎪⎩
(cid:7)Nu(cid:7) ≤C(cid:7)f(cid:7) .
Lq(∂V) Lq(∂V)
The bottom two lines are interpreted as follows. If X ∈∂V, let
(1.3) γ (X)={Y ∈U :|X −Y|<(1+a)dist(Y,∂V)}
V,a
forsomefixedpositivenumbera. Wesaythatu=f on∂V iff isthenontangential
limit of u; that is, limY→X,Y∈γV,a(X)u(Y)=f(X) for almost every X ∈∂V. The
nontangential maximal function Nu is defined by
(1.4) Nu(X)=N u(X)=esssup{|u(Y)|:Y ∈γ (X)}.
U,a U,a
If there exists a number C > 0 such that (D)A holds in V with constant C, we
q
simply say that (D)A holds in V.
q
Similarly, if 1<p<∞, we say that the Neumann problem (N)A or regularity
p
problem(R)A holdsinV withconstantC if, foreveryg ∈Lp(∂V)∩H1(∂V), there
p
1
2 1. INTRODUCTION
is a unique function u such that
⎧ ⎧
⎪⎨divA∇u=0 in V, ⎪⎨divA∇u=0 in V,
(N)A ν·A∇u=g on ∂V, or (R)A ∂ u=g on ∂V,
p ⎪⎩ p ⎪⎩ τ
(cid:7)N(∇u)(cid:7) ≤C(cid:7)g(cid:7) (cid:7)N(∇u)(cid:7) ≤C(cid:7)g(cid:7) .
Lp(∂V) Lp(∂V) Lp(∂V) Lp(∂V)
If VC is bounded we additionally require that lim|X|→∞u(X) exist. Here H1(∂V)
denotes the atomic Hardy space of harmonic analysis, and ν and τ are the unit
outwardnormalandunittangentvectorstothedomainV. Wesaythatν·A∇u=g
weakly if
ˆ ˆ
(1.5) A∇u·∇η = gTrηdσ for all η ∈C∞(V).
0
V ∂V
Iff isthenontangentiallimitofu,then∂ u=gifg isthederivativeoff along∂V.
τ
This monograph has two main results. The first main result is that under
certain conditions, the boundary value problems above hold in V.
Theorem 1.6. Suppose that A and A are bounded, elliptic matrices defined
0
on R2 which depend only on one of the two coordinates. Assume that A is real-
0
valued; A may be complex-valued. Let V be a Lipschitz domain in R2 which has
connected boundary.
Then there exist some constants (cid:10) > 0 and p > 1, depending only on V and
0
A0, such that if (cid:7)A−A0(cid:7)L∞ <(cid:10), 1<p≤p0, and 1/p+1/q =1, then (D)Aq, (N)Ap,
and (R)A hold in V.
p
Our second main result is an endpoint result for the Neumann and regularity
problems. We say that (N)A or (R)A holds in V with constant C if, for every
1 1
g ∈H1(∂V), there exists a unique function u defined in V such that
⎧ ⎧
⎪⎨divA∇u=0 in V, ⎪⎨divA∇u=0 in V,
(N)A ν·A∇u=g on ∂V, or (R)A ∂ u=g on ∂V,
1 ⎪⎩ 1 ⎪⎩ τ
(cid:7)N(∇u)(cid:7) ≤C(cid:7)g(cid:7) , (cid:7)N(∇u)(cid:7) ≤C(cid:7)g(cid:7) .
L1(∂V) H1(∂V) L1(∂V) H1(∂V)
That is, we consider only boundary data g in H1(∂V)(cid:2)L1(∂V).
Theorem 1.7. Let A , A, and V be as in Theorem 1.6. There is some (cid:10) > 0
0
depending only on V and A0 such that if (cid:7)A−A0(cid:7)L∞ < (cid:10), then (N)A1 and (R)A1
hold in V.
Conversely, if V is simplyconnected, divA∇u=0 inV and N(∇u)∈L1(∂V),
then the boundary values ν · A∇u and ∂ u exist and lie in H1(∂V). Further-
τ
more, there is a constant C such that (cid:7)ν ·A∇u(cid:7) ≤ C(cid:7)N(∇u)(cid:7) and
H1(∂V) L1(∂V)
(cid:7)∂ u(cid:7) ≤C(cid:7)N(∇u)(cid:7) .
