POISSON KERNEL CHARACTERIZATION OF REIFENBERG FLAT CHORD ARC DOMAINS

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1 Ann. Scient. Éc. Norm. Sup., e série, t. 36, 2003, p. 323 à 0. POISSON KERNEL CHARACTERIZATION OF REIFENBERG FLAT CHORD ARC DOMAINS BY CARLOS E. KENIG AND TATIANA TORO 2 ABSTRACT. In this paper we prove the conjecture stated by the authors in Free boundary regularity for harmonic measures and Poisson kernels Ann. of Math ) 369 5) concerning the free boundary regularity problem for the Poisson kernel below the continuous threshold. We show that if Ω is a Reifenberg flat chord arc domain, and the logarithm of the Poisson kernel has vanishing mean oscillation then the unit normal vector to the boundary also has vanishing mean oscillation Éditions scientifiques et médicales Elsevier SAS RÉSUMÉ. Dans cet article, on démontre la conjecture proposée par les auteurs dans Free boundary regularity for harmonic measures and Poisson kernels Ann. of Math ) 369 5) concernant la régularité de la frontière libre pour le noyau de Poisson au-dessous du seuil de continuité. On prouve que si Ω est un domaine corde-arc Reifenberg plat tel que le logarithme du noyau de Poisson appartienne à VMO, alors le vecteur unitaire normal à la frontière appartient aussi à VMO Éditions scientifiques et médicales Elsevier SAS. Introduction The main goal of this paper is to present a general blow up argument see Section ) which combines geometric and analytic information about the free boundary regularity problem for the Poisson kernel. This technique allows us to provide a complete characterization of Reifenberg flat chord arc domains via potential theory. In particular we prove the conjecture stated in [8], and show that the weak regularity of the Poisson kernel of a domain fully determines the geometry of its boundary. Namely we show that if Ω is a δ-reifenberg flat chord arc domain for δ>0 small enough, and the logarithm of its Poisson kernel has vanishing mean oscillation then the unit normal vector to the boundary also has vanishing mean oscillation. In our context the mean oscillation of the logarithm of the Poisson kernel, or of the unit normal vector replace stronger notions of regularity. As in Alt and Caffarelli s work see []) we show that at flat points of the boundary, the oscillation of the Poisson kernel controls the geometry of the boundary. The difference between our work and the work in [] is that we measure the oscillation in an integral sense BMO estimates) while they do so in a pointwise sense Hölder estimates). We now introduce formally the definitions needed to state our main results. We indicate how the main theorem follows from the other results, and sketch briefly the contents of each section of the paper. We always assume that n 2. The author was partially supported by the NSF. 2 The author was partially funded by the NSF and the Alfred P. Sloan Foundation /03/03/ 2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved

2 32 C.E. KENIG AND T. TORO DEFINITION.. Let Σ R n+ be a locally compact set, and let δ>0. We say that Σ is δ-reifenberg flat if for each compact set K R n+, there exists R K > 0 such that for every Q K Σ and every r 0,R K ] there exists an n-dimensional plane LQ, r) containing Q such that.) r D[ Σ BQ, r),lq, r) BQ, r) ] δ. Here BQ, r) denotes the n +)-dimensional ball of radius r and center Q,andD denotes the Hausdorff distance. Recall that for A, B R n+, D[A, B]=sup { da, B): a A } +sup { db, A): b B }. Note that the previous definition is only significant for δ>0 small. This notion was initially introduced by Reifenberg who proved the following remarkable theorem. THEOREM [2,23]. There exists δ>0 depending only on n so that if Σ is δ-reifenberg flat then locally Σ is a topological disc. We denote by { θq, r)=inf L r D[ Σ BQ, r),l BQ, r) ] }.2), where the infimum is taken over all n-planescontaining Q. DEFINITION.2. Let Σ R n+, we say that Σ is Reifenberg flat with vanishing constant if it is δ-reifenberg flat for some δ>0 and for each compact set K R n+ lim sup r 0 Q Σ K θq, r)=0. DEFINITION.3. A measure µ in R n+ is said to be Ahlfors regular if there exists C> such that for Q spt µ and r>0.3) C r n µ BQ, r) ) Cr n. DEFINITION.. Let Ω R n+ be a set of locally finite perimeter see [7]), is said to be Ahlfors regular if the surface measure to the boundary, i.e., the restriction of the n-dimensional Hausdorff measure to, σ = H n, is Ahlfors regular. DEFINITION.5. Let Ω R n+. We say that Ω has the separation property if for each compact set K R n+ there exists R>0 such that for Q K and r 0,R] there exists an n-dimensional plane LQ, r) containing Q and a choice of unit normal vector to LQ, r), n Q,r satisfying.) { T + Q, r)= X =x, t)=x + tn Q,r BQ, r): x LQ, r), t> } r Ω, and.5) { T Q, r)= X =x, t)=x + tn Q,r BQ, r): x LQ, r), t< } r Ω c. e SÉRIE TOME N 3

