UNIVERSIDAD DE SONORA

Divisi´ on de Ciencias Exactas y Naturales

Programa de Posgrado en Matem´ aticas

The A

property of elliptic measures for operators with coefficients supported in Whitney-type cubes

T E S I S

Que para obtener el grado de:

Doctora en Ciencias (Matem´ aticas)

Presenta:

Marysol Navarro Burruel

Director de Tesis: Dr. Jorge Rivera Noriega

Hermosillo, Sonora, M´exico, Septiembre 26 de 2014

Dr. Jorge Rivera Noriega

Universidad Aut´onoma del Estado de Morelos, Cuernavaca, M´exico.

Dra. Martha Dolores Guzm´ an Partida

Universidad de Sonora, Hermosillo, M´exico.

Dr. Mart´ın Gildardo Garc´ıa Alvarado

Universidad de Sonora, Hermosillo, M´exico.

Dr. Salvador P´ erez Esteva

Universidad Nacional Aut´onoma de M´exico, Unidad Cuernavaca.

Dr. Georgy Omelianov Medvedev

Universidad de Sonora, Hermosillo, M´exico.

“Al Rey de los Siglos, inmortal, invisible, al ´unico y sabio Dios, sea honor y gloria por los siglos de los siglos. Am´en”.

1 Timoteo 1:17

Dedicado a t´ı mi Se˜nor Jesucristo.

Las letras, frases o expresiones no alcanzan para agradecer a aqu´el que es mi motivo, mi maestro, mi mejor amigo, mi amor, mi sost´en, quien me mantiene con vida y es mi vida entera. Una vez m´as, esta tesis, este nuevo logro, es por t´ı y para t´ı.

Gracias a mis padres Vicente y Ana Mar´ıa, por sus oraciones, su confianza, por su apoyo incondicional, porque a pesar de que en su coraz´on quisieran que est´e siempre con ustedes me han permitido salir de la seguridad del hogar y as´ı poder crecer en cada ´area de mi vida y convertirme en la mujer que ahora soy. Gracias por amarme y por demostr´armelo. Definitivamente forman parte de esta tesis, y lo mejor es que forman parte de mi vida. Los amo.

A mi hermano Paul, gracias por alegrar mi vida, por levantarme el ´animo, por amarme a pesar de todo. Tu vida misma me ha dado el impulso para concluir esta etapa de mi vida y me da ´animo para seguirme esforzando en lo que vendr´a. Te amo manito.

A mi tutora, asesora, maestra y gran amiga Dra. Martha Guzm´an Partida, muchas gracias por haber invertido su tiempo en m´ı, por cada palabra de aliento,

5

consejo y por cada vez que levant´o mi ´animo. Mi formaci´on acad´emica se la debo en gran manera a usted. La quiero much´ısimo.

A mi director de tesis, Jorge Rivera. Su guianza y ayuda me permitieron concluir esta etapa. Gracias por brindarme su tiempo, conocimiento, por abrirme las puertas de su casa y permitirme conocer a su hermosa familia a los cuales quiero mucho.

Gracias a cada uno de mis sinodales, por cada sugerencia y tiempo invertido en revisar esta tesis. Gracias al Instituto de Matem´aticas de la UNAM-Cuernavaca, por su hospitalidad durante 18 meses de visita acad´emica, a University of Kentucky y por ´ultimo pero no menos importante gracias al Departamento de Matem´aticas de la Universidad de Sonora y a CONACYT-M´exico por el apoyo econ´omico y acad´emico durante este trayecto.

Introduction 9

1 Background Material 15

1.1 Basic Theory of Muckenhoupt weights and BM O Space . . . 16

1.2 Solutions of Divergence form elliptic equations . . . 20

1.3 Some basic estimates for solutions . . . 23

1.4 The Classical Dirichlet Problem . . . 29

1.5 Green’s function . . . 31

2 Elliptic Measure and Dirichlet Problems 39 2.1 Estimates for Elliptic Measure . . . 40

2.2 Definition and basic Properties of Kernel Function . . . 46

2.3 Estimates between S_ru(Q), N u(Q) and M_ωu(Q) . . . 48

2.4 L^p and BM O Dirichlet Problems . . . 51

2.5 Singularity and perturbation of Elliptic Measure . . . 68

3 A New Result of Perturbation about the A∞ property of Elliptic Measures 71 3.1 A Whitney type decomposition of the unit ball and Rⁿ+ . . . 71

7

3.2 Description of Main Theorem . . . 73 3.3 Proof of Theorem 3.2.1 . . . 75 3.4 A class of operators whose elliptic measure is in A∞ . . . 90

The aim of this dissertation is to present an example about the A∞property of el- liptic measure (with respect to surface measure) associated to second order divergence form operators with bounded measurable coefficients, when the discrepancy between the main coefficients of the operators is supported on Whitney type rectangles of the unit ball of Rⁿ. In particular the discrepancy between the coefficients does not satisfy a Carleson measure condition, nor the coefficients of the operators involved are radi- ally independent, as in former well known papers [8, 17, 18, 2]. As an application of the perturbation results, we provide a class of operators with coefficients supported on Whitney type rectangles, for which the A∞ property holds.

We consider elliptic operators of the form Lu = div A∇u

where A(X) = (aij(X)) is a symmetric n × n matrix of bounded and measurable functions satisfying

λ⁻¹|ξ|² < hAξ, ξi < λ|ξ|², λ > 1 for all ξ ∈ Rⁿ and X ∈ Rⁿ.

Let Ω ⊂ Rⁿbe a bounded and open set. The solutions u of Lu = 0 are understood in a weak sense, that is, the function u ∈ W^1,2(Ω) is solution of Lu = 0 if

¨

Ω

hA(X)∇u(X), ∇ϕ(X)i dx = 0 for all ϕ ∈ C_c^∞(Ω).

9

We will start out by making some historical remarks, in order to put our work in perspective. Littman, Stampacchia and Weinberger in 1963 [30] proved that there exists the solution to the classical Dirichlet problem. That is, let f be a continuous function on the boundary of Ω, we have a local solution to a Dirichlet problem

Lu = 0 in Ω u|_∂Ω= f (0.0.1)

Moreover, when Ω is a regular domain (for instance, if it satisfies the exterior cone condition), u = f on ∂Ω means

lim

X→Q

X∈Ω

u(X) = f (Q).

This solution is local H¨older continuous and it satisfies the Harnack principle and the maximum principle [11, 31, 35]. Then, by the maximum principle and the Riesz Representation theorem there exists a Borel measure ω^X on ∂Ω such that

u(X) = ˆ

∂Ω

f dω^X

where ω^X is called the elliptic measure associated with L and Ω.

Note that when A(X) is the identity matrix the elliptic operator L is actually the Laplace operator

∆u =X

j

∂_j²u.

