Lp Regularity of the dirichlet problem for elliptic equations with singular drift

Publ. Mat. 50 (2006), 475–507 LpREGULARITY OF THE DIRICHLET PROBLEM FOR ELLIPTIC EQUATIONS WITH SINGULAR DRIFT Cristian Rios Abstract Let L0and L1be two elliptic operators in nondivergence form, with coefficients Aℓand drift terms bℓ,ℓ= 0,1 satisfying sup |Y−X|≤ δ(X) 2 |A0(Y)−A1(Y)|2+δ(X)2|b0(Y)−b1(Y)|2 δ(X)dX is a Carleson measure in a Lipschitz domain Ω ⊂Rn+1,n≥1, (here δ(X) = dist (X, ∂Ω)). If the harmonic measure dωL0∈A∞, then dωL1∈A∞. This is an analog to Theorem 2.17 in [8] for divergence form operators. As an application of this, a new approximation argument and known results we are able to extend the results in [10] for divergence form operators while obtaining totally new results for nondivergence form operators. The theorems are sharp in all cases. 1. Introduction and Background Given a bounded Lipschitz domain Ω ⊂Rn+1,n≥1, and an operator Lgiven by L=(div A∇(divergence fom) or A· ∇2(nondivergence form), the harmonic measure at X∈Ω, dωX L, is the unique Borel measure on ∂Ω such that for all continuous functions g∈ C (∂Ω), u(X) = Z∂Ω g(Q)dωX L(Q) 2000 Mathematics Subject Classification. 35J15, 35J25, 35J67, 35C15, 35R05, 35B30. Key words. Dirichlet problem, harmonic measure, absolute continuity, divergence, nondivergence, singular drift. 476 C. Rios is continuous in Ω and it is the unique solution to the Dirichlet problem (1.1) (Lu= 0 in Ω u=gon ∂Ω. Where we assume that the equality Lu= 0 holds in the weak sense for divergence form operators and in the strong a.e. sense for nondivergence form operators. Here A=A(X) is a symmetric (n+ 1) ×(n+ 1) matrix with bounded measurable entries, satisfying a uniform ellipticity condition (1.2) λ|ξ|2≤ξ·A(X)ξ≤Λ|ξ|2, X, ξ ∈Rn+1, for some positive constants λ, Λ. In the nondivergence case the entries of the matrix Aare assumed to belong to BMO (Ω) with small enough norm. For a given operator L, the harmonic measures dωX L,X∈Ω, are regular probability measures which are mutually absolutely continuous with respect to each other. That is, k(X, Y, Q) = dωY L dωX L (Q)∈L1dωX L, ∂Ω, X, Y ∈Ω, Q ∈∂Ω. By the Harnack’s principle the kernel function k(X, Y, Q) is positive and uniformly bounded in compact subsets of Ω×Ω×∂Ω. As a consequence, to study differentiability properties of the family dωX LX∈Ωwith respect to any other Borel measure dν on ∂Ω, it is enough to fix a point X0∈Ω and study dω dν , where dω =dωX0 Lis referred as the harmonic measure of Lon ∂Ω. If there is unique solvability of the continuous Dirichlet problem and a boundary Maximum Principle is available, then the well definition of the harmonic measure follows from Riesz’s representation theorem. Definition 1.1 (Continuous Dirichlet problem - CD).Given an elliptic operator L, we say that the continuous Dirichlet problem is uniquely solvable in Ω, and we say that CD holds for L, if for every continuous function gon ∂Ω, there exists a unique solution uof (1.1), such that u∈ C0ΩTW2,p (Ω) for some 1 ≤p≤ ∞. Remark 1.2.By Theorem 3.2 below [16], a sufficient condition for CD to hold for a nondivergence form operator L=A· ∇2is that there exists ρ > 0 depending on nand the ellipticity constants such that (1.3) kAkBMO(Ω) ≤ρ, where k·kBMO(Ω) denotes the BMO norm in Ω (see Definition 3.1 below). It is not known whether or not the continuous Dirichlet problem Dirichlet Problem 477 is uniquely solvable in the case of elliptic nondivergence form operators with just bounded measurable coefficients. On the other hand, even under the restrictions (1.3) for any ρ > 0 it is known [18] that the continuous Dirichlet problem has non-unique “good solutions”. That is, for any ρ > 0 there exists A(X)∈BMO (Ω) with kAkBMO(Ω) ≤ρand two sequences of C∞symmetric matrices A0,j and A1,j with the same ellipticity constants as A, such that Aℓ,j (X)→A(X) as j→ ∞ for a.e. X,ℓ= 0,1, and such that for some continuous function gon ∂Ω the solutions u0,j