3 REIFENBERG FLAT CHORD ARC DOMAINS 325 Moreover if Ω is an unbounded domain we also require that R n+ \ divide R n+ into two distinct connected components Ω and int Ω c. The notation x, t)=x + t n Q,r is used to denote a point in R n+. The first component, x, of the pair belongs to an n-dimensional affine space whose unit normal vector is n Q,r. The second component t belongs to R. From the context it will always be clear what affine hyperplane x belongs to, and what the orientation of the unit normal vector is. DEFINITION.6. Let δ 0,δ n ),whereδ n is chosen appropriately see note below) and let Ω R n+. We say that Ω is a δ-reifenberg flat domain or a Reifenberg flat domain if Ω has the separation property and is δ-reifenberg flat. Moreover if Ω is an unbounded domain we also require that.6) sup sup r>0 Q θq, r) <δ n. When we consider δ-reifenberg flat domains in R n+ we assume that δ n > 0 is small enough, in order to ensure that we are working on NTA domains see definition in Appendix A, see also [] and [9, Theorem 3.]). DEFINITION.7. A set Ω R n+ is said to be a Reifenberg flat domain with vanishing constant if Ω is a Reifenberg flat domain, and for every compact set K R n+.7) lim sup r 0 Q K θq, r)=0. DEFINITION.8. A set of locally finite perimeter Ω R n+ see [7]) is said to be a chord arc domain,ifω is an NTA domain whose boundary is Ahlfors regular. DEFINITION.9. Let δ 0,δ n ). A set of locally finite perimeter Ω R n+ is said to be a δ-reifenberg flat chord arc domain,ifω is a δ-reifenberg flat domain whose boundary is Ahlfors regular. Remarks. )SinceΩ is a δ-reifenberg flat domain with δ>0 small enough, then for each compact set K R n+ so that K there exists R K > 0 so that for every Q K and every r 0,R K ) there exists an n-plane LQ, 2r) containing Q andsuchthat.8).9) and 2r D[ BQ, 2r); LQ, 2r) BQ, 2r) ] 2δ, { X =x, t)=x + t } n Q, 2r): x LQ, 2r),t>δr BQ, 2r) Ω,.0) { X =x, t)=x + t n Q, 2r): x LQ, 2r),t< δr } BQ, 2r) Ω c. Here n Q, 2r) denotes the appropriate unit normal vector to LQ, 2r), where we choose LQ, 2r) to be the best possible approximating n-plane to at Q and at radius 2r. See Remark. in [8].) 2) By Remark.2 in [8] we have that if Ω is a set of locally finite perimeter which is a Reifenberg flat domain then the topological boundary of Ω and its measure theoretic boundary agree. DEFINITION.0. Let δ 0,δ n ). A set of locally finite perimeter Ω see [7]) is said to be a δ-chord arc domain or a chord arc domain with small constant if Ω is a δ-reifenberg flat domain,

4 326 C.E. KENIG AND T. TORO is Ahlfors regular and for each compact set K R n+ there exists R>0 so that.) sup n Q, R) <δ. Q K Here n denotes the unit normal vector to the boundary,.2) n Q, R)= sup 0<s<R BQ,s) ) n n Q,s 2 2 dσ and n Q,s = BQ,s) ndσ. We only use the notation δ-reifenberg flat domain, δ-reifenberg flat chord arc domain or δ-chord arc domain when we want to emphasize the dependence on δ, otherwise we simply refer to them as Reifenberg flat domain, Reifenberg flat chord arc domain or chord arc domain with small constant. Note that a chord arc domain with small constant is a Reifenberg flat chord arc domain. DEFINITION.. A set of locally finite perimeter is said to be a chord arc domain with vanishing constant if it is a chord arc domain with small constant and for each compact set K R n+.3) lim sup r 0 Q K n Q, r)=0. We now present the definition of bounded resp. vanishing) mean oscillation functions on the boundary of a chord arc domain Ω; i.e.,bmo) resp. VMO)). DEFINITION.2. Let Ω R n+ be a chord arc domain. Let f L 2 loc dσ), we say that f BMO) if ) f =sup sup f f Q,r 2 2.) dσ <. r>0 Q Here f Q,r = BQ,r) fdσ,andσ = H n BQ,r). DEFINITION.3. Let Ω R n+ be a chord arc domain. We denote by VMO) the closure in BMO) of the set of uniformly continuous bounded functions defined on. The reader will remark that Definition.3 is slightly different than the one used in [8] see Definition.8 in [8]). These 2 definitions coincide in the case when Ω is bounded. In the case when Ω is unbounded, Definition.3 above provides good control on the behavior of f in large balls see discussion below). This is not the case for the definition used in [8]. Let Ω be a Reifenberg flat chord arc domain either bounded or unbounded), and let X Ω; then the harmonic measure with pole at X, ω X and σ = H n are mutually absolutely continuous see [] and [25]). The Radon Nikodym theorem ensures that the corresponding Poisson kernel k X Q)= dωx dσ Q)= GX, ) Q) L n locdσ). Here GX, ) denotes the Green s function of Ω with pole at X and n = n denotes the normal derivative at the boundary. We prove that if Ω is a Reifenberg flat chord arc domain, and log k X VMOdσ) then Ω is a Reifenberg flat domain with vanishing constant. e SÉRIE TOME N 3

