Divergence form operators in Reifenberg flat domains.

Transcription

1 Divergence form operators in Reifenberg flat domains. Emmanouil Milakis Tatiana Toro University of Washington Department of Mathematics P.O ox Seattle, WA 9895, USA Abstract We study the boundary regularity of solutions of elliptic operators in divergence form with C 0,α coefficients or operators which are small perturbations of the Laplacian in non-smooth domains. We show that, as in the case of the Laplacian, there exists a close relationship between the regularity of the corresponding elliptic measure the geometry of the domain. AMS Subject Classifications: 35J25, (305) Keywords: Reifenberg flat domain, Chord arc domain, Elliptic measure. Introduction The basic aim of this paper is to study the relationship between the elliptic measure of operators in divergence form with C 0,α coefficients or operators which are small perturbations of the Laplacian, the geometry of the boundary of the domain. We concentrate on domains whose boundary is locally flat, where this notion will be understood in a weak sense. Let R n+ be an open set. Loosely speaking we say that is locally flat if locally it can be well approximated by affine spaces. In particular, such domains are non-tangentially accessible therefore their harmonic measure ω is doubling (see [2], [3]). We prove that if is well approximated by n-planes in the Hausdorff distance sense then the doubling constant of the elliptic measure of divergence form operators with Hölder coefficients, milakis@math.washington.edu toro@math.washington.edu The second author was partially supported by NSF grant DMS

2 ω, asymptotically approaches the doubling constant of the n-dimensional Lebesgue measure. If moreover the unit normal vector to has small (resp. vanishing) mean oscillation, then the logarithm of the corresponding elliptic kernel has small (resp. vanishing) mean oscillation. In [4], Dahlberg showed that if is Lipschitz domain then the harmonic measure the surface measure are mutually absolutely continuous. In addition the Poisson kernel is a 2 weight with respect to the surface measure to the boundary, which implies that the logarithm of the Poisson Kernel is a function of bounded mean oscillation with respect to the surface measure on (i.e it is in MO( )). Jerison Kenig [2], showed that if is a C domain then the logarithm of the Poisson kernel is in V MO( ). In the case when the boundary,, is locally flat, Kenig Toro gave a full description of the relation between the harmonic measure of the domain the geometry of its boundary (see [5]). More precisely if R n+ is an open connected Reifenberg flat domain with vanishing constant then the harmonic measure behaves asymptotically like the Euclidean measure. If is a chord arc domain with vanishing constant then the logarithm of the Poisson kernel has vanishing mean oscillation (i.e. it is in V MO( )). The regularity of the elliptic kernel for divergence form operators which are perturbations of the Laplacian on Lipschitz domains has been studied by several authors. Dahlberg [5], showed that if the difference between the coefficients of an elliptic, divergence form operator L the Laplacian satisfies a Carleson condition with vanishing trace then the corresponding elliptic kernel is a 2 weight with respect to surface measure. In [9], Fefferman, Kenig Pipher studied the case when the same Carleson condition is satisfied but without the smallness assumption. In that case, the elliptic measure of L is an A weight with respect to surface measure. In [7] Escauriaza proved that on a C domain if the difference between the coefficients of L the Laplacian satisfies a Carleson condition with vanishing trace then the logarithm of the elliptic kernel is in V M O( ). In this paper we extend the results of [5] to more general uniformly elliptic operators in divergence form. In section 2 we present the preliminaries, define the two classes of operators we intend to study state our main results. In section 3 we prove that the elliptic measure of a divergence form elliptic operator with Hölder coefficients on a Reifenberg flat domain with vanishing constant is asymptotically optimally doubling. The proofs in this section follow the arguments presented in [5]. In section 4 we show that, in a chord arc domain with vanishing constant, the logarithm of the corresponding elliptic kernel is in V M O. In section 4, we also extend some of the results in [9] to chord arc domains with small constant. A natural question is 2

3 whether Escauriaza s result (see [7]) generalizes to chord arc domains with vanishing constant. We expect this to be the case. 2 Preliminaries results. In this section we recall some definitions state our main results. First we introduce the class of Reifenberg flat domains, which are domains whose boundary can be well approximated by planes. In particular Lipschitz domains with small constant are Reifenberg flat. Definition 2.. Let R n+ be a bounded domain, we say that separates R n+ if there exist δ > 0, R > 0 such that for each Q, there exist an n-dimensional plane L(Q, R) containing Q a choice of unit normal vector to L(Q, R), n Q,R satisfying T + (Q, R) = {X = (x, t) = x + tn Q,R (Q, R) : x L(Q, R), t > 2δR}, (2.) T (Q, R) = {X = (x, t) = x + tn Q,R (Q, R) : x L(Q, R), t < 2δR} c. (2.2) Here (Q, R) denotes the (n + )-dimensional ball of radius R center Q. Definition 2.2. Let R n+, δ > 0, R > 0. We say that is a (δ, R)-Reifenberg flat domain if separates R n+, for each Q, for every r (0, R] there exists an n-dimensional plane L(Q, R) containing Q such that where D denotes the Hausdorff distance. We denote by D[ (Q, r), L(Q, r) (Q, r)] δ. (2.3) r { } θ(r) = sup inf D[ (Q, r), L (Q, r)], (2.4) Q L r where the infimum is taken over all n-planes containing Q. Definition 2.3. Let R n+, we say that is a Reifenberg flat domain with vanishing constant if it is (δ, R)-Reifenberg flat for some δ > 0 R > 0, lim sup θ(r) = 0. (2.5) r 0 3

4 Note that definitions are only significant for δ > 0 small. Thus when talking about (δ, R)-Reifenberg flat domains we assume that δ is small enough. In particular, we assume that δ is small enough so that if is a (δ, R) Reifenberg flat domain it is also an NTA domain (see [5]). Definition 2.4. Let R n+. We say that is a chord arc domain (CAD) if is an NTA set of locally finite perimeter such that there exists C > so that for r (0, diam ) Q C r n H n ( (Q, r)) Cr n. (2.6) Here H n denotes the n-dimensional Hausdorff measure. Definition 2.5. Let R n+, δ > 0 R > 0. We say that is a (δ, R)-chord arc domain (CAD) if is a set of locally finite perimeter such that sup θ(r) δ (2.7) 0<r R H n ( (Q, r)) ( + δ)ω n r n Q r (0, R]. (2.8) Here ω n is the volume of the n-dimensional unit ball in R n. Definition 2.6. Let R n+, we say that is a chord arc domain with vanishing constant if it is a (δ, R)-CAD for some δ > 0 R > 0, lim sup r 0 θ(r) = 0 (2.9) lim sup H n ( (Q, r)) r 0 Q ω n r n =. (2.0) For the purpose of this paper we assume that R n+ is a bounded domain. We consider elliptic operators L of the form Lu = div(a(x) u) (2.) defined in the domain with symmetric coefficient matrix A(X) = (a ij (X)) such that there are λ, Λ > 0 satisfying for all X ξ R n+. λ ξ 2 n+ i,j= a ij (X)ξ i ξ j Λ ξ 2 (2.2) 4