and u1,j to the Dirichlet problems (L0,ju0,j = 0 in Ω u0,j =gon ∂Ω,and (L1,ju1,j = 0 in Ω u1,j =gon ∂Ω, converge uniformly in ¯ Ω to different continuous limits u0and u1. Given two regular Borel measures µand ωin ∂Ω, dω ∈A∞(dσ) if there exist constants 0 < ε,δ < 1 such that for any boundary ball ∆ = ∆r(Q) and any Borel set E⊂∆, µ(E) µ(∆) < δ =⇒ω(E) ω(∆) < ε. The relation dω ∈A∞(dµ) is an equivalence relation [15], and any two measures related by the A∞property are mutually absolutely continuous with respect to each other. From classic theory of weights, if dω ∈ A∞(dµ) then there exists 1 < q < ∞such that the density h=dω dµ satisfies a reverse H¨older inequality with exponent q: 1 µ(∆) Z∆ hqdµ1 q ≤C1 µ(∆) Z∆ h dµ. This property is denoted dω ∈Bq(dµ), and dω ∈Bq(dµ) is equivalent to the fact that the Dirichlet problem (1.1) for the operator Lis solvable in Lp(dµ, ∂Ω), 1 p+1 q= 1 (see [7] for details). When µ=σ, the Euclidean measure, we write A∞for A∞(dσ). Definition 1.3 (LpDirichlet problem, Dp).Let Lbe an elliptic operator that satisfies CD and let µbe a doubling measure in ∂Ω. We say that the Lp(dµ)-Dirichlet problem is uniquely solvable in Ω, and we write that Dp(dµ) holds for L, if for every continuous function gon ∂Ω, the unique solution uof (1.1) satisfies kNukLp(dµ)≤CkgkLp(dµ), 478 C. Rios for some constant independent of g. Here Nu denotes the nontangential maximal function of uon ∂Ω. When µ=σis the Lebesgue measure on ∂Ω we simply say that Dpholds for L. Definition 1.4 (Carleson measure).Let Ω be an open set in Rn+1 and let µbe a nonnegative Borel measure on ∂Ω. For X∈∂Ω and r>0 denote by △r(X)={Z∈∂Ω : |Z−X|< r}and Tr(X)={Z∈Ω : |Z−X|< r}. Given a nonnegative Borel measure νin Ω, we say that νis a Carleson measure in Ωwith respect to µ, if there exist a constant C0such that for all X∈∂Ω and r > 0, ν(Tr(X)) ≤C0µ(△r(X)) . The infimum of all the constants C0such that the above inequality holds for all X∈∂Ω and r > 0 is called the Carleson norm of νwith respect to µin Ω. For conciseness, we will write ν∈C(dµ, Ω) when vis a Carleson measure in Ω, and we denote by kνkC(dµ,Ω) its Carleson norm. When µ=σis the Lebesgue measure on ∂Ω we just say that νis a Carleson measure in Ω. Definition 1.5. Throughout this work, Qγ(X) denotes a cube centered at Xwith faces parallel to the coordinate axes and sidelength γ; i.e. Qγ(X)=nY= (y1,...,yn+1)∈Rn+1 :|yi−xi|<γ 2, i=1,...,n+ 1o. When Xbelongs to a domain Ω, we write δ(X) = dist (X, ∂Ω). In particular, when Ω = Rn+1 +and X∈Ω, it follows that δ(X) = xn+1. In the remarkable work [7], the authors established a perturbation result relating the harmonic measures of two operators in divergence form. The analogue result was later obtained by the author in [17] for nondivergence form operators. Theorem 1.6 ([7]–[17]).Let Ld,0= div A0∇and Ld,1= div A1∇be two elliptic operators with bounded measurable coefficients in Ω, and let ωd,0and ωd,1denote their respective harmonic measures. Let σbe a doubling measure on ∂Ωand suppose that (1.4) δ(X)−1sup Y∈Qδ(X) 2√n |A0(Y)−A1(Y)|2dX ∈C(dσ, Ω) . If dωd,0∈A∞then dωd,1∈A∞. Also, if CD holds for the operators Ln,0=A0·∇2and Ln,1=A1·∇2(see Definition 1.1), then their respective harmonic measures ωn,0,ωn,1, satisfy dωn,0∈A∞⇒dωn,1∈A∞. Dirichlet Problem 479 The theorems above were stated in terms of the supremum of the differences |A0(Y)−A1(Y)|for Yin Euclidean balls |Y−X|<δ(X) 2. The above formulation is equivalent. In [8], Theorem 1.6 was extended to elliptic divergence form operators with a singular drift: Theorem 1.7 (Theorem 1.9, Chapter III of [8]).If LD,0= div A0∇+ b0· ∇ and LD,1= div A1∇+b1· ∇ where A0and A1satisfy (1.4), and bi=bi jn+1 j=1 ,i= 0,1satisfy (1.5) δ(X) sup Y∈Qδ(X) 2√n (X) |b1(Y)−b0(Y)|2dX ∈C(dσ, Ω) , then dωD,0∈A∞⇒dωD,1∈A∞. The results in Theorems 1.6 and 1.7 concern perturbation of elliptic operators. They provide solvability for the LqDirichlet problem (for some q > 1) for an operator L1given that there