τ H1(∂V) L1(∂V)
If A = A is real-valued, then the conclusions of Theorem 1.6 are known to
0
hold; the conclusion regarding (D)A0 was proven in [KKPT00] by Kenig, Koch,
q
Pipher and Toro, and the conclusions for (N)A0 and (R)A0 were proven in [KR09]
p p
and [Rul07] by Kenig and Rule. The conclusions of Theorem 1.7 were proven in
[DK87] by Dahlberg and Kenig in the case of harmonic functions; the conclusions
(N)A and (R)A were proven in [KP93] by Kenig and Pipher under the conditions
1 1
that A is real symmetric and (N)A and (R)A hold for some p>1.
p p
Theorganizationofthismonographisasfollows. Themainresultswerestated
aboveinTheorem 1.6andTheorem 1.7. Wewillconcludethischapterbyreviewing
1.1. HISTORY 3
thehistoryofboundaryvalueproblemsinLipschitzdomainswithLp(∂V)boundary
data. In Chapter 2, we define our terminology. The proof of our main results is
by the classic method of layer potentials; for the reader’s convenience, we provide
an outline of this method in Section 2.4. In Chapter 3, we provide a number of
usefultheoremsregardingnontangentialmaximalfunctionsandsolutionstopartial
differential equations.
We work out the details of the method of layer potentials in Chapters 4–7.
In Chapter 4, we construct a fundamental solution for the operator divA∇. In
Chapter 5, we use this fundamental solution to construct layer potentials. In
Chapter 6, we show that layer potentials are bounded operators. In Chapter 7,
we prove some consequences of boundedness, including a perturbative invertibility
result. Theresultsofthesechaptersletusprovethatsolutionsto(D)A,(N)A and
q0 p0
(R)A exist.
p0
We prove uniqueness of solutions in Chapter 8. In Chapter 9, we use existence
and uniqueness of solutions to (N)A and (R)A to prove existence and uniqueness
p0 p0
ofsolutionsto(N)A and(R)A. Wemaytheninterpolatetoprovethat(D)A,(N)A
1 1 q p
and (R)A hold if 1<p<p .
p 0
Most of the results of Chapters 4–9 assume that the coefficient matrices A are
smooth. In Chapter 10, we pass from smooth coefficients to bounded measurable
coefficients. We also prove the converses mentioned in Theorem 1.7.
ThismonographisarevisionofmythesiswrittenattheUniversityofChicago.
MyadvisorwasCarlosE.Kenig,towhomIwouldliketoextendmygratefulthanks;
without his guidance and advice, the work here would not have been possible.
1.1. History
The study of second-order elliptic boundary value problems in Lipschitz do-
mains has a long and rich history. The study began with harmonic functions, that
is, with solutions u to (1.1) where A≡I, the identity matrix. Many of the results
of this study can be extended to more general problems under some conditions on
thecoefficientsA. Thismonographconcernsoneparticular suchcondition, namely
that the coefficients A be independent of some specified coordinate.
In this section, we begin by discussing some known results for harmonic func-
tions. We then discuss how these results have been extended under various other
assumptions on A, before focusing on the study of coefficients independent of one
coordinate. We conclude this section by briefly reviewing the history of boundary
value problems with data in Hardy spaces.
In [Dah77] and [Dah79], Dahlberg showed that if V is a Lipschitz domain,
then there is some (cid:10) > 0 depending on V such that if 2−(cid:10) < q < ∞, then (D)I
q
holdsinV. Thisrangeissharpinthesensethatgivenq <2,thereissomeLipschitz
domain V such that (D)I does not hold in V; see [FJL77].
q
In [JK81b], Jerison and Kenig showed that (N)I and (R)I hold in Lipschitz
p p
domains in all dimensions provided p = 2. This was extended to the case 1 <
p < 2+(cid:10), (cid:10) again depending on the domain, in [Ver84] (the regularity problem)
and in [DK87] (the Neumann problem in three or more dimensions). It had been
observed by Kenig and Fabes that (N)I holds, for p in this range, in Lipschitz
p
domains contained in R2. The same results hold if A is an arbitrary constant-
coefficient elliptic matrix; in [She06] and [She07], Shen has proven similar results
for systems of constant-coefficient elliptic operators.