5 We say that a function u in is a solution to Lu = 0 in provided that u W,2 loc () for all φ C c () A(x) u, φ dx = 0. A domain is called regular for the operator L, if for every g C( ), the generalized solution of the classical Dirichlet problem with boundary data g is a function u C(). Definition 2.7. Let be a regular domain for L as above g C( ). For X consider the linear functional g u(x) on C( ), where u is the generalized solution of the classical Dirichlet problem with boundary data g. y the Riesz representation theorem, there exists a family of regular orel probability measures {ωl X} X such that u(x) = For X, ω X L g(q)dω X L (Q). is called the L elliptic measure of with pole X. When no confusion arises, we will omit the reference to L simply called it as the elliptic measure. To state our results we introduce two classes of operators. We say that elliptic operator L L(λ, Λ, α) if it satisfies (2.), (2.2) the modulus of continuity of the corresponding matrix is given, up to the boundary, by w(r) = sup A(X) A(Y ) c 0 r α (2.3) X Y r for some α (0, ], that is A C α (). Without loss of generality we assume that A is defined in R n+ since A can be extended to a new matrix in the following way. If we start with A C α () then there exists an open set U such that U A C α (U). Consider now a smooth function φ C c (R n+ ) which is equal to in 0 outside U. We then extend A to = φa + ( φ)i in which gives that C α (R n+ ) = A in. An elliptic operator Lu = div(a(x) u) defined on a chord arc domain R n+ is a perturbation of the Laplacian for the purposes of this paper if the deviation function a(x) = sup{ Id A(Y) : Y (X, δ(x)/2)} (2.4) where δ(x) is the distance of X to, satisfies the following Carleson measure property: there exists C > 0 such that { sup sup σ((q, r)) 0<r<diam Q (Q,r) a 2 } (X) δ(x) dx C, (2.5) 5

6 where σ = H n. Note that in this case L = on therefore by letting L = in c we may assume that L is an elliptic operator in R n+. We now state some of our results: Theorem 2.8. Let R n+ be a Reifenberg flat domain with vanishing constant, let L L(λ, Λ, α) let ω be its elliptic measure. Then for all τ (0, ), lim inf ρ 0 Q ω((q, τρ)) ω((q, ρ)) = lim sup ω((q, τρ)) ρ 0 Q ω((q, ρ)) = τ n. In section 4 we show that if is a chord arc domain with vanishing constant L L(λ, Λ, α) then ω A (dσ). Furthermore we obtain the following results. Theorem 2.9. Given ε > 0, θ > 0 there exists δ > 0 such that if L L(λ, Λ, α) R n+ is a (δ, R)-CAD there exists r 0 > 0, so that for any Q r < r 0, if k(q) = dω dσ (Q) denotes the elliptic kernel of L, then for any β (0, /θ). ( ) k +β (+β) dσ (Q,r) ( + ε) kdσ, (Q,r) Theorem 2.0. Let R n+ be a chord arc domain with vanishing constant. L L(λ, Λ, α). Then log k V MO( ). Assume that We now recall some of the results concerning the regularity of the elliptic measure of perturbation operators in Lipschitz domains. The results in the literature are more general than those quoted below. Theorem 2.. [5] Let = (0, ). If a is as in (2.4), { h(q, r) = σ((q, r)) (Q,r) a 2 } (X) δ(x) dx (2.6) lim sup h(q, r) = 0. r 0 Q = Then the elliptic kernel of L, k = dω/dσ q (dσ) for all q >. In [8], Fefferman made the first step toward removing the smallness condition of h(q, r) in Theorem 2. by defining an appropriate quantity A(Q). 6

7 Theorem 2.2. [8] Let = (0, ). Let Γ(Q) denote a non-tangential cone with vertex Q ( A(Q) = Γ(Q) a 2 ) (X) /2 δ n (X) dx, where a is as in (2.4). If A L C then ω A (dσ). The main results in [5] in [8] are proved using a differential inequality for a specific family of harmonic measures which was introduced by Dahlberg. In [9], Fefferman, Kenig Pipher presented a new direct proof of these results without the use of this differential inequality. Theorem 2.3. [9] Let be a Lipschitz domain. Let L be such that (2.5) holds then ω A (dσ). In this paper we generalize Theorem 2.3 to chord arc domains with small constant. Theorem 2.4. Let be a chord arc domain. Let L be such that (2.5) holds. There exists δ(n) > 0 such that if R n+ is a (δ, R) CAD with 0 < δ δ(n) then ω A (dσ). The various constants that will appear in the sequel may vary from formula to formula, although for simplicity we use the same letter(s). If we do not give any explicit dependence for a constant, we mean that it depends only on the usual parameters such as ellipticity constants, NTA character of the domain dimension. Moreover throughout the paper we shall use the notation a b to mean that there is a constant c > 0 such that ca b. Similarly a b means that a b b a. Next we recall the main theorems about the boundary behavior of L elliptic functions in nontangentially accessible (NTA) domains. We refer the reader to [3] for the definitions more details regarding elliptic operators of divergence form defined in NTA domains. Lemma 2.5. Let be an NTA domain. If Lu = 0 in (Q, 2r) with 0 < 2r < R, u 0 vanishes continuously on (Q, 2r) then there exists β > 0 such that for all Q for X (Q, r), ( ) X Q β u(x) C sup{u(y ) : Y (Q, 2r) }. r Lemma 2.6. Let be an NTA domain, Q, 0 < 2r < R. If u 0, Lu = 0 in u vanishes continuously on (Q, 2r), then u(y ) Cu(A(Q, r)), for all Y (Q, r). Here C only depends on the NTA constants. 7

8 Lemma 2.7. Let be an NTA domain, Q, 0 < 2r < R, X \(Q, 2r). Then C < ω X ((Q, r)) r n G(A(Q, r), X) < C, where G(A(Q, r), X) is the L Green function of with pole X. Lemma 2.8. Let be an NTA domain with constants M > R > 0, Q, 0 < 2r < R, X \(Q, 2Mr). Then for s [0, r] ω X ((Q, 2s)) Cω X ((Q, s)), where C only depends on the NTA constants of. Lemma 2.9. Let be an NTA domain, 0 < Mr < R. Suppose that u, v vanish continuously on (Q, Mr) for some Q, u, v 0 Lu = Lv = 0 in. Then there exists a constant C > (only depending on the NTA constants) such that for all X (Q, r), u(a(q, r)) C v(a(q, r)) u(x) u(a(q, r)) C v(x) v(a(q, r)). Theorem Let be an NTA domain. There exists a number α > 0, such that for all Q, 0 < 2r < R, all u, v 0 satisfying Lu = Lv = 0 in (Q, 2r) which vanish continuously on (Q, 2r), the function u(x) v(x) is Hölder continuous of order α on (Q, r). In particular, u(x) for every Q, lim X Q v(x) exists, for X, Y (Q, r), u(x) v(x) u(y ) ( ) u(a(q, r)) X Y α v(y ) C. v(a(q, r)) r We finish this section by recalling a result concerning the regularity of elliptic measure on Lipschitz domains, as well as some doubling properties of the elliptic measure of a cylinder. Let H R n+ be an open half space, for M >, s > 0, Q 0 H = L we denote by C + (Q 0, Ms) = {(x, t) R n+ : x L; x Q 0 Ms n +, t Ms n + } H, the cylinder with basis (Q 0, Ms/ n + ) L height Ms/ n + contained in H. Note that C + (Q 0, Ms) (Q 0, Ms). Lemma 2.2. Given ε > 0 there exists M 0 = M 0 (n, ε) >, so that if M M 0, if ω denotes the L elliptic measure of C + (Q 0, Ms) as defined above, then for Q, Q 2 (Q 0, s) = L (Q 0, s), r, r 2 (0, s] ( ε) ( r r 2 ) n ωx ( (Q, r )) ω X ( + ε) ( (Q 2, r 2 )) as long as X = (x, t) C + (Q 0, Ms/2) C + (Q 0, Ms). 8 ( r r 2 ) n, (2.7)