exists an operator L0 for which the LpDirichlet problem is solvable for some p > 1and the disagreement of their coefficients satisfy the Carleson measure conditions (1.4) and (1.5). In [10], the authors answer a different question: What are sufficient conditions on the coefficients Aand bso that a given operator LD= div A∇+bhas unique solutions for the Lp-Dirichlet problem for some p > 1? See also [9]. Theorem 1.8 ([10]).Let LD= div A∇+b· ∇, where Asatisfies (1.6) δ(X) sup |Y−X|≤ δ(X) 2 |∇A(Y)|2dX ∈C(dσ, Ω) , and bsatisfies δ(X) sup Y,Z∈Qδ(X) 2√n (X) |b(Y)−b(Z)|2dX ∈C(dσ, Ω) . Then dωLD∈A∞. 2. Statement of the results One of the main results in this work is an analog of Theorem 1.7 for nondivergence form operators. 480 C. Rios Theorem 2.1. Let LN,0=A0·∇2+b0·∇ and LN,1=A1∇2+b1·∇ where Aℓ=Aℓ ij n+1 i,j=1 and bℓ=bℓ jn+1 j=1 ℓ= 0,1are measurable coefficients and Aℓsatisfy the ellipticity condition (1.2) for ℓ= 0,1in a bounded Lipschitz domain Ω. Suppose that CD holds for LN,ℓ,ℓ= 0,1and that (2.1) sup Y∈Qδ(X) 2√n (X) |A1(Y)−A0(Y)|2+δ2(X)|b1(Y)−b0(Y)|2 δ(X)dX ∈C(dσ, Ω) , where σis Lebesgue’s measure on ∂Ω. Then dωN,0∈A∞⇒dωN,1∈A∞. That is, if the Lp-Dirichlet problem is uniquely solvable for LN,0in Ω, for some 1< p < ∞, then there exists 1< q1<∞such that the Lq-Dirichlet problem is uniquely solvable for LN,1in Ω,1≤q≤q1. As an application of this result, Theorem 1.6 (nondivergence case) and a simple averaging of the coefficients argument we obtain an analog to Theorem 1.8. Moreover, this averaging argument and Theorem 1.7 yield an extension of Theorem 1.8 to the case when condition (1.6) is replaced by the weaker assumption (1.4). Remark 2.2.In the divergence form case considered in [8], it was only necessary to assume that condition CD held for just the operator LD,0, and not both of LD,0and LD,1. As mentioned in Remark 1.2, in the nondivergence case there is no well defined notion of solution for operators with measurable coefficients satisfying the hypotheses of Theorem 2.1. Hence, without the CD assumption the harmonic measure for LN,1would not be defined in general. To make the statements of the results below more concise, we introduce the following definition. Definition 2.3 (Oscillation).For r > 0, the r-oscillation of a measurable function f(X) (scalar or vector-valued) at a point X, denoted oscrf(X), is given by oscrf(X) = sup Y,W ∈Qr(X) |f(W)−f(Z)|. Dirichlet Problem 481 Theorem 2.4. Let LD= div A∇+b· ∇ and LN=A· ∇2+b· ∇ be uniformly elliptic operators in divergence form and nondivergence form, respectively, with bounded measurable coefficient matrix Aand drift vector bin a bounded Lipschitz domain Ω. In the nondivergence case we assume that CD holds for LN. Suppose that the coefficients A,bsatisfy (2.2) oscδ(X) 2√n A(X)2 +δ2(X)oscδ(X) 2√n b(X)2 δ(X)dX ∈C(dσ, Ω) . Then dωLD∈A∞and dωLN∈A∞. That is, there exist indexes 1< pD, pN<∞such that the Lp-Dirichlet problem is uniquely solvable for LDin Ω,1≤p≤pD, and the Lq-Dirichlet problem is uniquely solvable for LNin Ω,1≤q≤pN. Remark 2.5.The following extensions and generalizations can be obtained (and might be subject for a subsequent work): (1) The techniques used to obtain Theorems 2.1 and 2.4 also yield analog results with the Euclidean measure dσ replaced by any doubling measure dµ on ∂Ω. (2) Appropriate parabolic versions of Theorems 2.1 and 2.4 are possible in the nondivergence case (the divergence case was considered in [8]). (3) Stronger conclusions can be obtained in Theorems 2.1 and 2.4 if the Carleson measure conditions are replaced by vanishing Carleson measures (the Carleson norm vanishes as the radius of the regions goes to zero). Under such assumptions, it can be shown that the harmonic measures dω·,0and dω·,1from Theorem 2.1 preserve the Apcondition for any 1 ≤p≤ ∞. That is, dω·,0∈Ap⇒dω·,1∈Ap. In the case of Theorem 2.4, it can be shown that under vanishing Carleson measure conditions dωLD∈A1and dωLN∈A1. 