Elliptic Partial Differential Equations with Complex Coefficients

# Elliptic Partial Differential Equations with Complex Coefficients

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Mathematics \divisionPhysical Sciences \degreeDoctor of Philosophy

ELLIPTIC PARTIAL DIFFERENTIAL EQUATIONS WITH COMPLEX COEFFICIENTS

BY

NOVEMBER 2009

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Abstract In this paper I investigate elliptic partial differential equations on Lipschitz domains in whose coefficient matrices have small (but possibly nonzero) imaginary parts and depend only on one of the two coordinates.

I show that for Dirichlet boundary data in for large enough, solutions exist and are controlled by the -norm of the boundary data.

Similarly, for Neumann boundary data in , or for Dirichlet boundary data whose tangential derivative is in (“regularity” boundary data), for small enough, I show that solutions exist and are controlled by the -norm of the boundary data.

I prove similar results for Neumann or regularity boundary data in , and for Dirichlet boundary data in or . Finally, I show some converses: if the solutions are controlled in some sense, then Dirichlet, Neumann, or regularity boundary data must exist.

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Acknowledgements I would like to thank my advisor Carlos Kenig, and my parents.

## Chapter \thechapter Introduction

Let be a uniformly elliptic matrix-valued function defined on . That is, assume that there exist constants (called the ellipticity constants of ) such that

 λ|η|2≤Re¯η⋅A(X)η,|ξ⋅A(X)η|≤Λ|η||ξ| (1)

for every and every .

Let . We say that , , or hold in the Lipschitz domain if, for every , there is a function such that

 (D)Ap ⎧⎨⎩divA∇u=0 in V,u=f on ∂V,∥Nu∥Lp≤C(p)∥f∥Lp (N)Ap ⎧⎨⎩divA∇u=0 in V,ν⋅A∇u=g on ∂V,∥N(∇u)∥Lp≤C(p)∥g∥Lp (R)Ap ⎧⎨⎩divA∇u=0 in V,u=f on ∂V,∥N(∇u)∥Lp≤C(p)∥∂τf∥Lp

Furthermore, is unique among functions with and either (for ) or (for or ).

Here are the unit normal and tangent vectors to . In the case , , we in addition require that or , respectively.

In this paper, we will instead be concerned with complex matrix-valued functions on which satisfy

 λ|η|2≤Re¯η⋅A(x,t)η,|ξ⋅A(x,t)η|≤Λ|η||ξ|,A(x,t)=A(x,s) (2)

for all , , and all , .

In [13] and [14], the following theorem was proven:

###### Theorem 3.

Suppose that is real-valued and satisfies (2). Let be a Lipschitz domain which is either bounded or special, and let its Lipschitz constants be at most . (See Definition 13.)

Let . Then if , hold in with constants at most , then and hold in the Lipschitz domain , with constants depending only on , , and .

In [12], it was shown that

###### Theorem 4.

Let be as in Theorem 3, and suppose that is a bounded or special Lipschitz domain with Lipschitz constants at most . Then there is some (possibly large) and some such that holds in with constant at most .

In this paper, we intend to prove the following theorem:

###### Theorem 5.

Suppose that satisfies the conditions of Theorem 3. Suppose that satisfies the same conditions, except that is allowed to be a complex-valued matrix instead of a real-valued matrix. Let be a good Lipschitz domain, as defined in Definiton 13.

Then there is some , depending only on , and the Lipschitz constants of , such that if and , then , and hold in the Lipschitz domain .

Furthermore, if , then there are two functions , unique among with and , such that

 (N)A1 ⎧⎨⎩divA∇uN=0 in Ω,ν⋅A∇uN=f on ∂Ω,∥N(∇uN)∥L1≤C∥f∥H1, (6) (R)A1 ⎧⎨⎩divA∇uR=0 in Ω,∂τuR=f on ∂Ω,∥N(∇uR)∥L1≤C∥f∥H1, (7)

We will also prove results for boundary data in and :

###### Theorem 8.

Suppose that and satisfy the conditions of Theorem 5. If , the dual to , then there is a unique function such that on , in , and such that is a Carleson measure, that is, for any and any ,

 1σ(B(X0,R)∩∂V)∫B(X0,R)∩V|∇u(X)|2dist(X,∂V)dX≤~C (9)

where and depends only on , , the Lipschitz constants of , and the constants in Theorem 5. Furthermore, this function is unique among functions with in , on , and which satisfy (9) for some constant .