9 Proof. After rescaling we may assume without loss of generality that Ms =, r i (0, /M], for i =, 2. First let us examine the case when X = (x, t) C + (Q 0, /2), with t /(2κ n + ) κ > 2 to be chosen later. If ω X denotes the L elliptic measure then ω X ( (Q, r )) = A(Q) Q G(Q, X), ν dσ(q) (Q,r ) or ω X ( (Q, r )) (Q,r ) A(Q) Q G(Q, X), ν A(Q 0 ) Q G(Q 0, X), ν dσ(q) (2.8) + A(Q 0 ) Q G(Q 0, X), ν r n where ν denotes the inward unit normal to H at Q L. y the Hopf maximum principle (see [2], []) there exists a constant C k = C k (n, λ, Λ, κ) > 0 such that A(Q 0 ) Q G(Q 0, X), ν C k > 0. Moreover from the C,α regularity up to the boundary ([0]), we estimate the first term of (2.8) to obtain ω X ( (Q, r )) ( + Cr α )r n A(Q 0 ) Q G(Q 0, X), ν where C = C(n, λ, Λ, κ). In a similar way, using the appropriate representation we have ω X ( (Q 2, r 2 )) ( Cr α 2 )r n 2 A(Q 0 ) Q G(Q 0, X), ν provided that (Q i, r i ) (Q 0, 2/M). Since r, r 2 < /M, we conclude that provided that M is large enough. ω X ( (Q, r )) ω X ( (Q 2, r 2 )) ω X ( (Q, r )) ω X ( (Q 2, r 2 )) ( + C/M α C/M α ( C/M α + C/M α )( ) n r r 2 )( ) n r Now if X = (x, t) C + (Q 0, /2), t /2κ n +, ω X ( (Q, r )) ω X ( (Q 2, r 2 )) vanish on ((x, 0), /4 n + ) L are non negative in C + (Q 0, ). Applying Theorem 2.20 we have that for κ > 8, ω X ( (Q, r )) ω X ( (Q 2, r 2 )) ω(x,/2κ n+) ( (Q, r )) ω (x,/2κ n+) ( (Q 2, r 2 )) C ω(x,/6 n+) ( ) ( (Q, r )) α. ω (x,/6 n+) ( (Q 2, r 2 )) κ 9 r 2

10 On the other h our new reference points (x, /2κ n + ) (x, /6 n + ) fall into the first case as described above, since κ > 8. Thus ( ( ) α )( C/M α C κ + C/M α ( ( + C κ )( r r 2 ) α )( + C/M α C/M α ) n ωx ( (Q, r )) ω X ( (Q 2, r 2 )) )( ) n r. To finish the proof, for a given ε > 0 choose κ > 8 such that ( C(/κ) α ) ε, ( + C(/κ) α ) + ε. Next for that selection of κ choose M 0 > 2 large enough so that for M M 0, ε C/M α + C/M α + C/M α C/M α + ε. r 2 3 Optimal Doubling on Reifenberg Flat Domains. As seen in Lemma 2.8, the L elliptic measure of an NTA domain is a doubling measure on. In the present section we prove that the L elliptic measure of a Reifenberg flat domain R n+ with vanishing constant, behaves asymptotically like the Euclidean measure in R n. Using the terminology introduced by M. Korey (see [6]) we say that the L elliptic measure of a Reifenberg flat domain with vanishing constant is asymptotically optimally doubling. Theorem 3.. Given ε > 0 there exists M(n, ε) >, so that if M M(n, ε) there exists δ(ε, M, r/s) = δ > 0 such that, for 0 < r s R/M any (δ, R)-Reifenberg flat domain R n+ we have: ( ) r n ( ) ( ε) ωx ((Q, r)) r n s ω X ( + ε), ((Q 2, s)) s where Q, Q 2 (Q 0, s) for some Q 0, X, X Q 0 Ms. Corollary 3.2. Let R n+ be a Reifenberg flat domain with vanishing constant. Then for any X τ (0, ) lim inf ρ 0 Q ω X ( (Q, τρ)) ω X ( (Q, ρ)) = lim sup ω X ( (Q, τρ)) ρ 0 Q ω X ( (Q, ρ)) = τ n. The main idea of the proof is to compare the elliptic measure of a Reifenberg flat domain with the elliptic measure of an appropriate cylinder. In order to do this we need to introduce some extra notation. 0

11 Let R n+ be a (δ, R)-Reifenberg flat domain, with δ δ 0. Let M > be a large number to be determined later, let s > 0 be so that Ms R. There exists an n-dimensional plane L(Q 0, Ms) containing Q 0 such that Ms D[ (Q 0, Ms), L(Q 0, Ms) (Q 0, Ms)] δ, T + (Q 0, Ms) T (Q 0, Ms) c. In particular if we define for r = Ms or r = Ms/2 then (Q 0, Ms) = {(x, t) R n+ : x L(Q 0, Ms), x Q 0 C + (Q 0, r) = {(x, t) R n+ : x L(Q 0, Ms), x Q 0 C (Q 0, r) = {(x, t) R n+ : x L(Q 0, Ms), x Q 0 C(Q 0, Ms/2) = {(x, t) R n+ : x L(Q 0, Ms), x Q 0 Ms n +, t r n +, 2δr t r n +, 2δr t Ms n + }, r n + }, r n + } Ms 2 Ms, t n + 2 n + }, C + (Q 0, Ms) (Q 0, Ms) C (Q 0, Ms), C + (Q 0, Ms) T + (Q 0, Ms). Note that the Hausdorff distance between C + (Q 0, Ms) C (Q 0, Ms) is 4δMs. esides if n Ms,Q0 denotes the unit normal to L(Q 0, Ms) chosen with the appropriate orientation, then for δ small enough A(Q 0, Ms) = Q 0 + Ms 4 n + n Ms,Q 0 C + (Q 0, Ms/2), ( ) Ms A(Q 0, Ms), 8 C + (Q 0, Ms/2) n + [ ( dist A(Q 0, Ms), ) ] Ms 8, C + (Q 0, Ms/2) n + Ms 6 n +. Remark 3.3. If Π denotes the orthogonal projection from R n+ onto L(Q 0, Ms) then Π( (Q 0, Ms)) = {x L(Q 0, Ms) : x Q 0 Ms n + }.

12 Next we introduce the sets which arise from the intersection of (Ms, Q 0 ) cylinders having direction n Ms,Q0. We denote by by Γ(Q 0, s) = {(x, t) R n+ : x L(Q 0, Ms), x Q 0 s, t Γ(Q, r) = {(x, t) R n+ : x L(Q 0, Ms), x Π(Q) r, t Ms n + } (Q 0, Ms) Mr n + } (Q 0, Ms), for Q Γ(Q 0, s) r > 0 small enough so that Γ(Q, r) Γ(Q 0, s). In particular if r = τs for some τ (4δM, ), then ( Γ Q, r ( ) 2δM 2 ) (Q, r) Γ(Q, r). (3.) τ If τ is relatively large with respect to 2δM, the projections of these 3 sets on L(Q 0, Ms) have almost the same area. In fact recall that Π(Γ(Q, r)) = ω n r n. Let us denote by ω the elliptic measure of (Ms, Q 0 ) by ω ± the elliptic measures of C ± (Q 0, Ms). Lemma 3.4. Given ε > 0, there exists M(n, ε) > such that if M M(n, ε) there exists δ(ε, M, r/s) = δ > 0 so that if is a (δ, R)-Reifenberg flat domain, then for 0 < r s R/M, Q 0, Q, (Q, r) (Q 0, s), X C(Q 0, Ms/2) C + (Q 0, Ms), ( ε)ω X + ( + (Q +, r)) ω X ( (Q, r)) ( + ε)ω X ( (Q, r)) (3.2) where Q ± = Π(Q) ± 2δMsn Ms,Q0, ± (Q ±, r) = (Q ±, r) C ± (Q 0, Ms). Proof. The basic idea is to compare the appropriate solutions of Lu = 0 in C ± (Ms, Q 0 ) (Ms, Q 0 ) in order to apply the maximum principle. Since Lemmata are valid we may adopt the proof of Lemma 4. in [5]. The following lemma gives the opposite estimate when the pole is far away from the boundary. Lemma 3.5. Given ε > 0, there exists M(n, ε) >, so that if M M(n, ε) for κ > 2 there exists δ(ε, M, κ, r/s) = δ > 0 such that if is a (δ, R)-Reifenberg flat domain, then for 0 < r s R/M, Q 0, Q, (Q, r) (Q 0, s), X = (x, t) C(Q 0, Ms/2) C + (Q 0, Ms), with t Ms/κ n +, ( ε)ω X ( (Q, r)) ω X ((Q, r)) ( + ε)ω X + ( + (r, Q + )). (3.3) where Q ± = Π(Q) ± 2δMsn Ms,Q0, ± (Q ±, r) = (Q ±, r) C ± (Q 0, Ms). 2