Dahlberg in [9] showed that if Ω is a Lipschitz domain in Rⁿ, the harmonic measure associated to the Laplace operator and the surface measure are mutually absolutely continuous. Moreover, k = dω/dσ the Radon-Nikodym derivative is in A∞(dσ). In fact he proved that k ∈ L²(∂Ω, dσ) and

 1

σ(∆r) ˆ

∆r

k²dσ

1/2

≤ C

 1

σ(∆r) ˆ

∆r

kdσ



(0.0.2) where ∆ is a surface ball of radius r centered in Q ∈ ∂Ω, that is, ∆_r= B(Q, r) ∩ ∂Ω.

When k satisfies (0.0.2) we say that ω verifies the reverse H¨older inequality, or ω ∈ RH₂.

Nevertheless, in 1981 Caffarelli, Fabes and Kenig [4], found examples of an elliptic operator whose elliptic measure is completely singular with respect to the surface measure.

Arguing as in [9, 6, 32], using the results in [5], and Hardy-Littlewood Maximal Theorem we prove (see Theorem 2.4.2) that the following statements are equivalent:

i) ω ∈ A∞(dσ)

ii) ω << σ and there exist 1 < q0 < ∞ such that if k is the Radon-Nikod´ym derivative of ω respect to σ, then

 1 σ(∆)

ˆ

∆

k^q0(P )dσ(P )

1/q0

≤ C

σ(∆) ˆ

∆

k(P )dσ(P ) where C > 0 is independent of ∆.

iii) If u solves the Classical Dirichlet problem

 Lu = 0 in D

u = f on ∂D, f ∈ C(∂D) Then

kN uk_{Lp0(∂D,dσ)}≤ Ckf k_{Lp0(∂D,dσ)},

where C > 0 does not depend on f . Moreover, in ii) and iii) the indexes p₀ and q₀ satisfy 1/p₀+ 1/q₀ = 1.

In a recent Master’s thesis (see [36]) there is a detailed exposition of these tech- niques in the case of harmonic functions.

On the other hand, if u is solution to the Classical Dirichlet problem (0.0.1) and for certain 1 < p < ∞

kN uk_Lp(∂Ω,dσ) ≤ Ckf k_Lp(∂Ω,dσ)

then we say that the L^p- Dirichlet problem is solvable, where N u(Q) = sup

X∈Γ(Q)

|u(X)|

denotes the non-tangential maximal function and Γ(Q) is a cone with vertex in Q ∈

∂Ω.

Observe from Theorem 2.4.2 that in order to solve the L^p-Dirichlet problem it suffices to prove that the elliptic measure associated to the elliptic operator L satisfies a Muckenhoupt weight property.

On the other hand, we say that the BM O-Dirichlet problem is solvable if any solution u for the classical Dirichlet problem satisfies the Carleson measure estimate

sup

∆⊂∂Ω

σ(∆)⁻¹

¨

T (∆)

|∇u|²δ(X)dX ≤ Ckf k²_{BM O} (0.0.3)

In 2011, Dindos, Kenig and Pipher proved that in order to solve the BM O Dirich- let problem it suffices to prove that the elliptic measure ω associated to the elliptic operator L is in A∞(dσ). As a consequence they obtained that the solvability of the BM O Dirichlet problem implies L^p solvability for all p > p0 for certain 1 < p0 < ∞.

Therefore it has been of interest for harmonic analysts to find conditions that ensure that ω^X and σ be mutually absolutely continuous in the A∞ sense, or that k = dω^X/dσ ∈ RH_q(∂Ω) for some 1 < q < ∞.

One approach was to try to obtain results of perturbation of the ω ∈ A∞(dσ) property. In a series of results by Dahlberg [8], R. Fefferman [17], that culminated with the work of Fefferman, Kenig and Pipher in [18], the goal was to find an optimal condition on the difference between the coefficients of two elliptic operators L0 and L₁, such that, if there exists the solution to the L^p or BM O Dirichlet problem for L₀ then we have the same result for L₁.

In other words, let ω0 and ω1 be the elliptic measures associated to the operators L₀ and L₁ respectively. If ω₀ ∈ A_∞(dσ), when can we ensure that ω₁ is also in A∞(dσ)?

In 1991, Fefferman-Kenig and Pipher in [18] proved that if L0 = div A0∇ and L₁ = div A₁∇ are elliptic operators and the disagreement function

a(X) = sup

Y ∈B(X,δ(X)/2)

|A₀(Y ) − A₁(Y )|

where δ(X) = dist(X, ∂Ω), verifies the “Carleson measure condition”, that is,

sup

∆⊂∂Ω

( 1 σ(∆)

¨

T (∆)

a²(X) δ(X) dX

)

≤ C

then ω_L1 ∈ A_∞(dσ) if ω_L0 ∈ A_∞(dσ).

As we said at the beginning, in the last chapter of this work we will present an example about the result of perturbation between two elliptic operators L0 and L1, where their corresponding coefficient matrices A₀ and A₁ coincide everywhere except in some rectangles of Whitney-type on D, the unit ball of Rⁿ.

It is important to emphasize that our main theorem is not an immediate conse- quence of the well known perturbation results in terms of Carleson measures cited and contained in [18]. The coefficients of the operators in our work do not necessarily satisfy the modulus of continuity condition over a non-tangential direction [15], nor the radial independence as in [2].

The original motivation for the result comes from an observation related to qua- siconformal mappings communicated to the Prof. Jorge Rivera Noriega by Prof.

Jang-Mei Wu. She originally posed the question of finding any relationship between the elliptic measures ω₀ and ω₁ associated respectively to the operators L₀ and L₁ as described before, based on the following remark:

If f₁ and f₂ are two quasiconformal homeomorphisms of Rⁿ+ such that f₁ = f₂ on certain rectangles of Whitney type of Rⁿ₊, then the pullback operators of the laplacian under f1 and f2 induce identical elliptic measures on Rⁿ⁻¹.

In order to be more precise as far as the relationship one could find between the elliptic measures involved, we apply a mapping from a local rectangle of Rⁿ+ onto D. This way our assumptions on the operators coefficients are stated in terms of Whitney-type rectangles of D.

Then we can ask whether it is true that ω₀ ∈ A_∞(∂D) implies ω₁ ∈ A_∞(∂D), where A∞(∂D) denotes the well known class of Muckenhoupt weights over ∂D. This class of weights entails a mutual absolute continuity.

Thus, our main result implies an answer in positive to this rephrasing of Prof.

Wu’s question.

In the first chapter we describe the general framework in which we will be working.

We state basic preliminaries, such as the theory of Muckenhoupt weights, the BM O space, the characterization of BM O space by a Carleson measure and some other basic results.

Also we introduce definitions of solutions of Divergence form elliptic equations and some basic estimates for solutions, like Harnack and Cacciopoli’s inequalities.

We present the solution to the Classical Dirichlet problem, following Littman, Stampacchia and Weinberger in [30], which allows us to define the elliptic measure associated to an elliptic operator in divergence form. Finally we define the Green’s function and so we obtain another representation of the solution to a Dirichlet problem through the Green’s function.