2.1. The theorems are sharp. In [7] it was shown that Theorem 1.6 is sharp for divergence form equations in two fundamental ways (see Theorems 4.11 and 4.2 in [7]). The examples provided in that work were constructed using Beurling-Ahlfors quasiconformal mappings on the half plane R2 +[2]. Quasi-conformal mappings preserve the divergence formstructure of an elliptic operator LD, but when composed with a nondivergence form operator LNthe transformed operator has first order drift terms. The more recent work [10] for divergence form operators (Theorem 1.8) can be applied to obtain regularity of elliptic equation in nondivergence form in the special case that the coefficient matrix satisfies (1.6). 482 C. Rios The approximation technique to be introduced in the next section, together with Theorem 2.1, show that Theorem 1.8 can be extended to operators in divergence and nondivergence form with coefficients satisfying the weaker condition (2.2). At the same time, this opens the door to extend the scope of the examples provided in Theorem 1.6 to this wider class of operators. The following theorem is a nondivergence analog to Theorem 4.11 in [7]. Theorem 2.6. Given any nonnegative function α(X)in R2 +={(x, t) : t > 0}such that α(X)satisfies the doubling condition: α(X)≤Cα (X0) for all X= (x, t),X0= (x0, t0) : |X−X0|<t0 2and such that sup Y∈Qδ(X) 2√n (X) α(Y)2 δ(X)dX /∈Cdσ, [0,1]2, where dσ is the Euclidean measure in ∂[0,1]2and δ((x, t)) = t. There exists a coefficients matrix Asuch that (1) the function a(X) = supY∈Qδ(X) 2√n (X)|A(Y)−I|, satisfies that for all I⊂R, 1 |I|ZZT(I) a2(x, y)dxdy y≤C"1 |I|ZZT(2I) α2(x, y)dxdy y+ 1#; (2) the function ˜a(X) = oscQδ(X) 2√n A(X), satisfies that for all I⊂R, 1 |I|ZZT(I) ˜a2(x, y)dxdy y≤C"1 |I|ZZT(2I) α2(x, y)dxdy y+ 1#; and (3) if LN=A· ∇2on R2 +, the elliptic measure dωLNis not in A∞(dx, [0,1]). The above theorem shows that the Carleson measure condition (1.4) in Theorem 1.6 is sharp also in the nondivergence case. In particular, it shows that the main result in [17] is sharp. The proof is a simple application of Theorem 4.11 in [7] and the approximation argument given in the section below (Lemma 3.9). Indeed, by Theorem 4.11 in [7] there exists a coefficients matrix Asuch that (1) and (2) hold for Aand (3) holds for dωLD, with LD= div A∇. For simplicity, let say that two elliptic operators L0and L1are simultaneously in A∞, and write L0≈ L2if their respective harmonic measures dωL0and dωL1satisfy: Dirichlet Problem 483 dωL0∈A∞⇔dωL1∈A∞. Using Lemma 3.9 we can construct a coefficient matrix A∗such that if L∗= div A∗∇=A∗· ∇2+b· ∇ and L∗ N=A∗· ∇2then Theorems 2.1 and 2.4 can be applied to show that LD≈ L∗≈ L∗ N≈ LN. See the proof of Theorem 2.1 in Section 4 for a detailed application of this technique. 2.2. Organization of the paper. In the following section we define several function spaces where the coefficients or the solutions to our equations will belong. The basic objects associated to the geometry of Lipschitz domains are also introduced. In this section we also list useful known properties of solutions and the harmonic measure, and we establish some auxiliary results that will allow us to treat the problems locally. In Section 4 we prove Theorem 2.1 by reducing it to a special case (Theorem 2.1). Theorem 2.4 then follows from the result just established and previous theory. Finally, in Section 4.1 we prove Theorem 2.1 by implementing the techniques from [17] (originally adapted from [7]) to this special case. Acknowledgement. We are grateful to the referee for useful comments and insights that added clarity and elegance to the exposition. In particular we wish to acknowledge the referee’s suggestions leading to a simplification of the proof of Theorem 2.4. 