13 Proof. Let ε = ε (ε) > 0 to be chosen later. We first prove that, for M > 2 there exists 0 < δ(ε, M, κ, r/s) so that ω X ( (Q, r)) + ε ε ωx + ( + (Q +, r)). (3.4) Let us first show how to obtain (3.3) from (3.4). Choose M large as in Lemma 3.4. Denote δ := δ(ε, M, r/s) the constant in that lemma. Then for δ min{δ, δ(ε, M, κ, r/s)}, inequality (3.2) holds with ε instead of ε. Combining (3.2) (3.4) we obtain ( ε ) 2 + ε ω X ( (Q, r)) ω X ((Q, r)) ( + ε ) 2 ε ω X + ( + (Q +, r)). Choosing ε > 0 so that ε ( ε ) 2 /( + ε ) ( + ε ) 2 /( ε ) + ε we obtain inequality (3.3). Now we continue with the proof of (3.4). Recall that C + (Q 0, Ms) (Q 0, Ms) C (Q 0, Ms). Assume that δ δ define u (x, t) = ω (x,t) ( (Q, r)) for (x, t) C (Q 0, Ms), u 2 (x, t) = ω (x,t) + ( +(Q +, r)) for (x, t) C + (Q 0, Ms). We compare u (x, t 4δMs) u 2 (x, t) for (x, t) C + (Q 0, Ms). First note that if t = 2δMs or x Q 0 = Ms/ n + then u (x, t 4δMs) = u 2 (x, t). Indeed, if t = 2δMs then u (x, t 4δMs) vanishes for x Π(Q) r it is equal to one otherwise. The function u 2 has the same behavior. When x Q 0 = Ms/ n + >> r both functions vanish. Since C (Q 0, Ms) is an NTA domain, u is non negative on C (Q 0, Ms) u (x, t) = 0 for we apply Lemma 2.5 to get (x, t) C (Q 0, Ms) {(x, t) R n+ : t > u (x, Ms n + 4δMs) K δ β 3Ms 4 n + }, where K depends on the NTA constants of C (Q 0, Ms). Now consider a bounded function v(x, t) such that Lv(x, t) = 0, v(x, t) = 0, in C + (Q 0, Ms) on t = 2δMs v(x, t) =, on t = Ms/ n + v(x, t) 0, on x Q 0 = Ms/ n +. 3

14 Therefore for (x, t) C + (Q 0, Ms) u (x, t 4δMs) u 2 (x, t) + K v(x, t)δ β. (3.5) y the maximum principle inequality (3.5) holds for all (x, t) C + (Q 0, Ms). Let { R = C + (Q 0, Ms) (x, t) R n+ : x L(Q 0, Ms), x Q 0 Ms κ n + t Ms ( )}. n + κ Ms ) ( /κ, n + From the Hopf maximum principle Harnack s inequality for δ small enough for Y R u 2 (Y ) K 2 = K 2 (n, λ, Λ, κ, τ, M). Choosing δ > 0 even smaller we have K v(x, t)δ β K Cδ β ε K 2, therefore for (x, t) R we obtain u (x, t 4δMs) ( + ε )u 2 (x, t). (3.6) Applying classical interior estimates (see [0], chapter 8) we conclude that there exists β > 0 such that for δ > 0 small enough for (x, t) R, u (x, t 4δMs) ( C(δκ) β )u (x, t) ( ε )u (x, t). (3.7) Combining (3.6) (3.7) we have for (x, t) R the proof is concluded. u (x, t) + ε ε u 2(x, t) The next theorem is an immediate consequence of Lemmata We refer the reader to Theorem 4.2 of [5] for the details of the proof. Theorem 3.6. Given ε > 0 there exists M(n, ε) >, so that if M M(n, ε) there exists δ(ε, M, r/s) = δ > 0 such that for any (δ, R)-Reifenberg flat domain R n+, 0 < r s R/M we have ( ) r n ( ) ( ε) ωx ((Q, r)) r n s ω X ( + ε), ((Q 2, s)) s where Q, Q 2 (Q 0, s) for some Q 0, X (Q 0, Ms)\C(Q 0, Ms/2). Here ω denotes the elliptic measure of (Ms, Q 0 ). 4

15 We now show that as long as X (Q 0, Ms) is far away from Q 0, ω X (E)/ ω X (E ) ω X (E)/ω X (E ) are comparable, whenever E, E (Q, 2s). Lemma 3.7. Given ε > 0, τ (0, ) there exists M(n, ε) > 0 such that for M M(n, ε) there exists δ = δ(ε, M, τ) > 0 such that if is (δ, R)-Reifenberg flat domain, 0 < r < s R/M, Q 0, Q, (Q, r) (Q 0, S) X (Q 0, Ms) \ C(Q 0, Ms/2), then ( + ε) lim Y Q G(X, Y ) G(X, Y ) d ωx (Q) dω X ( + ε) lim (Q) Y Q G(X, Y ) G(X, Y ). Here ω (resp. ω) denotes the elliptic measure of (resp. (Q 0, Ms)) with pole at X, G (resp. G) denotes the Green s functions of (resp. (Q 0, Ms)). Proof. The Lebesgue differentiation theorem for Radon measures, ensures that, for ω-almost every Q (Q 0, 2s) d ω X ω X ((Q, r)) (Q) = lim dωx r 0 ω X ((Q, r)). (3.8) Consider a smooth function Ψ r such that Ψ r = on (Q, r), spt(ψ r ) (Q, 2r), Ψ r C r D 2 Ψ r C r 2. Let u r satisfy Lu r = 0 in u r = Ψ r on. Let ũ r satisfy Lũ r = 0 in (Ms, Q 0 ) ũ r = Ψ r on (Q 0, Ms). Then u r (X) = Ψ r (Q)dω X (Q) = A(Y ) G(X, Y ), Ψ r dy ũ r (X) = Ψ r (Q)d ω X (Q) = A(Y ) G(X, Y ), Ψ r dy. e (Q 0,Ms) e(q 0,Ms) An argument similar to the one used to prove the Lebesgue differentiation theorem ensures that d ω X ũ r (X) (Q) = lim dωx r 0 u r (X). (3.9) Let A(Q) = A Q, then A(Y ) G(X, Y ), Ψ r dy = G(X, Y ), A Q Ψ r dy + (A(Y ) A Q ) G(X, Y ), Ψ r dy. We estimate the last term by appealing (2.3), Hölder s inequality, a boundary Cacciopoli estimate 5