In Chapter 2, we present some estimates for elliptic measure and basic properties of Kernel function, which were proved by Caffarelli, Fabes, Mortola and Salsa in [5].

Among the fundamental results that we present here is the doubling property of the elliptic measure.

Also we introduce the Hardy Littlewood maximal operator, the non-tangential maximal function, the Area integral and some estimates between these [5, 23, 10].

On the other hand we study the L^p and BM O Dirichlet problems, and the equiva- lence between solving these problems and the fact that the elliptic measure associated to each of those operators satisfies an A∞ property. Also we will present a pertur- bation result where the discrepancy of these coefficients satisfy a Carleson measure condition, which was given in 2011 by Dindos, Kenig an Phiper in [12].

Finally in that chapter we present the result of Caffarelli, Fabes and Kenig [4], where they proved that there exists an elliptic operator whose elliptic measure is completely singular with respect to the surface measure. Besides, we display a little review about perturbation results satisfying a Carleson measure condition [8, 17, 18].

In Chapter 3 we show the example about perturbation result as we mentioned above, which was published in 2014 in [34].

Basic Notation:

D = {X ∈ Rⁿ: |X| < 1} denotes the unit ball in Rⁿ.

∂Ω is the boundary of the point set Ω.

Ω is the closure of Ω.

Rⁿ+= {(x, t) : x ∈ Rⁿ⁻¹, t > 0} is the half-space in Rⁿ.

C_c^∞(Ω) = {u : Ω → R/u is infinitely differentiable with compact support}.

B_r(y) is a open ball of radius r centered at y.

If Q ∈ ∂Ω:

∆r(Q) = Br(Q) ∩ ∂Ω denotes the surface cube.

T (∆) = B_r(Q) ∩ Ω denotes the Carleson region associated to ∆_r(Q).

If Q ∈ Rⁿ⁻¹, say Q = (q₁, ..., q_n−1):

∆(Q, r) = {(x1, ..., xn−1) ∈ Rⁿ⁻¹ : |xi− q_i| < r/2, 1 ≤ i ≤ n − 1}.

T (∆) = ∆(Q, r) × (0, r).

σ(F ) denotes the surface measure of a borel set F ⊂ ∂D.

δ(X) = dist(X, ∂Ω), where Ω ⊂ Rⁿ is a bounded and open set.

1

Chapter

Background Material

For wisdom is more precious than rubies, and nothing you desire can compare with her.

Proverbs 8:11

The goal of this chapter is to describe the general framework in which we will be working. Although our aim is to give an example of a result of perturbation about the A∞property of elliptic measures, which is equivalent to the existence of solutions to Dirichlet problems for second order divergence form elliptic operators, and these are questions from the area of partial differential equations, the techniques that we use are basically from harmonic analysis.

Our main theorem will be stated on D = {(θ, r) : θ ∈ Sⁿ⁻¹, 0 ≤ r < 1}, the unit ball in Rⁿ centered at the origin expressed in spherical coordinates.

We want to perform a decomposition of Whitney type for D. It is for this purpose that, let us start by associating points in a sector of D with points in a rectangle in Rⁿ, the same way as the points in one octant of the 3-dimensional ball are associated to points in [0, π/2) × [0, π/2) × [0, 1)

Let R₀ ⊂ Rⁿ be the rectangle given by

R₀= [0, π/2)ⁿ⁻¹× [0, 1), and define ρ : R0→ D as

ρ(x, t) = (x, 1 − t), (1.0.1)

15

where the n-tuple (x, 1 − t) is understood as spherical coordinates on D.

We may think of ρ as a mapping from R₀ to a sector of D, which is 1/2ⁿ the size of D.

We call R the union of the 2ⁿ disjoint rectangles of the form Rk = [θk, θk+ π/2)ⁿ⁻¹× [0, 1), for appropriately chosen θ_k, k = 0, 1, . . . , 2ⁿ− 1, with θ₀ = 0, such that

2ⁿ−1

[

k=0

ρ(Rk) = D.

In the last Chapter we will give a Whitney type decomposition of R, which in turn define the Whitney type cubes of D through ρ.

We will prove our main theorem in R, by simplicity of geometry. In the same way we will obtain the result for the unit ball D, with appropriate changes. It is important to emphasize that at the beginning of each section we will set the framework where we state definitions, basic estimates or results. In any case, our framework will be an open and bounded set from the the semispace or D the unit ball in Rⁿ.

We use the notation ˜

for the integral in the interior of the domain and ´ for the integral on the boundary.

1.1 Basic Theory of Muckenhoupt weights and BM O Space

In this section we introduce the theory of Muckenhoupt weights. Also we present the BM O space, the characterization of BM O space by a Carleson measure and some other basic results. For a basic reference see for instance [20].

We start introducing some notation. If Q ∈ Rⁿ⁻¹, say Q = (q₁, . . . , q_n−1), then

∆(Q, r) denotes the surface cube, ∆(Q, r) = {(x1, . . . , xn−1) ∈ Rⁿ⁻¹ : |xi − q_i| <

r, 1 ≤ i ≤ n − 1}. Similarly we may define the surface cube ˜∆ in D through the map ρ, namely ˜∆ = ρ(∆). Let υ a positive Borel measure on ∂D, υ is called doubling measure if for ˜∆(Q, 3r) ⊂ ˜∆, Q ∈ ∂D, then υ( ˜∆(Q, 2r)) ≤ C υ( ˜∆(Q, r)). We denote by σ(F ) the surface measure of a Borel set F ⊂ ∂D. In what follows we assume that ω is a regular and doubling measure.

A∞ Class

Definition 1.1.1. If ω is a positive Borel measure, we say that ω ∈ A∞(dσ) if given

 > 0 there exists η > 0 such that for every surface ball ˜∆ ⊂ ∂D and subset F ⊂ ˜∆ whenever σ(F )/σ( ˜∆) < η implies ω(F )/ω( ˜∆) < .

A∞ is an equivalence relation, and if ω ∈ A∞(dυ) then ω must be a doubling measure (see [38]).

Definition 1.1.2. A nonnegative Borel measure ω is said to belong to RHq with respect to σ or is in RH_q(dσ) if ω is absolutely continuous with respect to σ and k = ^dω_dσ the Radon-Nikodym derivative verifies that there exist C > 0 such that for all balls ˜∆ ⊂ ∂D

 1 σ( ˜∆)

ˆ

∆˜

k^q(Q)dσ(Q)

1/q

≤ C

σ( ˜∆) ˆ

∆˜

k(Q)dσ(Q). (1.1.1)

If ω satisfies (1.1.1) we say that ω verifies the reverse H¨older inequality with exponent q. The following theorem collects some results. For its demonstration see [20].