5 REIFENBERG FLAT CHORD ARC DOMAINS 327 THEOREM.. Assume that ) Ω R n+ is a δ-reifenberg flat chord arc domain for some δ>0 small enough; 2) log k X VMOdσ). Then Ω is a Reifenberg flat domain with vanishing constant. As mentioned above, under the previous assumptions we conclude also that the harmonic measure is asymptotically optimally doubling see Definition.5 in [8] and Theorem. in [9]). Hence combining Theorem. above with Theorems 5.3 or 5. in [8] and taking into account our modified version of VMOdσ)) we conclude that the following results hold both for bounded and unbounded domains. THEOREM.2. Assume that ) Ω R n+ is a chord arc domain with small enough constant. 2) log k X VMO). Then Ω is a chord arc domain with vanishing constant. Furthermore when Ω is an unbounded Reifenberg flat chord arc domain, the harmonic measure with pole at infinity, ω and σ = H n are mutually absolutely continuous. The Radon Nikodym theorem ensures that the Poisson kernel with pole at infinity hq)= dω dσ Q) L loc dσ). As before we prove that if Ω is an unbounded Reifenberg flat chord arc domain, and log h VMOdσ) then Ω is a Reifenberg flat domain with vanishing constant. THEOREM.3. Assume that ) Ω R n+ is an unbounded δ-reifenberg flat chord arc domain for some δ>0 small enough; 2) log h VMOdσ). Then Ω is a Reifenberg flat domain with vanishing constant. Moreover if h =H n -a.e. in, then Ω is a half space. Combining Theorem.3 above with the Main Theorem in [8] and taking into account our modified version of VMOdσ)) we conclude that the following result holds. THEOREM.. Assume that ) Ω R n+ is an unbounded chord arc domain with small enough constant; 2) log h VMO). Then Ω is a chord arc domain with vanishing constant. A more in depth analysis of the blow-up sequence described in Section allows us to prove that the conjecture stated in [8] holds. MAIN THEOREM. Assume that ) Ω R n+ is a unbounded) δ-reifenberg flat chord arc domain for some δ>0 small enough; 2) log k X VMOdσ) log h VMOdσ)). Then Ω is a chord arc domain with vanishing constant, i.e., n VMOdσ). Remark. Note that in [9] we have shown the converse of this, namely that if Ω R n+ is a δ-reifenberg flat chord arc domain and n VMOdσ) then log k X VMOdσ) for every X Ω. Jerison see [3]) introduced this end point problem in higher dimensions, but treated it under more restrictive assumptions, namely that the boundary is given locally as a Lipschitz graph, and the normal derivative data is continuous as opposed to having vanishing mean oscillation. His paper is based on the work of Jerison Kenig [5] and first points out the

6 328 C.E. KENIG AND T. TORO connection with the work of Alt and Caffarelli []. There is an error in Lemma of Jerison s paper. Nevertheless in our previous work see [8]) we made considerable use of the ideas in [3]. In this paper we bypass this approach. The basic difference between the Main Theorem above and the Main Theorem in [8] see Section 5) is that in [8] we needed to assume that the harmonic measure was asymptotically optimally doubling and that n had small BMO norm. The main ingredient of the proof in [8] was a decay-type argument. The assumption that the BMO norm of n was small gave us a starting point for the argument. The main ingredient of the proofs in this paper is a blow-up and hence the assumption on the BMO norm of n is not necessary. It is interesting to compare our results with those of Alt and Caffarelli []. In both cases the oscillation of the logarithm of the Poisson kernel controls the geometry i.e., the flatness ) of the boundary and the oscillation of the unit normal. THEOREM []. Assume that ) Ω R n+ is a δ-reifenberg flat chord arc domain for some δ>0 small enough; 2) log k X C 0,β or log h C 0,β ) for some β 0, ). Then Ω is a C,α domain for some α 0, ) which depends on β and n. Moreover if Ω is unbounded and h then Ω is a half-plane. Jerison showed that α = β see [3]). We would like to emphasize that the hypothesis above is necessary. Keldysh and Lavrentiev see [7] and [6]) constructed a domain in R 2 whose boundary is rectifiable but not Ahlfors regular, whose Poisson kernel is identically equal to and which is not C. Moreover there are examples of domains in R 2 whose boundary is Reifenberg flat with vanishing constant, rectifiable but not Ahlfors regular, for which the logarithm of the Poisson kernel is Hölder continuous and which are not even C domains see [6]). Furthermore if n 2 there are examples of chord arc domains satisfying hypothesis 2, whose boundaries are not C, they contain a neighborhood of the vertex a double cone see [] and [8]). These results should also be compared with Pommerenke s theorem [22]: THEOREM [22]. Let Ω R 2 be a chord arc domain. Then Ω is a chord arc domain with vanishing constant if and only if log k X VMO). We would like to point out that our proofs use a modified version of Alt and Caffarelli s result see Theorem 2.2 and [20] for a proof). We now sketch the content of each one of the sections. In Section 2 we prove some technical lemmas which play a central rôle in Sections 3 and. These results are of two types: either boundary regularity of non-negative harmonic functions on Reifenberg flat domains, or regularity statements for functions of vanishing mean oscillation. The proofs of Theorem. and the Main Theorem are accomplished in 2 main stages, described in Sections 3 and. In Section 3 we prove gradient bounds for the Green s function in terms of the integral of the corresponding Poisson kernel, provided its logarithm has vanishing mean oscillation. In Section we describe a general construction of a blow up sequence for a Reifenberg flat chord arc domain whose Poisson kernel has logarithm in VMO. In Section we also prove the Main Theorem. The estimates obtained in Section 3 ensure that the limit of this blow up sequence satisfies the hypothesis of Theorem 2.2 see [20]). Section constitutes the core of this paper. In Appendix A we prove Lemma 3.2 and Rellich s identity for chord arc domains with small constant, verifying a point left open in [8]. In particular in Appendix A we construct an approximation of Reifenberg flat chord arc domains by interior chord arc domains. This is a very useful tool in potential theory. We finish this introduction by briefly sketching the proof of Theorem. and Theorem.3. This is an application of the blow up technique described in Section. Let K R n+ be a e SÉRIE TOME N 3