3. Preliminary results Given a weight win the Muckenphout class Ap(Ω), we denote by Lq(Ω, w), 1 ≤p≤q < ∞, the space of measurable functions fsuch that kfkLq(Ω,w)=ZΩ |f(x)|qw(x)dx<∞. And for a nonnegative integer k, we define the Sobolev space Wk,q (Ω, w) as the space of functions fin Lq(Ω, w) such that fhas weak derivatives up to order kin Lq(Ω, w). Under the assumption w∈Ap, the space Wk,q (Ω, w) is a Banach space and it is also given as the closure of C∞ 0(Ω) (smooth functions of compact support in Ω) under the norm kfkWk,q (Ω,w)= k X ℓ=0  ∇ℓf Lq(Ω,w), see [6], [11]. We recall now some definitions. 490 C. Rios from the definition of ˜µand since diam (Qi)≈dist Qi, ∂Rn+1 +, ˜µPj\△=ZPjT△ M(X)d˜σ(X)≈µ△Xj σ△XjZPjT△ d˜σ(X) ≈Cµ△Xj diam (Pi)nZPjT△ d˜σ(X)≈Cµ △Xj. Then ˜µ(△)=µ△\Ω+ ∞ X j=1 ˜µPj\△≈µ△\Ω+ ∞ X j=1 µ△Xj ≥Cµ Qd0(x0,0) \Rn× {0}, (3.4) the last inequality follows from a simple geometrical argument. From this and (3.3) we have RT(△)d˜ν(X)≤C˜µ(△) as wanted. From (3.4) we also have ˜µ(△)≈˜µ(2△) in this case. The following lemma estates the local character of the regularity of the harmonic measure. Lemma 3.10. Let Lbe an elliptic operator in divergence form or nondivergence form with drift bin a Lipschitz domain Ω; i.e. L=A·∇+b·∇ or L=A· ∇2+b· ∇ where Asatisfies the ellipticity condition (1.2). Suppose that bis locally bounded in Ωand it satisfies (3.5) δ(X) oscδ(X) 2√n b(X)2∈C(Ω) . Then dωL∈A∞if and only if there exists a finite collection of Lipschitz domains {Ω}N i=1 and compact sets Ki⋐∂ΩiT∂Ωsuch that Sn i=1 Ωi⊂Ω, ∂Ω⊂Sn i=1 Ki, and dωLi|Ki∈A∞(dσ), i = 1,...,N, where Lidenotes the restriction of Lto the subdomain Ωi. Proof: If b= 0, the result is an immediate consequence of the “main lemma” in [4] for the divergence case and the analog to the main lemma in the nondivergence case, contained in [5] (see also Lemma 3.8). The case b6= 0 then follows from Theorem 1.7 for the divergence case and Theorem 4.1 from next section for the nondivergence case. Indeed, by the mentioned theorems, if Lis the operator with drift band L0is the operator with the same second order coefficients but without a drift term, then dωL∈A∞(dσ)⇔dωL0∈A∞(dσ). On the other hand, by Lemma 3.9 with g(X) = b(X), the restriction of bto any Lipschitz Dirichlet Problem 491 subdomain Ω′⊂Ω also satisfies (3.5) in Ω′. Hence, by Theorems 1.7 and 4.1, for any doubling measure dσ′on ∂Ω′, we have dωL|Ω′∈A∞(dσ′)⇔ dωL0|Ω′∈A∞(dσ′). Now, for Ωi,Kias in the statement of Lemma 3.10, dσ|Ki, the restriction of dσ to the compact set Ki, can be extended to a doubling measure dσion ∂Ωiwith the same doubling constant. Then dωL|Ωi∈A∞dσi⇔dωL0|Ωi∈A∞dσi, which implies that dωL|Ωi|Ki∈A∞(dσ|Ki)⇔dωL0|Ωi|Ki∈A∞(dσ|Ki), where dωL|Ωi|Kidenotes the restriction to Kiof the harmonic measure of Lin Ωi, with a similar definition for dωL0|Ωi|Ki. This shows that Lemma 3.10 in the case b6= 0 follows from the case b= 0. 4. Proofs of the Theorems We recall that Qr(X) denotes a cube centered at Xwith sidelength r, and δ(x) denotes the distance of Xto the boundary (see Definition 1.5). The proof of Theorem 2.1 relies on the following special case. Theorem 4.1. Let ˜ LN,0=A· ∇2and ˜ LN,1=A∇2+b· ∇ where A= (Aij )n+1 i,j=1 and b= (bj)n+1 j=1 are bounded, measurable coefficients and Asatisfy the ellipticity condition (1.2) in a Lipschitz domain Ω. Suppose that CD holds for ˜ LN,ℓ,ℓ= 0,1in Ωand that (4.1) δ(X) sup Y∈Qδ(X) 2√n (X) |b(Y)|2dX ∈C(dσ, Ω) . Then dω ˜ LN,0∈A∞⇒dω ˜ LN,1∈A∞. We defer the proof of this result (which contains the main substance of Theorem 2.1) to next section. Now we obtain Theorem 2.1 from Theorem 4.1. Proof of Theorem 2.1: Let LN,0=A0·∇2+b0·∇ and LN,1=A1∇2 +b1·∇ where Aℓ,bℓand LN,ℓ satisfy the hypotheses of Theorem 2.1 for ℓ= 0,1. Then if dωLN,0∈A∞, by Theorem 4.1 it follows that dωL∗ N,0∈A∞where L∗ N,0=A0· ∇2. By Theorem 1.6 [17] and Theorem 4.1 again, we have that if L∗ N,1=A1· ∇2, then dωL∗ N,0∈A∞⇒dωL∗ N,1A∞⇒dωLN,1∈A∞, as wanted. Theorem 2.4 will follow by reduction to the special case Ω = Rn+1 +, the localization given by Lemma 3.10 and an approximation of the coefficients matrix Aby appropriate smooth matrices. 