Theorem 1.1.1. Let ˜∆ be the surface ball ˜∆ ⊂ ∂D and F a subset of ˜∆. The following conditions are equivalent:

i) ω ∈ A∞(dσ).

ii) There exist  > 0 and η < 1 such that whenever σ(F )

σ( ˜∆) < η implies ω(F ) ω( ˜∆) <  iii) There exist constants C > 0 and δ > 0 such that

σ(F )

σ( ˜∆) ≤ C ω(F ) ω( ˜∆)

δ

iv) ω << σ and there exist  > 0 such that the Radon-Nikod´ym derivative k = ^dω_dσ ∈ RH1+.

Theorem 1.1.2. i) Let ε > 0. If ω ∈ RH1+ε then there exists ε⁰ > ε such that ω ∈ RH_1+ε⁰.

ii)

A∞(dσ) = [

q>1

RHq(dσ)

Similarly we have the definitions and results in Rⁿ⁻¹, with | · | the Lebesgue measure in Rⁿ⁻¹.

BM O Space

Definition 1.1.3. A function f : Rⁿ⁻¹→ R belongs to BMO(Rⁿ⁻¹) if sup

∆⊂Rⁿ⁻¹

1

|∆|

ˆ

∆

|f − f_∆|²dx < ∞, (1.1.2) where f_∆= |∆|⁻¹´

∆f dx, and ∆ denotes any surface cube in Rⁿ⁻¹.

In fact, using balls in Rⁿ⁻¹ instead of cubes gives the same definition. The BMO norm of f (modulo additive constants) is given by

kf k_{BM O} = sup

I⊂Rⁿ⁻¹

 1

|I|

ˆ

I

|f − f_I|²dx

1/2

. (1.1.3)

If f : ∂D → R we can define similarly the space BMO(∂D) and the BMO(∂D) norm of f , but now using the surface measure dσ and surface cubes on ∂D instead of Lebesgue measure and surface cubes on Rⁿ⁻¹.

In general, if Ω is a Lipschitz domain in Rⁿ and ω a doubling measure in ∂Ω we define for 1 ≤ p < ∞

kf k_{BM O(p,dω)} = sup

I⊂∂Ω

 1 ω(I)

ˆ

I

|f − f_I|^pdω

1/p

(1.1.4) In fact by John-Niremberg’s inequality there exist C > 0 such that

C⁻¹kf k_{BM O(p,dω)}≤ kf k_{BM O(dω)} ≤ Ckf k_{BM O(p,dω)}, (1.1.5) where kf k_{BM O(dω)} is defined as in (1.1.3) but with the measure ω. Moreover, if we define

kf k_{BM O}∗(p,dω)= sup

I⊂∂Ω ainfI∈R

 1 ω(I)

ˆ

I

|f − a_I|^pdω

1/p

This gives an equivalent norm with kf k_{BM O(dω)}. As we mentioned before to prove (1.1.5) one can use the following theorem adapted to ∂Ω and the doubling measure ω.

Theorem 1.1.3 (John-Nirenberg). Let f ∈ BM O; there exist constants C1 and C2

depending only of n such that for all cube Q ⊂ Rⁿ y θ > 0

|{X ∈ Q : |f (X) − f_Q| > θ}| ≤ C₁|Q| exp^{−C2θ/kf kBM O}

Now we will prove that if ω₁ ∈ A_∞(dω₂) then a BM O function with respect to the measure ω2 is also a BM O function with respect to the measure ω1.

Lemma 1.1.1. Let ω1, ω2 be doubling measures such that ω1 ∈ A_∞(dω2). For f ∈ BM O(dω₁) there exists C > 0 and 1 ≤ p < ∞ such that

sup

∆⊂∂Ω



ainf∆∈R

1 ω2(∆)

ˆ

∆

|f − a_∆|dω₂



≤ C sup

∆⊂∂Ω



ainf_∆∈R 1 ω1(∆)

ˆ

∆

|f − a_∆|^pdω1

1/p

,

that is, kf k_{BM O}^∗_(1,dω2) ≤ Ckf k_{BM O}∗(p,dω1). Proof. Let k = ^dω_dω²

1. The fact ω₁ ∈ A_∞(dω₂) implies that ω₂∈ A_∞(dω₁) =S

q>1RH_q(dω₁).

Hence there exist q > 1 such that k satisfies the reverse H¨older inequality

 1

ω1(∆) ˆ

∆

k^qdω₁

1/q

≤ C

ω1(∆) ˆ

∆

kdω₁ for all ∆ ⊂ ∂Ω It follows from here and H¨older inequality for 1/p + 1/q = 1

ω2(∆)⁻¹ ˆ

∆

|f − a_∆|dω₂ = ω2(∆)⁻¹ ˆ

∆

|f − a_∆|kdω₁

≤ ω₂(∆)⁻¹

ˆ

∆

k^qdω1

1/qˆ

∆

|f − a_∆|^pdω1

1/p

≤ Cω₂(∆)⁻¹ω₁(∆)^1/q−1

ˆ

∆

kdω₁

 ˆ

∆

|f − a_∆|^pdω₁

1/p

= C



ω1(∆)⁻¹ ˆ

∆

|f − a_∆|^pdω1

1/p

.

Finally we present a fundamental property of the harmonic extension to Rⁿ of a function of bounded mean oscillation on Rⁿ⁻¹, proved by Fefferman and Stein in [19].

In this result we have a characterization of f ∈ BM O using the fact that t|∇u|²dxdt is a Carleson measure (see definition below), where u(x, t) = P_t∗ f (x) is the Poisson extension of f .

Definition 1.1.4. For Q ∈ Rⁿ⁻¹ we denote by T (Q, r, s) the Carleson region in Rⁿ+

T (Q, r, s) = {(x, t) : |x − Q| < r, 0 ≤ t < sr} , s ≥ 1 (1.1.6) When s = 1 we denote simply T (∆(Q, r)), T_r(Q) or T (∆).

Ar(Q) denote the top part of T (Q, r, s), that is

A_r(Q) = (C_I, sr) (1.1.7)

where CI is the center of ∆(Q, r).

Fix r0 > 0 and Q0 ∈ Rⁿ⁻¹, and denote by ∆ and T the surface cube ∆(Q0, r0) and its corresponding Carleson region. A measure µ in Rⁿ+ is Carleson measure if there exists a constant C > 0, C = C(r₀) such that for all r ≤ r₀

µ (T (∆(Q, r))) ≤ Cσ(∆(Q, r)).

For such measure µ we denote the norm of the Carleson measure kµk_Car = sup

∆⊂Rⁿ⁻¹

σ(∆)⁻¹µ (T (∆(Q, r)))
1/2

. (1.1.8)

We can give the corresponding definition if our domain is a Lipschitz domain in Rⁿ. The following theorem gives a relation between Carleson measure of the solution to the Dirichlet problem and the BMO norm of data at the boundary. This is the first step to obtain a conjecture concerning the solvability of the Dirichlet problem for elliptic operators with data in BMO. This result was eventually proved by Dindos, Kenig and Pipher in [12]. In the next chapter will discuss this issue carefully.