7 REIFENBERG FLAT CHORD ARC DOMAINS 329 compact set, and let.5) l = lim sup r 0 Q K θq, r). Our goal is to show that l =0. There exist sequences {Q i } i K, and{r i } i R such that lim i Q i = Q, 0 <r i, lim i r i =0and.6) lim θq i,r i )=l. i We consider the blow up sequences Ω i = r i Ω Q i ), i = r i Q i ), u i, ω i and h i associated with Q i and r i as described in Section. Theorem. ensures that there exists a subsequence which we relabel) satisfying Ω i Ω, i in the Hausdorff distance sense uniformly on compact sets see Definition 2.) and u i u uniformly on compact sets, where u satisfies hypothesis 2.35). Furthermore ω i ω. Theorems.2 and.3 guarantee that if h = dω dσ then u and h satisfy hypothesis 2.36) and 2.37). Theorem 2.2 allows us to conclude that Ω is a half plane in R n+ and is an n-plane. Since i converges to in the Hausdorff distance sense uniformly on compact sets and O k, for each k,givenε>0 there exists k 0 so that for k k 0.7) D [ k B0, ); B0, ) ] ε. Hence.8) θq k,r k ) r k D [ BQ k,r k ); L k BQ k,r k ) ] ε, where L k = + Q k is an n-plane through Q k. Since by.6) l = lim k θq k,r k ),we conclude that l =0. 2. Preliminaries In this section we prove some technical lemmas that will be useful in the rest of the paper. DEFINITION 2. Uniform Hausdorff convergence on compact sets). Given a sequence of closed sets {A i } i in R n+ we say that A i converges to a closed set A R n+ i.e., A i A) in the Hausdorff distance sense uniformly on compact sets of R n+ if for any compact set K R n+ and any ε>0 there exists i 0 so that i i 0 2.) sup { distx, A): x A i K } +sup { distx, A i ): x A K } ε. Given a sequence of open sets {U i } i in R n+ we say that U i converges to an open set U R n+ i.e. U i U) in the Hausdorff distance sense uniformly on compact sets of R n+ if Ui c U c in the Hausdorff distance sense uniformly on compact sets of R n+. For A, B, C closed subsets of R n+, we use the convention that distx, B)=+ when B = but sup{distx, A): x C} =0when C =. DEFINITION 2.2. Let µ be a Radon measure on R n+. We say that µ is a doubling measure if there exists C>sothat every Q spt µ and every r>0 2.2) µ BQ, 2r) ) Cµ BQ, r) ). Here spt µ denotes the support of the measure µ.

8 330 C.E. KENIG AND T. TORO The following lemma gives an improvement of the conclusion of Lemma. in [] in the Reifenberg flat case. LEMMA 2.. Given ε>0 there exists δ = δn, ε) > 0 so that if Ω is a δ-reifenberg flat domain, then for every K R n+, there exists R K > 0 so that if r 0,R K ), Q K, and u is a non-negative harmonic function in Ω BQ, r) which vanishes continuously on BQ, r), we have for X BQ, r) Ω 2.3) ) ε X Q ux) C sup uy ) r Y BQ,2r) Ω where C depends only on K, n and ε. Proof. Let v 0 satisfy v 0 =0in Ω BQ, 2r), v 0 =on BQ, 2r) Ω and v 0 =0on BQ, 2r). By the maximum principle for X Ω BQ, r) 2.) ux) [ sup uy ) ] v 0 X). Y BQ,2r) Ω Since Ω is a δ-reifenberg flat domain Remark. in [] holds. Let 2.5) Λ= { X = x + t n Q, 2r); x LQ, 2r); t δr }. Let h 0 satisfy 2.6) h 0 =0 h 0 =0 h 0 = on Λ BQ, 2r), on Λ BQ, 2r), on Λ BQ, 2r). By the maximum principle v 0 X) h 0 X) for X Ω BQ, 2r). Consider the function g 0 defined by g 0 x + t n Q, 2r)) = t +δr; g 0 is a non-negative harmonic function on Λ BQ, 2r), g 0 = h 0 =0 on Λ BQ, 2r), and therefore by the Comparison principle Lemma.0 in []) we have that for X BQ, r) Ω 2.7) h 0 X) g 0 X) C h 0Q + r 2 n Q, 2r)), r and if X = x + t n Q, 2r) 2.8) h 0 X) C t +δr. r Thus for X BQ, θr) Ω with θ< 2.9) v 0 X) h 0 X) Cθ + δ). An iteration process ensures that for θ< 2.0) v 0 X) [ Cθ + δ) ] k for X B Q, θ k r ) Ω. In particular 2.) v 0 X) 2Cδ) k for X B Q, δ k r ) Ω. e SÉRIE TOME N 3