492 C. Rios Proof of Theorem 2.4: Given P0∈∂Ω let X= (x, t) be a coordinate system such that P0= (x0, t0)∈∂Ω and there exists a Lipschitz function ψ:Rn→Rdefining a local coordinate system of Ω in a neighborhood of P0. That is, for some r0=r0(Ω) >0 we have ∂Ω\{|x−x0|< r0} × R={(x, ψ (x)) : |x−x0|< r0} Ω′={(x, t) : |x−x0|< r0, ψ (x)< t < ψ (x) + r0} ⊂ Ω. (4.2) Let ηs(y) = s−nηy s, where ηis an even C∞approximate identity in Rnsupported in |y| ≤ 1 2. Set ρ(y, s) = (y, c0s+F(y, s)) with F(y, s) = ηs∗ψ(y) = RRnηs(y−z)ψ(z)dz. We have ∇Yρ=I∇yF 0c0+∂F ∂s . Since  ∂F ∂s  ∞≤Cnk∇yψk∞, where Cn=nRη(y)|y|dσ, (note that, for appropriate η,Cnis a universal constant), taking c0= 1+Cnk∇yψk∞,ρ is a 1-1 map of Rn+1 +onto {(x, t) : t > ψ (x)}, moreover, ρis bi-Lipschitz and 1 ≤ |det ∇Yρ| ≤ 1 + 2Cnk∇yψk∞. This transformation gives rise to the Dahlberg-Kenig-Stein adapted distance function. For α > 0, let Φα={y:|y−x0|< α} × 0,α c0, and Ωα=ρ(Φα). For α=α0small enough depending only on k∇yψk∞and n, we have Ωα0⊂Ω′, where Ω′is given by (4.2). Moreover, Ωα0is a Lipschitz domain with Lipschitz constant depending only on the constant of Ω. We note that because of Theorem 1.7 from [8] in the divergence case and because of Theorem 2.1 in the nondivergence case we might assume b≡0. We will first consider the divergence case, suppose that LD= div A∇is a uniformly elliptic operator in divergence form with bounded measurable coefficient matrix Asatisfying (2.2), i.e. (4.3) δ−1(X) oscQδ(X) 2√n (X)A(X)2dX ∈C(dσ, Ω) . Since P0is an arbitrary point on ∂Ω and ∂Ω is compact, by Lemma 3.10, to prove Theorem 2.4 it is enough to prove that if ωis the harmonic measure for LDin Ωα0, then (4.4) ω|K∈A∞(dσ|K), where K⊂∂Ωα0is the compact set given by (4.5) K=ρnY= (y, s) : |y−x0| ≤ α0 6, s = 0o⊂∂Ωα0. Dirichlet Problem 493 For simplicity, we will write Ω = Ωα0and Φ = Φα0. Since ψis Lipschitz, it follows that the transformation ρ: Φ →Ω can be extended to a homeomorphism from Φ to Ω and such that the restriction of ρto ∂Φ is a bi-Lipschitz homeomorphism from ∂Ω to ∂Ω. Indeed, since {ηs}s>0is a smooth approximation of the identity, it easily follows that ρrestricted to ∂Φ is given by (4.6) ρ(Y)=ρ(y, s)=           (y, ψ (y)) , s = 0 (y, c0s+F(y, s)) ,0< s < α0 c0 ,|y−x0|=α0 y, c0 α0 c0 +Fy, α0 c0 s=α0 c0 whenever Y∈∂Φ. If ˜ δ(Y) = dist (Y, ∂Φ), then for some constant ˜ C depending only on k∇yψk∞and n, the following estimate holds for the distance functions δand ˜ δ: (4.7) ˜ C−1˜ δ(Y)≤δ(ρ(Y)) ≤˜ Cδ (Y), Y ∈Φ. To see this, let Y0∈Φ and let ˜ δ0=δ(Y0). Now, δ(ρ(Y0)) = dist (ρ(Y0), ∂Ω) = |ρ(Y0)−X′|for some X′∈∂Ω. Let Y′=ρ−1(X′), and X0= ρ(Y0), thus, Y′∈∂Φ and since ρ−1is Lipschitz in Φ, we have ˜ δ(Y0)≤ |Y0−Y′|=ρ−1(X0)−ρ−1(X′)≤C|X0−X′|=Cδ (ρ(Y0)) , with C=Cn, k∇yψk∞. The other inequality in (4.7) follows in a similar manner. The Lebesgue measure σon ∂Ω induces a doubling measure ˜µon ∂Φ by the relation ˜µ(E) = σ(ρ(E)) ,for any Borel set E⊂∂Φ. Let ˜σbe the Lebesgue measure on ∂Φ, then from the definition of ˜µand the fact that ρis bi-Lipschitz it easily follows that d˜µ d˜σ≈1. Hence we can replace ˜µby ˜σin our calculations. Now, if u(x, t) is a solution of LDu= divXA∇Xu= 0 in Ω, then v(y, s) = u(ρ(y, s)), defined in Φ, is a solution of ˜ LDv= divY˜ A∇Yv= 0, where (4.8) ˜ A(Y) = (∇Yρ)−1(Y)t A(ρ(Y)) (∇Yρ)−1(Y) det (∇Yρ) (Y). We claim that ˜ Asatisfies (4.9) ˜ δ(Y)−1oscQδ(Y) 2√n (Y)˜ A(Y)2dY ∈C(d˜σ, Φ) . 494 C. Rios Let Y0∈Φ let Y1,Y2such that |Y1−Y0| ≤ 1 2δ0and |Y2−Y0| ≤ 1 2δ0, where δ0=˜ δ(Y0), then by (4.8)  ˜ A(Y1)−˜ A(Y2) 2 ˜ δ0 =˜ δ−1 0(∇Yρ)−1(Y1)t A(ρ(Y1)) (∇Yρ)−1(Y1) det (∇Yρ) (Y1) −(∇Yρ)−1(Y2)t A(ρ(Y2)) (∇Yρ)−1(Y2) det (∇Yρ) (Y2) 2 ≤C(∇Yρ)−1(Y1)t−(∇Yρ)−1(Y2)t 2 ˜ δ0 ×A(ρ(Y1)) (∇Yρ)−1(Y1) det (∇Yρ) (Y1) 2 +C(∇Yρ)−1(Y1)−(∇Yρ)−1(Y2) 2 ˜ δ0 ×(∇Yρ)−1(Y2)t A(ρ(Y1)) 2 |det (∇Yρ) (Y1)|2 +C|det (∇Yρ) (Y1)−det (∇Yρ) (Y2)|2 ˜ δ0 ×(∇Yρ)−1(Y2)tA(ρ(Y1)) (∇Yρ)−1(Y2) 2 +C|A(ρ(Y1)) −A(ρ(Y2))|2 ˜ δ0 ×(∇Yρ)−1(Y2)t 2(∇Yρ)−1(Y2) det (∇Yρ) (Y2) 2. (4.10) We will use the following fact: Dirichlet Problem 495 Lemma 4.2. For Ω,Φ,σ,˜µ,δ,˜ δand ρas above, the functions r1(Y) = 1 ˜ δ(Y) oscQ˜ δ(Y) 2√n (Y)(∇Yρ)−1(Y) 2 and r2(Y) = 1 ˜ δ(Y) oscQ˜ δ(Y) 2√n (Y)(det (∇Yρ) (Y)) 2 defined in Φ, satisfy [r1(Y) + r2(Y)] dY ∈C(d˜σ, Φ), where ˜σis the Lebesgue measure on ∂Φ. The lemma above follows from the fact that (4.11) ˜ δ(Y)∇2ρ 2dY ∈C(d˜σ, Φ) . This property is discussed in [10], and it can be obtained as an application of the characterization of A∞in terms of Carleson measures given in [7]. Then (4.9) follows by applying (4.3), (4.7) and Lemma 4.2 to (4.10). We recall that Φ=Φα0where Φαis given by Φα={y:|y−x0|< α}× 0,α c0. Denote by Φ± α={y:|y−x0|< α}×−α c0,α c0and let ν(Y)∈ C∞ 0Φ± α0such that 0 ≤ν≤1, ν≡1 in Φ± α0 3 and ν≡0 in Φ± α0\Φ± 2α0 3 . For Y∈Rn+1 +, let ˜ A∗(Y) = ν(Y)˜ A(Y) + (1 −ν(Y)) I, where Iis the (n+ 1)×(n+ 1) identity matrix. It follows that ˜ A∗(Y) is an elliptic matrix function, with the same ellipticity constants as ˜ A. The measure ˜σextends trivially from ∂ΦT∂Rn+1 +to ∂Rn+1 +, we dub this extension (which is just the Euclidean measure) d˜σ∗. With this definitions, because of (4.9), for Y= (y, t)∈Rn+1 +,˜ A∗satisfies (4.12) oscQt 2√n (Y)˜ A∗(Y)2 t∈Cd˜σ∗,Rn+1 +, (i.e.: is a Carleson measure in Rn+1 +with respect to ˜σ∗). Where Qγ(Y) is the cube centered at Ywith faces parallel to the coordinate axes and sidelength γt. Now we will construct a smooth approximation of ˜ A∗via an n+1 dimensional approximate identity. Let Ptf(x, t) = ZZ t−n−1ϕx−y t,t−s tf(y, s)dy ds, 496 C. Rios where ϕ∈ C∞ 0Rn+1is supported in the ball or radius α < 1 at the origin (αto be chosen later) and RRϕ= 1. Since Pt1≡1 we have ∇x,tPt1≡0 and therefore ˜ A∗∗ =Pt˜ A∗satisfies ∇˜ A∗∗ =ZZ t−n−1∇x,tϕx−y t,t−s t˜ A∗(y, s)dy ds =−ZZ t−n−1∇y,sϕx−y t,t−s t˜ A∗(y, s)dy ds =−ZZ t−n−1∇y,sϕx−y t,t−s t˜ A∗(y, s)−C(x, t)dy ds for any function C(X) at our disposal. Taking C(X) = ˜ A∗(X) it follows that for αsmall enough ˜ E∗(Y) = sup Z∈Q0(Y)∇˜ A∗∗ (Z)≤C oscQt 2√n (Y)˜ A∗ t and ˜ E∗∗ (Y) = sup Z∈Q0(Y) ˜ A∗∗ (Z)−˜ A∗(Z)≤CoscQt 2√n (Y)˜ A∗ where Q0(Y) = Qδ(Y) 6√n (Y) and Cis a universal constant (this constant depends on the Lipschitz norm of ϕ, which we can assume only depends on the dimension n). From (4.12) it follows that (4.13) tE∗(Y)2∈Cd˜σ∗,Rn+1 +and ˜ E∗∗ (X)2 tdX ∈Cd˜σ∗,Rn+1 +. Moreover, from the definitions it is easy to check that ˜ A∗∗ is elliptic with the same ellipticity constants as ˜ A. Let now ˜ L∗ D= divY˜ A∗∇Yand ˜ L∗∗ D= divY˜ A∗∗∇Y. By the Carleson measure property of ˜ E∗∗, and Lemma 3.9, ˜ L∗ Dand ˜ L∗∗ Dsatisfy the hypotheses of Theorem 1.7 in Φ (with respect to the measure ˜σ), therefore, if ˜ω∗and ˜ω∗∗ denote the harmonic measures of ˜ L∗ Dand ˜ L∗∗ Din Φ, respectively, we have that ˜ω∗∈A∞⇔˜ω∗∗ ∈A∞. On the other hand, because of the Carleson measure property of ˜ E∗, and Lemma 3.9, ˜ L∗∗ Dsatisfies the hypotheses of Theorem 1.8 in Φ; and therefore ˜ω∗∗ ∈A∞. From what we just proved it follows that ˜ω∗∈A∞. Dirichlet Problem 497 Let L∗ Ddenote the pull-back of ˜ L∗ Dfrom Φ to Ω through the mapping ρ. Since the mapping ρ:∂Φ→∂Ω is bi-Lipschitz, and if ω∗denotes the harmonic measure of L∗ Din Ω, then ω∗∈A∞. This follows directly from the definitions of the A∞class, the harmonic measure ω∗and L∗ D. On the other hand, the operator L∗ Dcoincides with LD= divXA∇X in ρΦα0 3, hence an application of Theorem 1.7 and the“main lemma” in [4] (see Lemma 3.10), implies that ω|K∈A∞(dω∗|K), where ωis the harmonic measure of LDrestricted to the compact set Kgiven by (4.5). This, in turn, implies that ω|K∈A∞(dσ|K) and proves (4.4), hence Theorem 2.4, for the divergence case. We now consider the nondivergence case. Let LN=A· ∇2where Asatisfies (4.3). Let Ω = Ωα0, Φ = Φα0, and ρbe as before. Also, for Y∈Φ, let ˜ A∗∗ (Y) be as above, and define L∗∗ =A∗∗ · ∇2, where A∗∗ (X) = (∇Yρ)ρ−1(X)t˜ A∗∗ ρ−1(X)(∇Yρ)ρ−1(X) ×det (∇Yρ)ρ−1(X)−1. Let Ω1 3=ρΦα0 3, we claim that A∗∗ (X) satisfies the following sup Z∈Qδ(X) 2√n (X) |A(Z)−A∗∗ (Z)|2 δ(X)dX ∈Cdˆσ, Ω1 3,and(4.14) sup Z∈Qδ(X) 