Theorem 1.1.4. The following conditions are equivalent i) f ∈ BM O

ii) ´

Rⁿ⁻¹

|f (X)|

1+|X|ⁿdX < ∞ and sup

Q⁰∈Rⁿ⁻¹

¨

T (∆(Q⁰,h))

t|∇u|²dxdt ≤ Chⁿ⁻¹, 0 < h < ∞

where |∇u|² = Pn−1 j=1

∂u

∂xj

2

+ ^∂u_∂t

2, and u(x, t) is the Poisson integral of f , namely u(x, t) = P_t∗ f (x) = ´

Rⁿ⁻¹P_t(x − y)f (y)dy for t > 0, where P_t(x) =

Ct

(t²+|x|²)^n/2 is the Poisson kernel in Rⁿ⁻¹. Moreover in case f ∈ BM O one has

sup

Q⁰∈Rⁿ⁻¹

¨

T (∆(Q⁰,h))

t|∇u|²dxdt ≤ Chⁿ⁻¹kf k_{BM O}

See [19] for the proof.

1.2 Solutions of Divergence form elliptic equations

Assume Ω ⊂ Rⁿ be a bounded open set (we can also assume that Ω = D and all definitions would be similar). For a fixed 1 ≤ p < ∞ we define the Sobolev space as W^1,p(Ω) =



u ∈ L¹_loc(Ω) : u_Xi exists for i = 1, ..., n and

¨

Ω

|u|^p+

¨

Ω

|∇u|^p < +∞

 .

In the above definition uXi there exists in a weak sense. That is, v ∈ L¹_loc(Ω) is the weak partial derivative of u if

¨

Ω

uϕ(X)_XidX = −

¨

Ω

vϕ(X)dX for all functions ϕ ∈ C_c^∞(Ω).

Let u ∈ W^1,p(Ω) we define its norm to be kuk_W1,p(Ω)= kuk_L^p_(Ω)+

n

X

i=1

ku_xik_Lp(Ω).

We say that u ∈ W_loc^1,p(Ω) if u ∈ W^1,p(Ω⁰) for every Ω⁰ ⊂⊂ Ω, namely Ω⁰⊂ Ω and Ω⁰ is in Ω.

We denote by W₀^1,p(Ω) the closure of C_c^∞(Ω) in W^1,p(Ω), where C_c^∞(Ω) denote the space of infinitely differentiable functions with compact support in Ω. Moreover u ∈ W₀^1,p(Ω) if and only if there exist functions um∈ C_c^∞(Ω) such that

u_m→ u in W^1,p(Ω).

It is well known that the Sobolev space W^1,p(Ω) is a Banach space, and if u ∈ W^1,p(Ω) for some 1 ≤ p < ∞, there exist functions u_m∈ C^∞(Ω) ∩ W^1,p(Ω) such that

um→ u in W^1,p(Ω).

Moreover, if ∂Ω is C¹, it is possible to approximate u by functions belonging to C^∞( ¯Ω).

The dual space of W^1,p(Ω) is denoted W^−1,q(Ω), for p > 1 and q the conjugate exponent of p. In other words T belongs to W^−1,q(Ω) if T is a bounded linear functional on W^1,p(Ω). It is important to have an explicit characterization of the dual space of W^1,p(Ω) as we will see later. W^−1,q consist of distributions T on Ω of the form

T v = T_f0v −

n

X

i=1

∂T_fi

∂x_i v =

¨

Ω

f0v −

n

X

i=1

fi

∂v

∂v_i

!

dx, (1.2.1)

where v ∈ W₀^1,p(Ω) and f0, f1, ..., fn∈ L^q(Ω). Furthermore, if T ∈ W^−1,q(Ω) then

kT k_W−1,q(Ω)= inf







¨

Ω

|f₀|^qdx −

¨

Ω n

X

i=1

|f_i|^qdx

!1/q





(1.2.2)

where the infimum is taken over all representations of T given by (1.2.1) for f0, ..., fn∈ L^q(Ω). We write T ≡ f₀−P_n

i=1(f_i)_xi whenever (1.2.1) holds.

The proof of the following inequality is a slight variant of the proof of Morrey’s inequality (see [14, p. 266-268]): For u ∈ W^1,2(B(X, 2r)), n < p < ∞ and r > 0

|u(Y ) − u(X)| ≤ Cr^1−1p

¨

B(X,2r)

|∇u(Z)|^pdZ

!1/p

. (1.2.3)

for Y ∈ B(X, r). We will use this inequality later in section 1.5.

To see in detail the definitions and previous results of this section the reader is referred to Chapter 5 of [14].

We consider elliptic operators of the form

Lu = div A∇u. (1.2.4)

Here A(X) = (a_ij(X)) is a symmetric n × n matrix of measurable functions satisfying the ellipticity condition

λ⁻¹|ξ|²< hAξ, ξi < λ|ξ|², λ > 1, (1.2.5) for every ξ = (ξ₁, . . . , ξ_n) ∈ Rⁿ and every X ∈ Ω, where hAξ, ξi =P

j

P

ia_ij(X)ξ_iξ_j. The constant λ is called ellipticity constant of L. We assume that the coefficients A(X) = (ai,j(X)) are bounded and measurable on Ω.

Definition 1.2.1. Given f0, f1, ..., fn ∈ L²(Ω), the function u ∈ W^1,2(Ω) is said to be a solution of the equation Lu = f₀−Pn

i=1(f_i)_xi, if

¨

Ω

hA(X)∇u(X), ∇φ(X)i dX =

¨

Ω

f0 φ +

n

X

i=1

fi φXi

!

dX (1.2.6)

for all φ ∈ W₀^1,2(Ω).

We can see that the right hand side term T = f0−Pn

i=1(fi)xibelongs to W^−1,2(Ω), the dual space of W^1,2(Ω). Also we can assume φ ∈ C_c^∞(Ω).

Definition 1.2.2. A function u(X) ∈ W_loc^1,2(Ω) will be called a local solution in Ω of Lu = 0, if

¨

Ω

hA(X)∇u(X), ∇φ(X)i dX = 0 for all φ ∈ C_c¹( ¯Ω).

Definition 1.2.3. A function u(X) ∈ W^1,2(Ω) is called a subsolution of the operator L in Ω if

¨

Ω

hA(X)∇u(X), ∇φ(X)i dx ≤ 0 (1.2.7)

for all non-negative functions φ ∈ W₀^1,2(Ω). Also u(X) is called a super solution in Ω if −u is a sub solution.

Remark 1.2.1. To prove that u ∈ W^1,2 is solution for Lu = 0 in Ω it is enough to prove that for each compact K ⊂⊂ Ω

¨

K

hA∇u, ∇ϕi dX = 0, ϕ ∈ C_c^∞(Ω).

In fact, for Kn= {X ∈ Ω : δ(X) ≥ 1/n} and by dominated convergence Theorem

¨

Ω

hA∇u, ∇ϕi dX = lim

n→∞

¨

Kn

hA∇u, ∇ϕi dX = 0.

Thus u is solution for Lu = 0 in Ω.