9 REIFENBERG FLAT CHORD ARC DOMAINS 33 By choosing δ>0 small enough we can ensure that 2Cδ δ ε, which implies that 2.2) ) ε X Q v 0 X) C for X BQ, r). r Combining 2.) and 2.2) we obtain 2.3). Notation. ForΩ R n+ as above and X Ω we denote by δx)=distx,). COROLLARY 2.. Given ε>0 there exists δ = δn, ε) > 0 so that if Ω is a δ-reifenberg flat domain then for every K R n+, there exists R K > 0 so that if r 0,R K ), Q K and u is a non-negative harmonic function in Ω BQ, r) which vanishes continuously on BQ, r), we have for X BQ, r 2 ) Ω 2.3) ) ε δx) ux) C sup uy ) r Y BQ,r) Ω where C depends only on K, n and ε. Proof. Apply Lemma 2. to K =K, 2R K )={X R n+, distx,k) 2R K },forr K as above. If r<min{r K, }, Q K, andp BQ, r) K ; 2.3) and the maximum principle yield that for X BP,r) Ω 2.) ) ε X P ux) C sup uy ), r Y BQ,r) Ω which implies 2.3). COROLLARY 2.2. Given ε>0 there exists δ = δn, ε) > 0 so that if Ω is an unbounded Reifenberg flat domain such that 2.5) sup Q r>0 sup θq, r) δ, and u is a non-negative harmonic function in Ω which vanishes continuously on, then for Q, R>0, and X BQ, R) Ω 2.6) ) ε δx) ux) C sup uy ), R Y BQ,2R) Ω where C depends only on n and ε. Proof. Note that since 2.5) holds for each compact set K R n+, we can take R K =, thus 2.6) follows from 2.3). COROLLARY 2.3. Given ε>0there exists δ = δn, ε) > 0 so that if Ω is a δ-reifenberg flat domain, Q 0, and u is a non-negative harmonic function on Ω BQ, R 0 ) which vanishes continuously on BQ, 6R 0 ), then for X BQ 0,R 0 ) Ω 2.7) ) ε δx) ux) C sup uy ), R 0 Y BQ 0,6R 0) Ω where C depends on R 0, ε and n.

10 332 C.E. KENIG AND T. TORO Proof. Let K = BQ 0, 6R 0 ), Corollary 2. ensures that there exists R K > 0 so that for r 0 = 2 min{r 0,R K }, Q BQ 0,R 0 ),andx BQ, r0 2 ) Ω, 2.8) δx) ux) C r 0 C R 0 r 0 ) ε sup Y BQ,R 0) Ω uy ) δx) R 0 sup uy ). Y BQ 0,6R 0) Ω Furthermore by Harnack s principle for X BQ 0,R 0 ) Ω with δx) r0 2 we have 2.9) ux) C sup Y BQ 0,6R 0) Ω ) ε δx) uy ) C sup uy ). r 0 Y BQ 0,6R 0) Ω Combining 2.8) and 2.9) we obtain 2.7). The following theorem is a consequence of the John Nirenberg inequality [6], see Garnett and Jones [0] or [9, Chapter ] in the Euclidean case. As they point out the result remains true on an Ahlfors regular set. This is not surprising since most of the proof relies on a Calderon Zygmund type decomposition, which is possible in this case thanks to the existence of a family of dyadic cubes see [2] or [5, Chapter 3]). THEOREM 2.. Let Ω R n+ be a chord arc domain f VMO) and h = e f then for all Q, r 0, diam Ω) and q< 2.20) ) q h q dσ Cq hdσ, 2.2) BQ,r) h q dσ ) BQ,r) q Cq h dσ. BQ,r) BQ,r) Here C q only depends on the VMO character of f,onn, q and the Ahlfors constant for σ. Proof. Since f VMO), thenf BMO) and there exits p> such that h, h A p.sincevmo) is the closure of the class of bounded uniformly continuous functions in BMO) in,then 2.22) distf,l )= inf g L { f g } =0 where denotes the norm in BMO) see Definition.2. Combining Corollary., and Lemma. in [0] we conclude that h, h A q for every q>. COROLLARY 2.. Let Ω R n+ be a chord arc domain and log h VMO), then for all ε>0, Q, r 0, diamω), and E BQ, r) 2.23) ) +ε ) ε σe) Cε ωe) σe) σbq, r)) ωbq, r)) C ε, σbq, r)) where ωa)= A hdσ.herec ε only depends on n, ε and the Ahlfors constant of σ. e SÉRIE TOME N 3