16 (see [3], Lemma.2) Lemma 2.6, ( (A(Y ) A Q ) G(X, Y ), Ψ r dy Cr α (Q,2r) Cr α r rn/2 r n/2 ( Cr α r n 2 ( ) /2 ( G 2 dy (Q,2r) ) /2 G 2 dy (Q,2r) G 2 dy (Q,4r) Cr n 2+α G(A(Q, r), X). ) /2 ) /2 Ψ r 2 dy To estimate the first term on the right h side note that G(X, Y ), A Q Ψ r dy = div(ga Q Ψ r )dy Gdiv(A Q Ψ r )dy thus u r(x) G(X, Y )div(a Q Ψ r )dy Crn 2+α G(A(Q, r)) + div(g(x, Y )A Q Ψ r )dy. Note also that div(g(x, Y )A Q Ψ r )dy = div(g(x, Y )A Q, Ψ r )dy. e(q 0,Ms) (3.0) If F = (Q 0, Ms) \ C + (Q 0, Ms) then div(g(x, Y )A Q Ψ r )dy = e(q 0,Ms) C + (Q 0,Ms) div(g(x, Y )A Q Ψ r )dy + F div(g(x, Y )A Q Ψ r )dy C + (Q 0,Ms) div(g(x, Y )A Q Ψ r )dy = {t= Msδ n+ } G(X, Y )A Q Ψ r, e n ds C sup Y (Q,2r) {t= Msδ n+ } r rn G(X, Y ) ( ) Msδ α C G(A(Q, r), X)r n 2. r 6

17 Now F G(X, Y )div(a Q Ψ r )dy C sup G(X, Y ) H n (F (Q, 2r)) F (Q,2r) r 2 ( ) Msδ α C G(A(Q, r), X) r r 2 rn Msδ ( ) Msδ α+ = C G(A(Q, r), X)r n 2. r In a similar way, A Q Ψ r, G dy C r F C r C rn/2 F (Q,2r) ( (Q,2r) G dy G 2 dy r 2 G(A(Q, r), X)(Msδrn ) /2 ( ) Msδ /2 = Cr n 2 G(A(Q, r), X). r ) /2 ( ) /2 H n (F (Q, 2r)) Therefore e(q 0,Ms) ( ) Msδ β div(g(x, Y )A Q Ψ r )dy C G(A(Q, r), X)r n 2 r for β = min{α, 2 }. We use this estimate in (3.0), ( ) u r(x) G(X, Y )div(a Q Ψ r )dy Msδ β Crn+α 2 G(A(Q, r), X) + C G(A(Q, r), X)r n 2. r Note that a similar estimate holds for ũ r (X) in terms of G. Next we write = e(q 0,Ms) e(q 0,Ms) G(X, Y )div(a Q Ψ r )dy = e(q 0,Ms) ( ) G(X, Y ) G(X, Y ) l(q) G(X, Y )div(a Q Ψ r )dy + l(q) G(X, Y ) G(X, Y ) G(X, Y )div(a Q Ψ r )dy e(q 0,Ms) G(X, Y )div(a Q Ψ r )dy where We now choose τs r < s G(X, Y ) l(q) = lim Y Q G(X, Y ). M α < ε 7

18 where ε = ε (ε). Then we choose δ such that ( ) δm β < ε. τ Combining the estimates above with Theorem 2.20 we have ũr(x) l(q)u r (X) ũ r (X) G(X, Y )div(a Q Ψ r )dy (3.) + G(X, Y )div(a Q Ψ r )dy l(q) + l(q) u r(x) G(X, Y )div(a Q Ψ r )dy r n+α 2 G(A(Q, r), X) + ε r n 2 G(A(Q, r), X) G(X, Y )div(a Q Ψ r )dy + l(q)r n+α 2 G(A(Q, r), X) + ε l(q)r n 2 G(A(Q, r), X) ( ) r β ( ) δms α+ + l(q) r n 2 G(A(Q, r), X) Ms r { r n 2 G(A(Q, r), X) (r α + ε ) G(A(Q, r), X) since by the maximum principle } G(A(Q, r), X) + (rα + ε )l(q) } { u r (X) (r α + ε ) G(A(Q, r), X) G(A(Q, r), X) + (rα + ε )l(q) r n 2 G(A(Q, r), X) ω X ((Q, r)) u r (X). Furthermore since lim r 0 G(A(Q, r), X) G(A(Q, r), X) = l(q) then ũ r (X) u r (X) l(q) εl(q). We conclude the proof by combining (3.9), (3.) choosing ε in terms of ε. Corollary 3.8. Given ε > 0, there exists M(n, ε) > so that if M M(n, ε) there exists δ(n, ε, M) > 0, such that if R n+ is a (δ, R)-Reifenberg flat domain with δ (0, δ(n, ε)], Q 0, 0 < s R/M, E, E (Q 0, 2s), X (Q 0, Ms)\C(Q 0, Ms/2) then ( ε) ωx (E) ω X (E ) ωx (E) ω X (E ) ( + ε) ωx (E) ω X (E ). Proof. We choose ε to depend on ε such that Lemma 3.7 is satisfied. From Theorem 2.20 we have l(q 0 ) l(q) C G(X, ( ) A(Q 0, Ms)) Q Q0 α G(X, A(Q 0, Ms)) Ms 8

19 for Q (Q 0, 2s). In addition Lemma 2.9 guarantees that there exists a constant C > so that Hence C l(q 0 ) G(X, A(Q 0, Ms)) G(X, A(Q 0, Ms)) Cl(Q 0). ( CM ) α l(q 0 ) l(q) ( + CM ) α l(q 0 ). Since ω X (E) = E d ω X dω X (Q)dωX (Q) ( + ε ) E l(q)dω X (Q) ( + ε ) ( + CM ) α l(q 0 )ω X (E) ω X (E) ( + ε ) ( C M α )l(q 0 )ω X (E) we have that ( + ε ) 2 (C/M α ) + (C/M α ) ωx (E) ω X (E ) ωx (E) ω X (E ) ( + ε ) 2 + (C/M α ) (C/M α ) ωx (E) ω X (E ). Choosing M ε appropriately we conclude the proof of Corollary 3.8. Proof of Theorem 3.: For ε = ε (ε), let M(n, ε ) > be as in Theorem 3.6 Corollary 3.8. For M M(n, ε ) there exists δ(n, ε, r/s, M) > 0 so that if is a (δ, R) Reifenberg flat domain with δ δ(n, ε, r/s, M) then Theorem 3.6, Lemma 3.7 Corollary 3.8 hold. Namely for 0 < r s R/M we have that ( ) r n ( ) ( ε ) ωx ((Q, r)) r n s ω X ((Q 2, r)) ( + ε ), s where Q, Q 2 (Q 0, s) for some Q 0, X (Q 0, Ms)\C(Q 0, Ms/2). Moreover ( ε ) ωx ((Q, r)) ω X ((Q 2, s)) ωx ((Q, r)) ω X ((Q 2, s)) ( + ε ) ωx ((Q, r)) ω X ((Q 2, s)). Therefore for ε > 0 so that ε ( ε ) 2 ( + ε ) 2 + ε, X (Q 0, Ms/2) ( ) r n ( ) r n ( ε) ω X ((Q 2, s)) ω X ((Q, r)) ( + ε) ω X ((Q 2, s)). s s The maximum principle guarantees that for all X \(Ms/2, Q 0 ) ( ) r n ( ) ( ε) ωx ((Q, r)) r n s ω X ( + ε). ((Q 2, s)) s 9