1.3 Some basic estimates for solutions

Now we introduce some known results for solutions of form divergence elliptic oper- ators. In several of these theorems we only give a sketch of the proof.

For X in Rⁿ+ we denote by B_r(X) = Y ∈ Rⁿ+: |X − Y | < r , and B_r(X) =

Y ∈ Rⁿ+: |X − Y | ≤ r . Sometimes we will use the notation

A

{ }dX = 1

|A|

¨

A

{ }dX.

We consider L any elliptic operator of the form (1.2.4) with symmetric coefficient matrix of bounded measurable functions satisfying (1.2.5). We also adopt the stan- dard notation that a constant can change from line to line, and will still be denoted by the same letter. One way we adopt to indicate the dependance of the constants using function notation C = C(. . . ), and otherwise it is assumed that the constant depends at most on dimension n and ellipticity constants of the operators involved.

Theorem 1.3.1 (Caccioppoli). Let u be a non-negative solution in B_3r(X) ⊂ Rⁿ+ of Lu = 0. Then

1

|B_r(X)|

¨

Br(X)

|∇u(Y )|²dY ≤ C r⁻²

|B_2r(X)|

¨

B2r(X)

u(Y )²dY where C depends only of n and the ellipticity constant.

Proof. Let ϕ ∈ C_c^∞B2r(X), ϕ ≥ 0 such that ϕ ≡ 1 in Br(X), supp(ϕ) ⊂ B2r(X) and

|∇ϕ| ≤ r⁻¹. Since u is solution of Lu = 0 we have that

¨

A∇u, ϕ²∇u dY +

¨

hA∇u, 2ϕu∇ϕi dY =

¨

A∇u, ∇(ϕ²u) dY = 0.

Hence

¨

ϕ²hA∇u, ∇ui dY ≤ 2

¨

|ϕu| hA∇u, ∇ϕi dY = 2

¨

|ϕu|D

A^1/2∇u, A^1/2∇ϕE dY

≤ 2

¨

|ϕ|D

A^1/2∇u, A^1/2∇uE1/2

|u|D

A^1/2∇ϕ, A^1/2∇ϕE1/2

dY

≤ 2

¨

ϕD

A^1/2∇u, A^1/2∇uE dY

1/2¨

u²D

A^1/2∇ϕ, A^1/2∇ϕE dY

1/2

= 2

¨

ϕ²hA∇u, ∇ui dY

1/2¨

u²hA∇ϕ, ∇ϕi dX

1/2

Thus ¨

ϕ²hA∇u, ∇ui dx ≤ 4

¨

u²hA∇ϕ, ∇ϕi dx.

From the ellipticity condition (1.2.5) and by our choice of ϕ we obtain λ⁻¹

¨

Br(X)

|∇u|²dY ≤ λ⁻¹

¨

ϕ²|∇u|²dY ≤ 4λ

¨

u²|∇ϕ|²dY ≤ 4λ r²

¨

B2r(X)

u²dY

Then ¨

Br(X)

|∇u|²dY ≤ 4λ² r²

¨

B2r(X)

u²dY

We will give a sketch proof of the Harnack’s inequality and H¨older continuity of a non-negative solution to Lu = 0, which was proved by Moser in [31].

Theorem 1.3.2 (Harnack Inequality). Let u be a non-negative solution to Lu = 0 in B4r(X) ⊂ Rⁿ+. Then there exists a constant depending only on the ellipticity constant of L and n such that

sup

Br(X)

u ≤ C inf

Br(X)u Proof. (Sketch)

Step 1. Let us prove that for p > 0 sup

Br

u ≤ C



B2r

|u|^p

1/p

(1.3.1)

where C depends only of n and the ellipticity constant. For this, first suppose that u is bounded subsolution in B_2r, and let v = u^k. Then one prove that for η ∈ C_c^∞(B_2r)

¨

B2r

|η|²|∇v|² ≤ C

 2k 2k − 1

2¨

B2r

|∇η|²|v|²,

where C depends only of n and ellipticity constant. Using this and the Sobolev inequality



Br

|v|^2γ

1/γ

≤ C

 r²

Br

|∇v|²+

Br

|v|²

 ,

where C depends of n, and γ = _n−2ⁿ , we can prove that for p_m = 2γ^m, m > 1 and hm= r + ₂¹m

B_hm+1

|u|^pm+1

!1/pm+1

≤ C



B2r

u²

1/2

(1.3.2)

If u is not bounded we can proceed similarly to above but now with v = u^β/2+1m

where u_m = min{u, m}, β > −1, and using again the Sobolev inequality and domi- nated convergence Theorem we also show (1.3.2). Observe that the the right hand side of (1.3.2) approaches to sup_Bru when m → ∞. Then if u is subsolution non-negative

sup

Br

u ≤ C



B2r

u²

1/2

. (1.3.3)

where C depends only of n and λ.

This way we obtain the inequality (1.3.1) for p > 2 using the H¨older inequality, and for 0 < p ≤ 2 see [22, p.80-81]

We can similarly obtain infBr

u ≥ C⁻¹



B2r

u^−p

−1/p

(1.3.4) for all p > 0

Step 2. Let v = log(u), we can prove that

¨

Q

|v − v_Q|²≤ C|Q| Q ⊂ Q⁰

where Q and Q⁰ are cubes such that B2r ⊂ Q⁰ and 2Q ⊂ B4r, vQ=ffl

Qv. This entails the fact that log u ∈ BM O. Then by John-Nirenberg inequality (Theorem 1.1.3) there exist p, β > 0 such that

¨

Q⁰

e^−pv

 ¨

Q⁰

e^pv



≤ β².

Then

¨

B2r

u^p

1/p

≤ β²

¨

B2r

u^−p

−1/p

From this, (1.3.1) and (1.3.4) we get sup

Br

u ≤ C inf

Br

u.

where C depends only of n and the ellipticity constant.

Theorem 1.3.3 (H¨older Continuity). Let u be a solution to Lu = 0 in B4r(X) ⊂ Rⁿ+, then there exist α > 0 such that for all Z, Y ∈ B_r⁰(X), 0 < r⁰< r

|u(Z) − u(Y )|

|Z − Y |^α ≤ Ckuk_L2(Br(X))

where C depends of r, n and the ellipticity constant of L.

Proof. Without loss of generality suppose r = 1. For 0 < ρ < 1 denote the oscillation of u in B_ρ(X) as O(ρ) = sup_Bρu − inf_Bρu ≡ M (ρ) − m(ρ). Using that M (ρ) − u and u − m(ρ) are positive solutions in Bρ(X), by Theorem 1.3.2

M (ρ) − m(ρ/2) ≤ C (M (ρ) − M (ρ/2)) M (ρ/2) − m(ρ) ≤ C (m(ρ/2) − m(ρ))

Then (1 + C) (M (ρ/2) − m(ρ/2)) ≤ (C − 1) (M (ρ) − m(ρ)), namely O(ρ/2) ≤ γO(ρ) 0 < γ < 1,

with γ = ^C−1_C+1 < 1. Iteration of this relation yields

O(2⁻ⁿρ) ≤ γⁿO(ρ) for all 0 < ρ < 1.