In [5], it was shown that for real coefficent matrices on bounded Lipschitz domains , this theorem is equivalent to an condition on harmonic measure; this condition implies that holds in for some .

###### Theorem 10.

If is bounded, then the maximum principle holds; that is, if , then there exists a unique with in , on and .

Finally, we will prove some converses:

###### Theorem 11.

Suppose that satisfy the conditions of Theorem 5. Suppose further that in .

If , then the boundary values and exist and are in with -norm at most .

If satisfies (9) for some , then exists and is in with norm at most . Furthermore, if , then ; functions in are unique, and so holds.

Theorem 10 will be proven in Elliptic Partial Differential Equations with Complex Coefficients. Uniqueness and converses for Theorems 5 and 8 will be proven in Elliptic Partial Differential Equations with Complex Coefficients. Existence for smooth coefficients for Theorem 8 will be proven in Elliptic Partial Differential Equations with Complex Coefficients, and we will pass to arbitrary elliptic (rough) coefficients in Theorems 145 and 156.

We now outline the proof of existence for Theorem 5; resolving the details will form the bulk of this work.

###### Proof of Theorem 5.

If is in for , then we can define (Definition 17) functions such that if , then and are well-defined complex numbers, and , in . (Here is the matrix transpose of .)

We will show that, if , are bounded then

 ∥N(Df)∥Lp(∂V)≤C(p)∥f∥Lp(∂V),∥N(∇STf)∥Lp(∂V)≤C(p)∥f∥Lp(∂V)

for all . (Theorem 64.) Furthermore, if , then , and so by density we may extend to all of . (Corollary 66.)

There exist operators on such that

 K±f =Df|∂V±,Ltf=τ⋅∇STf|∂V, Kt±f =ν⋅AT∇STf|∂V∓

in appropriate weak senses, where , .

If is invertible with bounded inverse on some , then for every , we may let . Then

 divA∇u=0,u|∂V=g,∥Nu∥Lp≤Cp∥K−1+g∥Lp≤Cp∥K−1+∥Lp↦Lp∥g∥Lp

and so is a solution to .

Similarly, if or is bounded and invertible on or , then or is a solution to , , , or .

So we need only show that the layer potentials are bounded and invertible on or . (Invertibility of follows from invertibility of its transpose .)

If is smooth, and

 Ω={X∈R2:φ(X⋅e⊥)

for some Lipschitz function (which we may assume, a priori, to be in ) and some unit vector , then we can prove that are bounded and for . We can then extend this to arbitrary Lipschitz domains . (Theorem 68, Theorem 77 and Theorem 91.)

There is some (in fact, the in Theorem 3) and some such that if , then has a bounded inverse on and , have bounded inverses on and . (Chapters Elliptic Partial Differential Equations with Complex Coefficients and Elliptic Partial Differential Equations with Complex Coefficients.)

Using standard interpolation techniques (Elliptic Partial Differential Equations with Complex Coefficients), we can show that have bounded inverses on for , and therefore has a bounded inverse on for .

Thus, we have that , , hold if is smooth, for small. We pass to arbitrary (rough) in Elliptic Partial Differential Equations with Complex Coefficients. ∎

## Chapter \thechapter Definitions

If , then we define in in the weak sense, that is,

 ∫A∇u⋅∇η=0

for all . Similarly, we define on in the weak sense, that is,

 ∫VA∇u⋅∇η=∫∂Vgη,

for all .

We say that on if is the non-tangential limit of a.e., that is, if

 limη→0sup{|u(Y)−f(X)|:|X−Y|<(1+a)dist(Y,∂V)<η}

holds for almost every .

We fix some notation. If is a domain, then , . The inner product between and will be given by

 ⟨G,F⟩=∫G(x)tF(x)dx.

(This notation is more convenient than the usual inner product .) A superscript of will denote the transpose of a matrix or the adjoint of an operator with respect to this inner product. (So if happen to be scalar-valued functions, then and , and if is an operator, then .)

If I have a function or operator defined in terms of , then a superscript of will denote the corresponding function or operator for . (So ; however, for most operators defined in terms of .)