20 4 Regularity on chord arc domains. In this section we prove that on a chord arc domain with small enough constant if L is either in L(λ, Λ, α) or if it is a perturbation of the Laplacian then the elliptic measure is an A weight with respect to surface measure. In the case that is a chord arc domain with vanishing constant L L(λ, Λ, α) we show that the logarithm of elliptic kernel (i.e. the density of the elliptic measure with respect to the surface measure) is in VMO. A key step in these proofs is Semmes Decomposition for chord arc domains with small constant (see [5], Theorem 2.2). Let be a (δ, R)-CAD for δ small enough so Theorem 2.2 in [5] holds. Let P let r > 0 small enough so the construction in Lemma 5. in [5] goes through. In this case there exist two Lipschitz functions h + h defined in L(P, r) such that h h + h ± η where η δ /4. Let + = {(x, t) R n+ : x L(P, r), t > h + (x)} = {(x, t) R n+ : x L(P, r), t > h (x)}. As in Lemma 5. in [5] the graphs Γ ± of h ± approximate in C(P, r) from above below respectively, in the sense that [ ] ( ) D Γ ± (P, r); (P, r) ηr H n ( Γ + Γ (P, r)) c exp{ c 2 /η} ω n r n where c, c 2 are positive constants as in ([5], Theorem 2.2). Moreover (4.) + C(P, r) C(P, r) C(P, r). Lemma 4.. Let L L(λ, Λ, α). There exists δ(n) > 0 such that if R n+ is a (δ, R) CAD with 0 < δ δ(n) X then ω X A (dσ) where σ = H n. Proof. Choose δ(n) > 0 such that Semmes decomposition applies as in Lemma 5. in [5]. For X let d = dist(x, ) ω X = ω. Let 0 < r min{r/2, d/4}. For P let = (P, r) A = A(P, r) be the non-tangential interior point of C(P, 2r). We may assume that + C(P, 2r) C(P, 2r). We denote by ω + be the L elliptic measure of + C(P, 2r). Since for E, ω A (E) ω(e)/ω( ). It is enough to prove that for α (0, ) small, there exists β (0, ) so that if ω A (E) < α then σ(e)/σ( ) < β. 20

21 Assume that ω A (E) < α. We decompose E as in Lemma 5.2 of [5], E = E E 2 where E = E + E 2 = E \ +. y the maximum principle ω A +(E ) ω A (E) < α. We write σ(e) σ( ) = σ(e ) σ( ) + σ(e 2) σ( ). (4.2) Since + is a Lipschitz domain, ω + A (dσ + ) so there are positive constants θ, C, C 2 such that ( ) σ+ (E ) /θ C ω ( ) +(E ) σ + ( + ) ω + ( + ) C σ+ (E ) θ 2 σ + ( + ) where + = (Π(P ), h + (Π(P )), r + η 2 ) σ + denotes the surface measure of +. Therefore the first term of (4.2) is estimated by σ(e ) σ( ) σ +(E ) σ + ( + ) σ+( + ) α θ ( + η 2 ) (n+)/2 ( + η). σ( ) Finally the second term of (4.2) is controlled using the Semmes Decomposition estimate for chord arc domains with small constant (see [5], Theorem 2.2). That is, σ(e 2 ) σ( ) c exp( c 2 /η)ω n r n σ( ) ( + η) exp( c 2 /η). Gathering all the estimates choosing α > 0 δ > 0 small enough, since η δ /4 we conclude that σ(e)/σ( ) < β <. An immediate consequence of Lemma 4. is the following Corollary. Corollary 4.2. Let L L(λ, Λ, α). There exist δ(n, λ, Λ) = δ 0 > 0, µ > 0 β > 0 such that if R n+ is a (δ, R)-CAD with 0 < δ δ 0, for X, = (Q, s) with Q, s min{dist(x, )/4, R/4}, E is a measurable set then ω X ( ) (E) σ(e) 2µ ω X ( ) c. σ( ) Moreover if k X = dω X /dσ then ( σ( ) /+2β X dσ) c σ( ) k +2β where c > denotes a constant that depends only on n, λ, Λ. k X dσ The next theorem states that the density satisfies a reverse Hölder inequality with optimal constant. The proof is very similar to the proof of Theorem 5.2 in [5], which we can adopt in our case due to the C α regularity of the coefficients. Here we present only the main steps of the proof. 2

22 Theorem 4.3. Let L L(λ, Λ, α). Given ε > 0, N > 0 there exists δ 0 = δ(ε, N, λ, Λ, n) > 0 such that if R n+ is a (δ, R)-CAD with δ (0, δ 0 ) there exists γ = γ(ε, N, λ, Λ, w) > 0, so that for any surface ball with radius s γ/2, if X with dist(x, ) N, k X (Q) = k(q) = dωx dσ (Q) then ( σ() where β > 0. /+β k dσ) +β ( + ε) σ() kdσ, Proof. We intend to apply Semmes decomposition in the set (Q 0, r) = (Q 0, r) with r = Ms for M >> large enough. Let X C(Q 0, r/2) + (Q 0, Ms) (Q 0, Ms) = C(Q 0, Ms). We denote by ω the elliptic measure of with pole at X, by ω the elliptic measure of (Q 0, Ms) with pole at X, by ω the elliptic measure of C(Q 0, Ms) with pole X by ω the elliptic measure of with pole X. Moreover we denote by k (Q) = dω /dσ, k (Q) = d ω /dσ. We need to estimate k +β dσ = k +β dσ + k +β dσ. (4.3) \ To estimate the first term of (4.3), we apply Semmes decomposition to get or σ( \ ) C exp( C 2 η )ω n(ms) n σ( \ ) σ( ) 2C exp( C 2 η )M n. where η δ /4. Applying Corollary 4.2 choosing δ > 0 small enough we conclude that k +β dσ σ( ) \ σ( ) ( \ ( σ( \ ) K σ( ) ) +β k +2β +2β dσ σ( \ ) β +2β ) β +2β σ( ) ( K 2C exp( C ) β ( 2 δ )M n +2β σ( ) ( ε σ( ) k +β dσ) +β. 22 ) +β k +β dσ ) +β k +β dσ

23 In order to estimate the second term of (4.3), we need to show that k ( Q) ( + ε ) 6 k (Q) ωx ( ) ω X ( ) (4.4) for every X C(Ms/2, Q 0 ), where k k denote the elliptic kernel with pole X, = ((Π(Q 0 ), h (Π(Q 0 ))), s) Q. The proof of (4.4) follows the same guidelines as the corresponding proof in [5]. We include the proof in the case that the pole is far from the boundary in order to illustrate which results need to be used in this case. We refer the reader to [5] for the proof of the case when the pole is close to the boundary. Let X = (x, t) with t Ms/κ n +.. y Lebesgue density theorem Let G 0 be the set of density points of k +β dσ = applying Corollary 4.2 for Q G 0, we have G 0 k +β dσ ω( 0 ) lim =, 0 Q ω( 0 ) ω( 0 ) k(q) = lim 0 Q σ( 0 ) = lim ω( 0 ) 0 Q σ( 0 ) where 0 is a surface ball centered at Q contained in. Let F = 0 apply the maximum principle to obtain where ω(f ) ω( ) ω (F ) ω ( ) ω ( ) ω( ) ε ω ( ) ω( ) + ε since Lemmata are valid. Now using Corollary 3.8 we obtain ω(f ) ω( ) ( + ε ) 3 ω (F ) ω ( ) ω(f ) σ( 0 ) ( + ε ) 3 ω (F ) σ ( σ ( 0 ) 0 ) σ( 0 ) ω( ) ω ( ) 23