For 0 < ρ < r⁰< 1, γ = 2^−α for α > 0 and n such that ρ = 2⁻ⁿr⁰ O(ρ) ≤ O(2⁻ⁿr⁰) ≤ γⁿO(r⁰) ≤ 2^−αnO(r⁰) ≤ρ

r⁰

α

O(r⁰) Then

O(ρ) ≤ρ r⁰

α

O(r⁰). (1.3.5)

which leads to for Z ∈ B_r⁰ and |Z − Y | ≡ ρ

|u(Z) − u(Y )|

|Z − Y |^α = ρ^−α|u(X) − u(Y )| ≤ ρ^−αO(ρ) ≤ 1 r⁰

α

O(r⁰) ≤ Ckuk_L2(B1). Equivalently

|u(Z) − u(Y )| ≤ C |Z − Y | r⁰

α

sup

B1(Y )

u or

O(ρ) ≤ρ r⁰

α

O(r⁰) ≤ Cρ r⁰



B1

u²

1/2

. This proves the H¨older continuity of the solutions.

Now we start out with some boundary estimates.

Theorem 1.3.4 (Boundary Caccioppoli inequality). If u is solution to Lu = 0 on T_2r(P ), u ∈ C(T_2r(P )) and u ≡ 0 in ∆(P, 2r), with P ∈ Rⁿ⁻¹such that ∆(P, 2r) ⊂ ∆, then

1

|T_r(P )|

¨

Tr(P )

|∇u(Y )|²dY ≤ C r²

1

|T_2r(P )|

¨

T2r(P )

|u(Y )|²dY.

The constant depends only on the ellipticity constant of L and n.

Proof. (Sketch) Let ϕ ∈ C_c^∞(Rⁿ), ϕ ≡ 1 on T_r^∗(P ), supp ϕ ⊂ T_2r^∗(P ), |∇ϕ| ≤ C/r, where T^∗ is the Carleson region extended, that is

T_r^∗(P ) = {(x, t) : |x − P | < r, |t| < r}

Since u ≡ 0 on ∆_2r(P ) we have that uϕ² ≡ 0 in Rⁿ⁻¹, moreover uϕ² ∈ W₀^1,2, in fact, observe that by H¨older inequality

¨

|∇(uϕ²)| ≤

¨

T2r(P )

|∇u · ϕ²| + 2

¨

T2r(P )

|uϕ∇ϕ| ≤ Ckuk_W1,2 < ∞,

and then we can proceed as in the proof of the Theorem 1.3.1.

The next theorem is a result of Caffareli-Fabes-Mortola and Salsa, see [5] for details.

Theorem 1.3.5 (Harnack principle at the boundary). If u is a non-negative solution to Lu = 0 on T = (Q₀, 4r, 8r) with Q₀ ∈ Rⁿ+ and u ≡ 0 in ∆(Q₀, 4r), then there exists a constant C depending only on the ellipticity constant of L and n such that

sup

T0

u(x, t) ≤ Cu(Z₀) (1.3.6)

where T₀ = T (Q₀, r, 2r) and Z₀= center of T₀. Proof. (Sketch)

Step 1. Without loss of generality we can assume that u(Z₀) = 1, r = 1 and Q₀ is the origin. Also by reflecting u across t = 0 as an odd function in t we may assume u is solution of a divergence equation which we call again L, in the domain

T = {(x, t) : |x˜ _i| < 4, i = 1, ..., n − 1, |t| < 8}

From Harnack’s inequality (Theorem 1.3.2) there exists a constant M depending only of λ such that for i = 1, ..., n − 1

u(x, t) ≤

 M u(x, 2t) for |x_i| ≤ 2, 0 < t ≤ 3/2 M = M u(Z0) for |xi| ≤ 2, 1 ≤ t ≤ 3 Step 2. Suppose there exists Y0 ∈ Q₀ ≡ ¯T0 such that

u(Y0) ≥ M^h+2 (1.3.7)

with h to be determined. We will obtain a contradiction which will imply that for all Y₀∈ ¯T₀

sup

T0

u(Y₀) ≤ Cu(Z₀) and hence we obtain the desired inequality.

From (1.3.7) and Harnack’s inequality we obtain that δ(Y0) ≤ 2^−hand O_Q(Y0,2^−h+N₎≥ 2M^h+4, where O_Q(Y0,2−h+N) denote the oscillation of u in the cube Q(Y₀, 2^−h+N)

Step 3. We can obtain by induction a sequence of points {Y_k} such that δ(Y_k) ≤ 2^−h−2k, Y_k ∈ Q(Y_k−1, 2−h−2(k−1)+N

) ∩ Rⁿ+

and

u(Y_k) ≥ M^h+2(k+1), k = 1, 2, ... (1.3.8) Step 4. We can assure that each Q(Y_k, 2^−h−2k+N) is contained in ˜Q a fixed closed subcube of ˜T and we obtain a contradiction with (1.3.8) because as each cube Q(Y_k+1, 2−h−2(k−1)+N

) ∩ Rⁿ⁻¹6= ∅ by Harnack’s inequality u(Yk) ≤ C sup

Y ∈ ˜Q

u(Y ) ≤ C inf

Y ∈ ˜Q

u(Y ) = 0 for all k = 1, 2, .... Therefore we conclude the proof.

1.4 The Classical Dirichlet Problem

In this section we again assume that Ω ⊂ Rⁿ is a bounded open set. Given a contin- uous function h : ∂Ω → R we want to solve the classical Dirichlet problem

 Lu = 0 in Ω

u = h on ∂Ω (1.4.1)

Using the Lax-Milgram Theorem it is proven the following

Theorem 1.4.1. Given f ∈ W^−1,2(Ω) and h ∈ W^1,2(Ω), there exists one and only one solution of the equation

Lu = f such that u − f ∈ W₀^1,2(Ω).

Recall that f ∈ W^−1,2 can be written as f = f₀−Pn

i=1(f_i)_Xi, where f_i ∈ L²(Ω), i = 1, ..., n.

u = h ∈ W₀^1,2(Ω) means u = h on ∂Ω in the trace sense, as we now describe. If u ∈ C( ¯Ω) clearly u has values on ∂Ω in the usual sense, but it is not possible to assign boundary values along ∂Ω to a function u ∈ W^1,2(Ω). If Ω is bounded and ∂Ω is C¹, for 1 ≤ p < ∞ then there exists a bounded linear operator T : W^1,p(Ω) → L^p(∂Ω) such that

i) T u = u|_∂Ωif u ∈ W^1,p(Ω) ∩ C( ¯Ω), and ii) kT uk_L^p_(∂Ω)≤ Ckuk_W1,p(Ω)

for each u ∈ W^1,p(Ω), with C depending only on p and Ω (see [14, p. 258]). We call T u the trace of u on ∂Ω. Moreover

u ∈ W₀^1,p(Ω) if and only if T u = 0 on ∂Ω.