11 REIFENBERG FLAT CHORD ARC DOMAINS 333 Proof. Let q = ε.fore BQ, r), applying 2.20) we have ωe) E hdσ σ BQ, r) ) q BQ,r) ) h q q dσ E ) q dσ ) h q q dσ σe) q C q σ BQ, r) ) q BQ,r) ) hdσ σe) q BQ,r) ) ε σe) 2.2) C ε ω BQ, r) ), σbq, r)) which shows that ) ε ωe) σe) 2.25) ωbq, r)) C ε. σbq, r)) Since σa) = A h dω, the argument above applied to h rather than h, yields the first inequality in 2.23). Let us finish this section by specifying our set up. Let Ω R n+ be a δ-reifenberg flat chord arc domain δ>0is chosen so that Ω is an NTA domain, see [9]). Let A Ω be fixed, and let u denote the Green s function of Ω with either pole at infinity see [8, Lemma 3.7]) or the Green s function of Ω with pole at A. By the results of [25] or [] we know that ω and ω A the harmonic measures of Ω with pole at infinity and pole at A respectively are A -weights with respect to σ, the surface measure to the boundary. Let k A = dωa dσ denote the Poisson kernel with pole at A and h = dω dσ denote the Poisson kernel with pole at infinity. Recall that if u denotes the Green s function with pole at infinity we have u =0 in Ω, 2.26) u =0 on, u>0 in Ω, and 2.27) Ω u ϕ = ϕdω= ϕh dh n for all ϕ C c R n+ ). Similarly note that if u denotes the Green s function with pole at A then we have u =0 in Ω BQ, R), 2.28) u =0 on BQ, R), u>0 on Ω BQ, R), and 2.29) u ϕ = ϕdω A = ϕk A dh n for all ϕ CC ) BQ, R), Ω

12 33 C.E. KENIG AND T. TORO for any Q and R>0 so that A/ BQ, R). In order to unify our presentation we denote by ω the harmonic measure with either finite or infinite pole, and by h the corresponding Poisson kernel. The following 2 lemmas are used in the proof of Lemma.2. We present them here to avoid interrupting the flow of ideas in Section. The first lemma is essentially Lemma 5. in [8]. LEMMA 2.2. Let Ω R n+ be a δ-reifenberg flat chord arc domain. Let X Ω then for H n a.e. Q 2.30) dω X Q) dω = k XQ) hq) = lim ω X BQ, r)) X 0 ωbq, r)) = lim Z Q GX,Z). uz) Here ω X denotes the harmonic measure, GX, ) denotes the Green s function, and k X the kx Q) Poisson kernel for Ω with pole at X.LetKX,Q)= hq). There exist constants C>, N 0 > and α 0, ) so that for s 0, diam Ω), and Q 0,ifX Ω\BQ 0, 2N 0 s), then for every P,Q BQ 0,s) 2.3) KX,Q) KX,P) ) α Q P CKX,Q). s Although the hypothesis above are somewhat weaker than those in the statement of Lemma 5. in [8], the reader will easily check that the proof presented in [8] works in this setting. Simply note that ω X, ω and σ are doubling measures on and ω X, ω A dσ). Thus ω X and ω are mutually absolutely continuous, and the proof presented in [8] goes through. LEMMA 2.3. Let Ω R n+ be a δ-reifenberg flat chord arc domain. Assume that h the Poisson kernel satisfies for all Q, and r 0, diam Ω) 2.32) ) 2 h 2 dσ C0 hdσ. BQ,r) BQ,r) There exist constants C> and N 0 > so that for r 0, diam Ω), and Q if X Ω\BQ, 2N 0 r) then 2.33) ) 2 kx 2 dσ C k X dσ. BQ,r) BQ,r) Proof. Let N 0 > be as in Lemma 2.2. Let Q, r 0, diam Ω) and X Ω\BQ, 2Nr), then using 2.3) and 2.32) we have BQ,r) = k 2 X P ) dσp ) ) 2 BQ,r) CKX,Q) kx 2 P ) ) 2 h 2 P ) h2 P ) dσp ) CKX,Q) BQ,r) hp ) dσp ) BQ,r) ) h 2 2 P ) dσp ) e SÉRIE TOME N 3

13 C REIFENBERG FLAT CHORD ARC DOMAINS 335 hp ) [ KX,Q) KX,P) ] dσp )+C hp )KX,P) dσp ) 2.3) C BQ,r) hp )KX,P) dσp ) C BQ,r) k X P ) dσp ). BQ,r) BQ,r) We finish this section with the statement of a theorem that plays a crucial role in our proof. It generalizes some of the results that appear in []. In Sections 7 and 8 of [], Alt and Caffarelli prove that if Ω is a Reifenberg flat chord arc domain and log h C 0,β ) for some β 0, ) then Ω is a C,α domain for some α 0, ). In particular they show that if h then Ω is a half space. THEOREM 2.2. There exists δ n > 0 so that if Ω R n+ is an unbounded δ-reifenberg flat chord arc domain for δ 0,δ n )) and v and k satisfy v =0 in Ω, 2.35) v>0 in Ω, v =0 on, and 2.36) Ω v ϕ = ϕk dh n for all ϕ C c R n+ with 2.37) sup vx) and kq) for H n a.e. Q, X Ω then Ω is a half space, and in suitable coordinates vx, x n+ )=x n+. Note that the uniqueness modulo multiplication by a positive constant) of the Green s function with pole at infinity for unbounded NTA domains allows us to conclude that k = on see [9]). The proof of Theorem 2.2 follows the same steps as the argument presented in Sections 7 and 8 of [], for a proof see [20]. 3. Gradient bound for the Green s function As mentioned in the introduction the proofs of our results are done in 2 stages. First we give a bound for the gradient of the Green s function in terms of the integral of the Poisson kernel. Second we use this estimate to produce a blow up sequence whose limit satisfies the hypothesis of Alt and Caffarelli s result as stated in Theorem 2.2. In this section we prove the gradient estimate. From now on we assume that Ω R n+ is a δ-reifenberg flat chord arc domain, where δ>0 is chosen so that, in the unbounded case Corollaries 2.2, and 2.3 hold for ε = and in the bounded case Corollary 2.3 holds for ε =. Moreover we assume that log h VMO). This hypothesis ensures that h L 2 loc dσ) and that for Q, r 0, diam Ω),ands 0,r) 3.) ) h 2 2 dσ C hdσ, BQ,r) BQ,r)