24 where 0 is a surface ball in centered at Q with the same radius as 0. Using the fact that is a Lipschitz graph with small constant less we conclude that for δ > 0 small enough, ω(f ) σ( 0 ) ( + ε ) 5 ω (F ) σ ( 0 ) Therefore letting 0 Q we conclude that ω( ) ω ( ) ( + ε ) 5 ω ( 0 ) σ ( 0 ) k(q) ( + ε ) 5 ω( ) ω ( ) k (Q). ω( ) ω ( ). The proof of the case when the pole is close to the boundary uses Theorem 2.20 the ideas of the proof of Theorem 4.2 in [5]. Next we estimate the second term in (4.3). For X C(Q 0, Ms/2), k +β dσ σ( ) G 0 [ ] ( + ε ) 6 ω( ) +β ω ( ) σ( ) G 0 k +β dσ [ ] ( + ε ) ( + ε ) 6 ω( ) +β ω ( ) σ ( ) [ ] ( + ε ) ( + ε ) 7 ω( ) +β ( ω ( ) σ ( ) ( ( + ε )( + ε ) 8(+β) σ( ) Combining all the estimates above choosing ε in term of ε kdσ) +β. k +β dσ k dσ ) +β we have that ( σ() /+β k dσ) +β ( + ε) σ() kdσ, The regularity result is a consequence of the following corollary. Corollary 4.4. Let L L(λ, Λ, α). Given ε > 0, N > 0 there exists δ 0 = δ(ε, N, λ, Λ, n) > 0 such that if R n+ is a (δ, R)-CAD with δ (0, δ 0 ) there exists γ = γ(ε, N, R, λ, Λ, w) > 0, so that for any surface ball with radius s γ/2, if X with dist(x, ) N, k X (Q) = k(q) = dωx dσ (Q), then σ() ( log k σ() ) log kdσ dσ ε. 24

25 Proof. We will use Sarason s lemma John-Niremberg s in the following manner. Let ε (ε) > 0 to be determined later. For ε N let δ γ be as in Theorem 4.3 dν = ( kdσ) kdσ. From Hölder s inequality we have Hence for ε small enough ( k β dσ k +β dσ) β/+β σ() 2β/+β. k β dν k β dν ( σ() ) 2 ( kdσ σ() k +β dσ) 2/+β + 3ε. Applying now Sarason s lemma (see [7]) together with John-Nirenberg s inequality guarantees that for p [, ), if s γ/8, ( ω() ) /p log k c p dω C p β ε /3, where c = /ω() log kdω. From the theory of A -weights we have for some p large enough ( k /p ) p C σ()p ω(). Thus applying Hölder s inequality we have σ() ( log k c dσ C ω() log k c p dω) /p C(β, p)ε /3. Choosing ε so that C(β, p)ε /3 ε/2 we conclude that for a surface ball with radius s γ/8 σ() ( log k σ() ) log kdσ dσ ε. Corollary 4.5. Let L L(λ, Λ, α). Let R n+ be a chord arc domain with vanishing constant. Then for any X, log k X V MO( ). We now concentrate in the case when L is perturbation of the Laplacian, i.e we assume that (2.5) holds. The crucial step in the proof of Theorem 2.4 is to compare the L-elliptic measures of the Lipschitz domains ± we constructed above. 25

26 Lemma 4.6. Let L be a uniformly elliptic operator in divergence form satisfying the assumptions of Theorem 2.4. Let Γ ± be defined as above. Then there exists θ > 0 such that for Q s > 0 if E Γ + Γ (Q, s) ( ) ω (E) ω ( ) C ω+ (E) θ (4.5) ω + ( + ) where ± = (Q ±, s) with Q ± = (Π(Q), h ± (Π(Q))) Π is the projection in L(P, r). Here ω ± denote the L harmonic measures of ± C(P, 2r) with pole outside (P, r). Proof. The proof will follow the lines of Main Lemma in [6]. Let G Γ + Γ (P, r) denote by Q ± = (q, h ± (q)). If X = (x, x n+ ) then dist(x, Γ ± ) x n+ h ± (x) for q such that h + (q) h (q) h + (q) h (q) dist(q +, Γ ) dist(q, Γ + ). (4.6) We proceed by constructing a Whitney decomposition of R n+ \ G. Extract a subfamily {Q i } such that Γ \ G Q i. y (4.6) note that dist(q i, Γ+ ) diam Q i. Since Lip(h+ ) η η << there exist a family {Q + i } obtained by vertical translations from {Q i } such that Q Q i only if Q + Q + i. Furthermore if diam Q i = diam Q + i dist(q + i, Γ ) dist(q i, Γ+ ) Γ + \ G Q + i. We can find Q i such that 2Q i 2Q + i Q i diam Q i diam Q ± i measure µ for F Γ (P, r) by. Next we define the µ(f ) = ω + (F G) + i ω (F Q i ) ω (Q i ) ω + (Q i ). We will prove the following claim. Claim. If Q Γ (Q, s) (P, r) for F Γ (P, r) then µ(f ) µ((q, s)) ω (F ) ω ((Q, s)) (4.7) µ((p, r)). (4.8) 26

27 From the claim, using the real variable lemma of Coifman Fefferman (see [3]) we conclude (4.5). Proof of claim. Let Q Γ (P, r) 0 < s r. If for all i, Q i (Q, s/2) = then (Q, s/2) G, Q Γ + by the doubling property of ω + µ((q, s)) ω + ((Q, s/2) G) = ω + ((Q, s/2)) ω + ((Q, s)). If there exists an i such that Q i (Q, s/2) by the doubling property of ω + µ((q, s)) ω + ((Q, s) G) + ω + (Q i ). Q i (Q,s/2) Moreover, if Q Γ + Γ, Q i (Q, s/2) for some i, µ((q, s)) ω + ((Q, s) G) + ω + ((Q, s/2) \ G) ω + ((Q, s/2)). Now if Q / Γ +, Q = Q Q i (for the same i as above) Q + Q + i. In addition (Q +, s/4) Γ + \ G 2Q + i Q i (Q, s/2) since for (x, h + (x)) (Q +, s/4) Γ + \ G, (x, h + (x)) (q, h (q)) (x, h (x)) (x, h + (x)) + (x, h (x)) (q, h (q)) diam Q i + s/4 + ηs/4. Thus ω + (Q i ) ω +((Q +, s/4) \ G) with Q i (Q, s/2) µ((q, s)) ω + ((Q, s) G) + ω + ((Q +, s/4) \ G) for Q Γ \ Γ +. Now if (Q, s/(2 0 6 )) G =, for all X = (x, h + (x)) = (x, h (x)) we have (q, h (q)) (x, h (x)) > s/(2 0 6 ). Hence if q x < s/(2 0 6 ) then for η small enough x q s ηs (q, h + (q)) (x, h + (x)) x q ηs/(2 0 6 ) 27 s