2√n (X) δ(X)|∇A∗∗ (Z)|2dX ∈Cdˆσ, Ω1 3 (4.15) where ˆσis the Lebesgue measure on ∂Ω1 3. Taking these properties for granted, by (4.15), (4.3) and Theorem 1.8 applied to the operator L∗∗, we have that if ω∗∗ is the harmonic measure of L∗∗ on ∂Ω1 3, then ω∗∗ ∈A∞. On the other hand, by (4.14) and Theorem 2.1 applied to the operators L and L∗∗, from ω∗∗ ∈A∞we conclude that ω∈A∞, where ωis the harmonic measure of Lon ∂Ω1 3. This finishes the proof of Theorem 2.4 in the nondivergence case. It only rests to establish properties (4.14) and (4.15). Let Z∈ Qδ(X) 2√n (X) and let W=ρ−1(Z), then from the definitions of ˜ Aand A∗∗ we have |A(Z)−A∗∗ (Z)|=((∇Yρ) (W))th˜ A(W)−˜ A∗∗ (W)i ×(∇Yρ) (W) (det (∇Yρ) (W))−1. 498 C. Rios From (4.7) and the fact that ρis bi-Lipschitz, and since ρ−1Ω1 3= Φα0 3, |A(Z)−A∗∗ (Z)|2 δ(Z)≈ ˜ A(W)−˜ A∗∗ (W) 2 ˜ δ(W)= ˜ A∗(W)−˜ A∗∗ (W) 2 ˜ δ(W). Applying the proof of Lemma 3.9 to ˜ δ(W)−1oscα˜ δ(W)˜ A∗−˜ A∗∗(W), from the second property in (4.13) it follows that for some 0 < c < 1 supW∈Qc˜ δ(Y)(Y) ˜ A∗(W)−˜ A∗∗ (W) 2 ˜ δ(Y)∈Cdµ, Φα0 3, where µis the Lebesgue measure on ∂Φα0 3; (4.14) then follows from the fact that ρis bi-Lipschitz. Now, by the product rule of differentiation ∇A∗∗ =∇Xn(∇Yρ)t˜ A∗∗ (∇Yρ) det (∇Yρ)−1o =n∇X(∇Yρ)to˜ A∗∗ (∇Yρ) det (∇Yρ)−1 + (∇Yρ)tn∇X˜ A∗∗ (∇Yρ)odet (∇Yρ)−1 + (∇Yρ)t˜ A∗∗ {∇X(∇Yρ)}det (∇Yρ)−1 + (∇Yρ)t˜ A∗∗ (∇Yρ)n∇Xdet (∇Yρ)−1o. Applying the chain rule in each term, we see that ∇A∗∗ satisfies (4.15) because of the first property in (4.13), (4.11), and the boundedness of |∇Yρ|. 5. Proof of Theorem 4.1 In the spirit of [7] (see also [17]) we will obtain Theorem 4.1 as a consequence of the following perturbation result. Theorem 5.1. Let ˜ LN,0=A· ∇2and ˜ LN,1=A∇2+b· ∇ where A= (Aij)n+1 i,j=1 and b= (bj)n+1 j=1 are bounded, measurable coefficients and Asatisfy the ellipticity condition (1.2). Suppose that CD holds for ˜ LN,ℓ, ℓ= 0,1. Let G0(X, Y )denote the Green’s function for ˜ LN,0in Ωand set G0(Y) = G0(0, Y ). There exists ε0>0which depends only on n,λ, Dirichlet Problem 499 Λand Ωsuch that if (5.1) G0(X) sup Y∈Qδ(X) 2√n (X) |b|2dX, X ∈Ω, is a Carleson measure in Ωwith respect to dω ˜ LN,0on ∂Ωwith Carleson norm bounded by ε0, i.e., G0(X) sup Y∈Qδ(X) 2√n (X) |b|2dX ∈Cdω ˜ LN,0,Ω,        G0(X) sup Y∈Qδ(X) 2√n (X) |b|2dX      C“dω ˜ LN,0,Ω” ≤ε0, then dω ˜ LN,1∈B2dω ˜ LN,0. Where B2dω ˜ LN,0denotes the reverse H¨older class of dω ˜ LN,0with exponent 2. We defer the proof of Theorem 5.1 to the next subsection, and prove now Theorem 4.1, we follow the argument in [7]. Let △r(Q) be the boundary ball △r(Q)={P∈∂Ω : |Q−P|< r}, and denote by Tr(Q) the Carleson region in Ω associated to △r(Q), Tr(Q)={X∈Ω : |X−Q|<r}. By Lemma 3.10 we may assume that b(X)≡0 if δ(X)> r0for some fixed (small) r0>0. To prove Theorem 4.1 it is enough to show that if ˜ω1=ω˜ LN,1with ˜ LN,1as in the statement of the theorem, then for all Q∈∂Ω, (5.2) ˜ω1|△r0(Q)∈A∞. For Q∈∂Ω, r > 0, α > 0, let Γα,r (Q) be a nontangential cone of fixed aperture αand height r, i.e. Γα,r (Q) = {X∈Ω : |X−Q|<(1 + α)δ(X)<(1 + α)r}. For a fixed α0>0 to be determined later, let Er(Q) be given by Eb,r (Q) =     ZΓα0,r(Q) δ(X)1−nsup Y∈Qδ(X) 2√n (X) |b|2dX     1 2 , Q ∈∂Ω. Fix α0,r0, such that Γα0,r0(Q)⊂T2r0(Q) for all Q∈∂Ω. 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Rios, Sufficient Conditions for the absolute continuity of the nondivergence harmonic measure, Ph.D. Thesis, University of Minnesota, Minneapolis (2001). [17] C. Rios, The LpDirichlet problem and nondivergence harmonic measure, Trans. Amer. Math. Soc. 355(2) (2003), 665–687 (electronic). [18] M. V. Safonov, Nonuniqueness for second-order elliptic equations with measurable coefficients, SIAM J. Math. Anal. 30(4) (1999), 879–895 (electronic). Department of Mathematics Trinity College 300 Summit Street Hartford, CT 06106 USA E-mail address:[email protected] Primera versi´o rebuda el 10 de novembre de 2005, darrera versi´o rebuda el 21 de mar¸c de 2006.