14 336 C.E. KENIG AND T. TORO ) + ) σbq, s)) C 8n ωbq, s)) σbq, s)) σbq, r)) ωbq, r)) C 8n 3.2), σbq, r)) where C is a constant that only depends on n, and the Ahlfors constant of σ. Recall that u denotes either the Green s function with pôle A Ω or with pôle at infinity if Ω is unbounded); h denotes the corresponding Poisson kernel and ω the associated harmonic measure dω = hdσ. We denote by l one quarter of the distance from the pôle of u to, i.e., l = δa)/ or l =+. THEOREM 3.. Let Ω R n+ be a δ-reifenberg flat chord arc domain satisfying Corollaries 2.2 and 2.3 with ε =.Letu denote the Green s function with pôle at infinity, ω the harmonic measure with pôle at infinity, and h = dω dσ the corresponding Poisson kernel. Assume that log h VMO), then for all X Ω we have 3.3) ux) hq) dω X Q). THEOREM 3.2. Let Ω R n+ be a δ-reifenberg flat chord arc domain satisfying Corollary 2.3 with ε =. Let GA, ) denote the Green s function with pole at A and k A = dωa dσ the corresponding Poisson kernel. Assume that log k A VMO), then for all X Ω {Y R n+ : δy ) <δa)/8} we have 3.) GA, X) k A Q) dω X + C ) 3 δx) ω A δa) n B Q X,δA) )), δa) for any Q X such that X BQ X,δA)/8) Ω. LEMMA 3.. Let X Ω.Letu, ω and h as above, and assume that h L 2 loc dσ). Then for ω a.e. Q, ux) converges non-tangentially to F Q), and F L loc ). dωx Proof. Let l =min{,l}.letk R n+ be a compact set, let K = { X R n+ :distx,k) l }. Let Q K, andx ΓQ) with δx) <l.hereγq) denotes a nontangential access region. By a standard estimate for non-negative harmonic functions we have 3.5) ux) C ux) δx). Furthermore by Lemma.8 in [] there is C> so that for every Q K, X ΓQ) with δx) <l,ify Ω\BQ, 2δX)) 3.6) C < ωy BQ, δx))) δx) n GX,Y ) <C. Since A Ω\BQ, 2δX)) for X Ω with δx) <l, 3.6) yields 3.7) C < ωa BQ, δx))) δx) n GX,A) <C. e SÉRIE TOME N 3

15 REIFENBERG FLAT CHORD ARC DOMAINS 337 By the construction described in the proof of Lemma 3.7 in [8], we know that letting Y tend to infinity for Q K and X ΓQ), 3.6) yields 3.8) C < ωbq, δx))) δx) n ux) <C. Combining 3.5), 3.7) and 3.8) we have that for X ΓQ) with δx) l, 3.9) so that if δx) l 3.0) where 3.) Since 3.2) ux) C ux) δx) C δx) n BQ,δX)) sup ux) CMl h)q), X ΓQ) δx)l M l h)q)= sup 0<rl r n K BQ,r) K hdσ. [ Ml h) ] 2 dσ C h 2 dσ <, hdσ, we see that the truncated non-tangential maximal function of u is in L 2 loc dσ) and hence in L loc ). By Fatou s theorem for NTA domains see [] Theorem 5.8 and Lemma 8.3 as dωx well as Lemma 3.3 in Appendix A) we know that u converges non-tangentially to F,and F L loc ). dωx LEMMA 3.2. Let F be the non-tangential limit of u. Thensinceh L 2 loc dσ), forhn a.e. Q we have that 3.3) F Q)=hQ) n Q). The proof of this lemma appears in Sections A. and A.2 of Appendix A. LEMMA 3.3. Let Ω R n+ be an unbounded δ-reifenberg flat chord arc domain satisfying Corollaries 2.2 and 2.3 with ε =. Assume that log h VMO), and that 0.FixR> large and let ϕ R Cc R n+ ), ϕ R for X R, spt ϕ R B0, 2R), 0 ϕ R and ϕ R C/R, ϕ R C/R 2.ForX Ω define 3.) ω R X)= GX,Y ) [ ϕ R Y ) uy ) ] dy, Ω where u denotes the Green s function of Ω with pole at. Thenω R 0, ω R C α Ω) for some α 0, ), and we have the following estimates for X Ω 3.5) ωr X) C δx) 3/ R /2 for X < R 2.