28 Thus by the doubling property of ω +, µ((q, s)) ω + ((Q +, s/(4 0 6 ))) ω + ((Q +, cs) ω + ((Q, cs)). On the other h, if (Q, s/(2 0 6 )) G, then (Q +, s/4) (Q, s) again by the doubling property of ω + µ((q, s)) ω + ((Q +, s/4)) ω + ((Q, cs)). Thus, in any case we have shown that µ((q, s)) ω + ((Q +, s/4)) ω + ((Q, cs)). (4.9) Let Q Γ (P, r), (Q, s) (P, r) consider two cases. Case. For every i, Q i (Q, s) 0 diam Q i 00s. Then, (Q, Cs) Γ +. For simplicity, let A = A + (Q +, Cs) = A (Q, Cs) the non-tangential points of ± at Q ± at radius Cs. Since Q i (Q, s) 0, the distance of Q = Q to Q i is less or equal to s 2Q i (Q, Cs), so Q i (Q, Cs) using the Carleson estimate in [] we have ω A ±(Q i ) ω A ± ((Q, Cs)) ω ± (Q i ) ω ± ((Q, Cs)) (4.0) Similarly Recall that ω A ±((Q, Cs)) thus (4.) (4.2) become ω A (F Q i ) ω A ((Q, Cs)) ω (F Q i ) ω ((Q, Cs)). (4.) ω A +(F G) ω A + ((Q, Cs)) ω +(F G) ω + ((Q, Cs)). (4.2) ω A (F Q i ) ω (F Q i ) ω ((Q, Cs)) (4.3) In addition, since ω + is a doubling measure, ω A +(F G) ω +(F G) ω + ((Q, Cs)). (4.4) ω A +(Q i ) ω A +(Q + i ). (4.5) If Z Q + i, then ωz (Q i ) = ωz +(Q + i ). For Z (+ (P, Cr)) \ Q + i, ωz (Q i ) 0 ω Z +(Q + i ) = 0. Therefore by the maximum principle for Z (+ (P, Cr)) ω Z +(Q + i ) ωz (Q i ). (4.6) 28

29 From (4.5), (4.6) we deduce that, Thus, by (4.9), (4.0), (4.), (4.3) (4.7) we have ω A +(Q i ) ω A (Q i ). (4.7) µ(f ) µ((q, s)) ω (F G) + ω (F Q i ) i ω (Q i ) ω + (Q i ) ω + ((Q, Cs)) ω A +(F G) + i ω A (F Q i ) ω A (Q i ) ω A +(Q i ) ω A (F G) + i ω A (F Q i ) ω A (F ) ω (F ) ω ((Q, s)). Case 2. Suppose there exists Q i such that Q i (Q, s) diam Q i > 00s. This implies (Q, s) G = (Q, s) Q i. If Q l (Q, s), then, dist(q i, Q l ) 2s, with diam Q i diam Q l. Since ω ω + are doubling measures Q i, Q l have large overlaps, ω ± (Q i ) ω ±(Q l ). Thus for F (Q, s) similarly which yields µ(f ) = i ω (F Q i ) ω (Q i ) ω + (Q i ) ω +(Q l ) ω (Q l ) ω (F Q i ) ω +(Q l ) ω (Q l )ω (F ) µ((q, s)) ω +(Q l ) ω (Q s)) l )ω((q, µ(f ) µ((q, s)) ω (F ) ω ((Q, s)). This concludes the proof of (4.7) in the claim. To prove (4.8) recall that by (4.) ( ) H n (Γ + Γ (P, r)) c exp{ c 2 /η} r n ω n. i Hence by the doubling property of ω + µ((p, r)) ω + ((P, r) G) ω + ((P, r)). Clearly by the doubling character of ω + we also have that µ((p, r)). 29

30 Proof of Theorem 2.4. Choose δ(n) > 0 such that Semmes decomposition applies as in Lemma 5. in [5]. For X let d = dist(x, ) ω X = ω. Let 0 < r min{r/2.d/4}. For P let = (P, r) A = A(P, r) be the non-tangential interior point of C(P, 2r). We may assume that + C(P, 2r) C(P, 2r). We denote by ω + be the L elliptic measure of + C(P, 2r). Since for E, ω A (E) ω(e)/ω( ) It is enough to prove that for α (0, ) small, there exists β (0, ) so that if ω A (E) < α then σ(e)/σ( ) < β. Assume that ω A (E) < α. We decompose E as in Lemma 5.2 of [5], E = E E 2 where E = E + E 2 = E \ +. y the maximum principle ω+(e A ) ω A (E) < α. We write σ(e) σ( ) = σ(e ) σ( ) + σ(e 2) σ( ) (4.8) In this case we do not know if ω + A (dσ + ). On the other h since is a Lipschitz domain L (extended to be the Laplacian in c ) is a perturbation of the Laplacian satisfying (2.5) we know that ω A (dσ ) by Theorem 2.3. Therefore there are positive constants γ, C, C 2 such that ( ) σ (E ) /γ C ω ( ) (E ) σ ( ) ω ( ) C σ (E ) γ 2 (4.9) σ ( ) where = (Π(P ), h (Π(P )), r + η 2 ) σ denotes the surface measure of. The first term of (4.8) is estimated combining (4.9) Lemma 4.6, σ(e ) σ( ) σ (E ) σ ( ) σ ( ) σ( ) ( ) ω (E ) γ σ ( ) ω ( ) σ( ) ( ) ω+ (E ) θ σ ( ) ω + ( + ) σ( ) α θ ( + η 2 ) n+/2 ( + η) (4.20) where θ = θ (θ, γ). Finally the second term of (4.8) is controlled using the Semmes Decomposition estimate for chord arc domains with small constant (see [5], Theorem 2.2). That is, σ(e 2 ) σ( ) c exp( c 2 /η)ω n r n σ( ) ( + η) exp( c 2 /η). (4.2) Combining (4.20) (4.2), choosing α > 0 δ > 0 small enough(recall η δ /4 ) we conclude that σ(e)/σ( ) < β <. 30

31 References [] L. Caffarelli, E. Fabes, S. Mortola, S. Salsa, oundary behavior of nonnegative solutions of elliptic operators in divergence form. Indiana Univ. Math. J. 30 (98), no. 4, [2] L. Caffarelli, C. Kenig, Gradient estimates for variable coefficient parabolic equations singular perturbation problems. Amer. J. Math. 20 (998), no. 2, [3] R. Coifman, C. Fefferman, Weighted norm inequalities for maximal functions singular integrals. Studia Math. 5 (974), [4]. Dahlberg, Estimates of harmonic measure. Arch. Rational Mech. Anal. 65 (977), no. 3, [5]. Dahlberg, On the absolute continuity of elliptic measure. American Journal of Mathematics 08 (986), [6]. Dahlberg, D. Jerison, C. Kenig, Area integral estimates for elliptic differential operators with nonsmooth coefficients. Ark. Mat. 22 (984), no., [7] L. Escauriaza, The L p Dirichlet problem for small perturbations of the Laplacian. Israel J. Math. 94 (996), [8] R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator. J. Amer. Math. Soc. 2 (989), no., [9] R. Fefferman, C. Kenig, J. Pipher, The theory of weights the Dirichlet problem for elliptic equations. Ann. of Math. (2) 34 (99), no., [0] D. Gilbarg, N. Trudinger, Elliptic partial differential equations of second order. Reprint of the 998 edition. Classics in Mathematics. Springer-Verlag, erlin, 200. xiv+57 pp. [] R. Hardt, L. Simon, oundary regularity embedded solutions for the oriented Plateau problem. Ann. of Math. (2) 0 (979), no. 3, [2] D. Jerison, C. Kenig, The logarithm of the Poisson kernel of a C domain has vanishing mean oscillation. Trans. Amer. Math. Soc. 273 (982), no. 2,

32 [3] C. Kenig, Harmonic analysis techniques for second order elliptic boundary value problems. CMS Regional Conference Series in Mathematics, 83. AMS Providence, RI, 994. xii+46 pp. [4] C. Kenig, J. Pipher, The Dirichlet problem for elliptic equations with drift terms. Publ. Mat. 45 (200), no., [5] C. Kenig, T. Toro, Harmonic measure on locally flat domains. Duke Math. J. 87 (997), no. 3, [6] M. Korey, Ideal Weights: Doubling Absolute Continuity with Asymptotically Optimal ounds, Ph.D Dissertation, Univ. of Chicago, (995). [7] D. Sarason, Functions of vanishing mean oscillation. Tran. Amer. Math. Soc. 207 (975),