If is a domain, then if and is a function defined on , we let be the generalization of the Littlewood-Paley maximal function given by connected.

If is defined on , we define the non-tangential cones and non-tangential maximal function by

 γU,a(X)={Y∈U:|X−Y|<(1+a)dist(Y,∂U)},NU,af(X)=supγU,a(X)|f(Y)| (12)

for some number . When no ambiguity will arise we suppress the subscripts  or ; we let .

###### Definition 13.

We say that the domain is a special Lipschitz domain if, for some Lipschitz function and unit vector ,

 Ω={X∈R2:φ(X⋅e⊥)

We refer to as the Lipschitz constant of .

We say that is a Lipschitz domain with Lipschitz constant if there is some such that, for every , there is some neighborhood of such that for some special Lipschitz domain with Lipschitz constant at most .

If is a special Lipschitz domain, then let . Otherwise, let be numbers such that may be covered (with overlaps) by at most such neighborhoods whose size varies by at most .

That is, there is a constant such that

 ∂U⊂k2⋃j=1B(Xj,rj)

for some and . We further require that there exist unit vectors and Lipschitz functions , with , such that if

 Ωj ={X∈R2:φj((X−Xj)⋅e⊥j)<(X−Xj)⋅ej}, Rj ={X∈R2:|(X−Xj)⋅e⊥j|<2rj,|(X−Xj)⋅ej|<(2+2k1)rj},

then

 U∩Rj=Ωj∩Rj.

We refer to , , as the Lipschitz constants of . For simplicity, when we write , we mean , .

If the are finite, and is connected, then we call a good Lipschitz domain.

We will reserve for good Lipschitz domains and for special Lipschitz domains. Note that if is a good Lipschitz domain, then either is special or is bounded.

Note that we do not care about the value of the positive constant in the definition of ; this is because , , are scale-invariant.

A number depending on these constants is very important. Let be such that, if and , then . We refer to as the Ahlfors-David constant of .

We can bound some integrals which will be needed later:

 ∫∂U∖B(X0,r)rp−1|X−X0|pdσ(X)≤∞∑j=0rp−1(bjr)pσ(B(X0,rbj+1))≤k4p−1 (14)

where . In particular, if then

 ∫∂U1|X−X0|pdσ(X)≤k4p−1dist(X0,∂U)1−p.

The letter will always represent a positive constant, whose value may change from line to line, but which depends only on the ellipticity constants of , the positive constant in the definition of non-tangential maximal function, and the Lipschitz constants of whatever domain we are dealing with. If a particular constant depends on another parameter, it will be indicated explicitly. We will occasionally use the symbols to indicate inequality up to a multiplicative constant (e.g.  as shorthand for ); we will use to mean that and  both hold.

###### Lemma 15.

Let be an elliptic matrix-valued function. Then, for each , there is a function , unique up to an additive constant, such that

 |ΓX(Y)|≤C+C|log|X−Y||

for every and

 ∫R2A(Y)∇ΓX(Y)⋅∇η(Y)dy=−η(X) (16)

for every .

We refer to this function as the fundamental solution for with pole at .

This lemma will be proven in Elliptic Partial Differential Equations with Complex Coefficients. By we mean the gradient in . We will sometimes wish to refer to the gradient in ; we will then write .

is defined to be is finite for some Schwarz function with , where . It can be shown that if , then , where , , for some , and . Functions satisfying these conditions are called atoms.

We may extend the definition of to , where is a Lipschitz domain. We say that if , where the are complex numbers, and , for some is connected, and . The norm is the smallest among all such representations of .

If is a good Lipschitz domain, then this is equivalent to defining atoms to be functions which satisfy , for some , and .

We consider to be the dual of . This means that

 ∥f∥BMO(∂U) =supΔ⊂∂U connected1σ(Δ)∫Δ∣∣f−\vbox−∫Δf∣∣dσ ≈supX∈∂V,R>0infC1R∫B(X,R)∩∂U|f−C|dσ.

Multiply connected domains are beyond the scope of this paper. However, many lemmas and theorems in this paper have obvious generalizations to multiply connected domains. For the most part, these require that functions integrate to 0 on each connected component of .