To solve the Classical Dirichlet problem (1.4.1), (see [30] for details) first we need to know in which sense u = h on ∂Ω. If h is the trace on ∂Ω of a function ¯h(X) in W^1,2(Ω) by Theorem 1.4.1 we would have a solution u ∈ W^1,2(Ω) for

 Lu = 0 in Ω

u = ¯h on ∂Ω (1.4.2)

where u = ¯h on ¯Ω means u − h ∈ W₀^1,2(Ω), then T (u − ¯h) = 0. But in general h ∈ C(∂Ω) is not the trace of a function in W^1,2(Ω). To solve this problem we consider a continuous linear mapping

B : W^1,2(Ω)W0^1,2(Ω) → W^1,2(Ω)

such that Bg = u, where u is the solution given by Theorem 1.4.1 associated to g ∈ W^1,2(Ω)W₀^1,2(Ω). We can get

|||Bg||| ≤ C(λ, Ω)kgk_L^∞_(∂Ω) where

|||v||| = sup

Ω¯⁰⊂Ω

δkvk_W1,2(Ω⁰)+ kvk_L^∞_(Ω), with δ = dist( ¯Ω⁰, ∂Ω).

Let {qn} ⊂ P(∂Ω) = {q : ∂Ω → R / q is a polynomial function} such that q_n→ h

uniformly on ∂Ω, h ∈ C(∂Ω). We observe that the trace of qn coincide with qnfor all n, then for each qnthere exist the solution H¨older continuous unfor (1.4.2), moreover

|||u_n||| ≤ C max

X∈∂Ω|q_n(X)|

Then

|||u_n− u_m||| ≤ C max

∂Ω |q_n− q_m| <  n, m → ∞

Thus {u_n} converges in W_loc^1,2 and uniformly in Ω to u ∈ C( ¯Ω). Furthermore, for each n un is solution to

 Lun= 0 in Ω

u_n= q_n on ∂Ω (1.4.3)

where q_n → h uniformly on ∂Ω to h ∈ C(∂Ω). It is easily proved that the function u(X) is a local solution of the equation Lu = 0.

Then given a continuous function h on ∂Ω there exists a local solution u of

 Lu = 0 in Ω

u = h on ∂Ω (1.4.4)

which is locally H¨older continuous, and u = h on ∂Ω in the sense (1.4.3). In this way Littman-Stampacchia and Weinberger in [30] proved.

Theorem 1.4.2. Given any continuous function f on ∂Ω there exists a mapping Bf which associates f to a local solution of Lu = 0 which is locally H¨older continuous.

Moreover, if f is the trace of a C¹( ¯Ω) function, then u = Bh coincides with the solution in W^1,2(Ω) obtained by Theorem 1.4.1.

Now we observe that when Ω is regular (definition below), u = h on ∂Ω means lim

X→Q

X∈Ω

u(X) = h(Y ).

Definition 1.4.1. A point Q ∈ ∂Ω is said to be regular if for any continuous function h(X) on ∂Ω the generalized solution u = Bh satisfies

lim

X→Q

X∈Ω

u(X) = h(Q). (1.4.5)

If there is at least one continuous function h on ∂Ω for which (1.4.5) is not satisfied, the point Y is said to be irregular.

In particular when Ω = D or Ω = R ⊂ Rⁿ we have that Ω is regular. We can see this by the following theorems proved in [30] and [1] respectively.

Theorem 1.4.3. A point Y ∈ ∂Ω is regular with respect to any uniformly elliptic operator L if and only if it is regular with respect to the Laplace operator.

Theorem 1.4.4 (Zaremba). Let Ω be an open bounded connected subset of Rⁿ. If Z ∈ ∂Ω is a vertex of a cone contained in the complement of Ω, then Z is regular for Laplace equation.

This condition is called condition of exterior cone. From Theorem 1.4.3 and 1.4.4 we have that if Ω = D then each Y ∈ ∂Ω is regular point and therefore the solution u ∈ C( ¯Ω) ∩ W_loc^1,2(Ω) of (1.4.4) satisfies (1.4.5)

Given a continuous function f ∈ C(∂Ω) we have a local solution to a Dirichlet problem

Lu = 0 in Ω u|∂Ω= f

Then, for each X ∈ Ω we can define the functional Λ_X : C(∂Ω) → R such that Λ_X(f ) = u(X)

The maximum principle implies that for each fixed X ∈ Ω, ΛX is continuous, moreover Λ_X is a positive linear functional on C(∂Ω). Therefore from Representation Riesz theorem there exists a Borel measure ω_L^X on ∂Ω such that

u(X) = ˆ

∂Ω

f (Q)dω_L^X(Q). (1.4.6)

We will call ω^X_L the elliptic measure associated with L and Ω, or simply the L-elliptic measure for Ω, evaluated at X.

1.5 Green’s function

In this section we want to obtain another representation of the solution to Dirichlet problem through the Green’s function. We continue assuming that Ω ⊂ Rⁿ is a bounded open set. We invite the reader to see [30] for details of this section.

Definition 1.5.1. For a measure µ of bounded variation on Ω we say that u ∈ L¹(Ω) is a weak solution of the equation

Lu = µ vanishing at the boundary ∂Ω if it satisfies

¨

Ω

uLφdx =

¨

Ω

φdµ

for every φ ∈ W₀^1,2(Ω) ∩ C⁰( ¯Ω) such that Lφ ∈ C⁰( ¯Ω).

We will define the Green’s function as a weak solution, vanishing on ∂Ω of the equation

LG = δ_Y

where δY is the Dirac measure at Y . For this reason we need first prove the following Theorem 1.5.1. For every measure µ of bounded variation on Ω there exists a unique weak solution u of the equation

Lu = µ

Proof. We observe that µ ∈ W^−1,2(Ω). In fact, for v ∈ W^1,2(Ω) hµ, vi =

¨

Ω

|v|dµ ≤

¨

Ω

|v|²dµ

1/2

[µ(Ω)]^1/2≤ Ckvk_W1,2

then from Theorem 1.4.1 there exist unique solution u to Lu = µ.

See [30, p- 59-60] for the proof of the next theorems.

Theorem 1.5.2. The weak solution u of Lu = µ is in W₀^1,2(Ω) if and only if µ ∈ W^−1,2(Ω).

Theorem 1.5.3. If the solution u of Lu = µ is in W₀^1,2 and 0 ≤ µ⁰ ≤ µ, then the solution v of Lv = µ⁰ is in W₀^1,2.

Now we shall show that a weak solution can be approximated by weak solutions of equations with continuous coefficients. Let αs(X) any approximate identity, namely a family having the properties:

1. α_s(X) ∈ C^∞(E), E ⊂ Rⁿ. 2. α_s(X) ≥ 0.

3. αs(X) = 0 for |X| > 1/s