###### Definition 17.

Let be a Lipschitz domain with unit outward normal and tangent vectors and . If is a function, we define

 ∇Sf(X) =∇SVf(X)=∫∂V∇XΓTX(Y)f(Y)dσ(Y) (18) Df(X) =DVf(X)=∫∂Vν(Y)⋅AT(Y)∇ΓTX(Y)f(Y)dσ(Y). (19)

This defines up to an additive constant.

We define the layer potentials , via

 K±f(X) =KAV,±f(X)=limZ→X,Z∈γ±(X)∫∂Vν(Y)⋅AT(Y)∇ΓATZ(Y)f(Y)dσ(Y) (20) Lf(X) =LAVf(X)=limZ→X,Z∈γ+(X)∪γ−(X)∫∂Vτ(Y)⋅∇ΓATZ(Y)f(Y)dσ(Y). (21)

When no confusion will arise we omit the subscripts and superscripts. By (50), ; hence, and converge for and , .

We will show that the limits in the definition of and are well-defined for smooth, . (Lemma 61.)

as operators on (Lemma 103), so ; however, if with , then at any point where the obvious analogies exist, they are equal. (Lemma 104).

It will be shown that

 Kt±f=ν⋅AT∇STf|∂V∓,Ltf=τ⋅∇STf.

(Lemma 62).

We also need the following definitions:

 A(X) =(a11(X)a12(X)a21(X)a22(X)),A0(X)=(a011(X)a012(X)a021(X)a022(X)) (22) ∇~ΓTX(Y) =(0−110)A(Y)∇ΓTX(Y) (23) BA6(X) =(a11(X)a21(X)01) (24) BA7(X) =(B6(X)t)−1(A(X)ν(X)τ(X)) (25) KA(X,Y) =(BA6(Y)∇ΓATX(Y)BA6(Y)∇ΓATX(Y))t (26) ~KA(X,Y) =BA6(X)(∇X~ΓATX(Y)∇X~ΓATX(Y)) (27) TAVF(X) =limZ→X n.t.,Z∈V∫∂VKA(Z,Y)F(Y)dσ(Y) (28) ~TVF(X) =limZ→X n.t.,Z∈V∫∂V~K(Z,Y)F(Y)dσ(Y) (29)

When considering special Lipschitz domains, we will need some terminology:

 e =(e1e2),e⊥=(e2−e1) (30) ψ(x) =xe⊥+φ(x)e∈∂Ω, (31) ψ(x,h) =ψ(x)+he=(xe2+(φ(x)+h)e1−xe1+(φ(x)+h)e2), (32) so (x,t)=ψ(e2x−e1t,e1x+e2t−φ(e2x−e1t)), ⟨G,T±F⟩ =limh→0±∫R2G(x)tKh(x,y)F(y)dydx (33) ⟨G,~T±F⟩ =limh→0∓∫R2G(x)t~Kh(x,y)F(y)dydx (34) Kh(x,y) =(B6(ψ(y))∇ΓTψ(x,h)(ψ(y))B6(ψ(y))∇ΓTψ(x,h)(ψ(y)))t (35) =(∇ΓTψ(x,h)(ψ(y))t∇ΓTψ(x,h)(ψ(y))t)B6(ψ(y))t=K(ψ(x,h),ψ(y)) ~Kh(x,y) =B6(ψ(x))(∇X~ΓTψ(x)(ψ(y,h))∇X~ΓTψ(x)(ψ(y,h))) (36) ν(x) =1√1+φ′(x)2(φ′(x)e⊥−e)=1√1+φ′(x)2(−e1+e2φ′(x)−e2−e1φ′(x)) (37) τ(x) =1√1+φ′(x)2(e⊥+φ′(x)e)=1√1+φ′(x)2(e2+e1φ′(x)−e1+e2φ′(x)) (38) B1(x) =B1(ψ(x)), B1(ψ(x,h)) =(B6(ψ(x,h))t)−1√1+φ′(x)2(A(ψ(x,h))ν(x)τ(x)). (39)

Note that

We will occasionally want slightly different forms of :

 K′h(x,y) =(B6(ψ(y,h))∇ΓTψ(x)(ψ(y,h))B6(ψ(y,h))∇ΓTψ(x)(ψ(y,h)))t (40) =(∇ΓTψ(x)(ψ(y,h))t∇ΓTψ(x)(ψ(y,h))t)B6(ψ(y,h))t ~K′h(x