16 338 C.E. KENIG AND T. TORO 3.6) 3.7) ωr X) CR n ω R X) C [ ] 2 ) 3 ωb0,r)) δx) R n X ωb0, X )) ) 3 δx) ωb0,r)) R R n for R X R. 2 for X R. Proof. Let V X)= ux) for X Ω.Then ϕ R V )= ϕ R )V +2 ϕ R V so that 3.8) ω R X)=ω R X)+ω2 R X) with 3.9) and 3.20) Note that 3.2) ωrx)= ω 2 R X)=2 Ω Ω V Y ) C uy ) δy ) also spt ϕ R, spt ϕ R {R< Y < 2R}.Let 3.22) I R = CLAIM. If Ω is as above then GX,Y ) ϕ R Y )V Y )dy, GX,Y ) ϕ R Y ) V Y )dy. {R< Y <2R} Ω and V Y ) C uy ) δ 2 Y ), ) 2 uy ) dy. δy ) 3.23) I R CR n+ u2 A 2R ) R 2. In fact note that by Harnack s principle and our assumption that δ is chosen so that Corollary 2.2 holds for ε = we have that for Y Ω B0, 2R)\B0,R) 3.2) Thus 3.25) ) 3 δy ) uy ) C ua2r ). R [ I R Cu 2 A 2R ) R 3/2 {R< Y <2R} Ω ] dy. δy ) /2 We want to show that the term in brackets is bounded above by C R 2 R n+. Scaling shows that it is enough to prove this for R =, i.e., we have to show that for Ω as in Corollaries 2.2 and ) {< Y <2} Ω dy C. δy ) /2 e SÉRIE TOME N 3

17 REIFENBERG FLAT CHORD ARC DOMAINS 339 Let j 0,and 3.27) A j =Ω { Y < 2: 2 j <δy ) 2 j+}. Cover B0, 2) by balls {BQ i, /2 j )} N i= centered in and so that Q i Q l /2 j for i l. Since is Ahlfors regular, it is straightforward that N C2 jn,wherec depends on n and on the Ahlfors regularity constant of. If Y A j there exists X so that X Y /2 j and Q i so that Q i Y /2 j 2. On the other hand since δy ) > /2 j, Q i Y > /2 j. Thus {BQ i, /2 j 2 )\BQ i, /2 j )} N i= covers A j and 3.28) which implies that 3.29) { ) n+ ) n+ } H n+ A j ) C n 2 nj 2 j 2 2 j C n 2 j, {< Y <2} Ω which proves the claim. Case. Let X R 2.Then 3.30) dy δy ) = /2 ω 2 R X) C R j=0 A j {R< Y <2R} Ω dy δy ) /2 C n 2 j/2 C, j=0 GX,Y ) uy ) δy ) 2 dy. Let A S = A0,S/2); i.e., S/M A S S and δa S ) S/M see Definition 3. of NTA domain in [8]). Then for Y Ω B0, 2R)\B0,R) we have, using Corollary 2.3, that 3.3) [ ] 3 δx) GX,Y ) C GAR,Y). R Moreover by the Comparison Principle Lemma.0 in []) we have that for Y {R< Y < 2R} 3.32) GA R,Y) uy ) C GA R,A 2R ) ua 2R ), hence 3.33) GA R,Y) CGA R,A 2R ) uy ) ua 2R ) C uy ) R n ua 2R ), and combining 3.30), 3.33), 3.23), 3.8), 3.2), and using the fact that R> we have that 3.3) ωr 2 X) C ) 3 δx) R n R ua 2R ) C δx) R n R δx) C R ) 3 ua 2R ) I R ) 3 ua 2R ) R {R< Y <2R} Ω δx) C R u 2 Y ) δy ) 2 dy ) 3 ωb0, 2R)) R n

18 30 C.E. KENIG AND T. TORO 3.35) ) 3 δx) ωb0,r)) C R σb0,r)) C δx) R ) 3 R ω B0, ) ). Hence for X R ) ωr 2 X) C δx)3/. R /2 We now estimate the term ωr X). Using 3.2), 3.3) and 3.33) we obtain ω R X) GX,Y ) ϕr Y ) V Y ) dy, 3.37) Ω ) 3 δx) C R R 2 C R n+ δx) R Ω {R< Y <2R} ) 3 Ω {R< Y <2R} Ω {R< Y <2R} GA R,Y) uy ) δy ) dy uy ) 2 δy ) dy ua 2R ). Since for Y Ω with R< Y < 2R, δy ) 2R, 3.37) becomes ω R X) ) 3 ) C δx) 2 uy ) 3.38) R n dy ua 2R ) R δy ) C ua ) 3 2R) δx) δx) 3 C, R R R 2 because of 3.23) and 3.3). This concludes the proof of 3.5). Case 2. Let X R. Assume that 2 j R X < 2 j+ R for some j 2.LetA j = A0, 2 j R) be a non-tangential point for 0 at radius 2 j R.ForY Ω with R< Y < 2R, by Corollary 2.3 the comparison principle and 3.8) we have ) 3 ) 3 δx) δx) GX,Y ) C GAj 2 j,y) C GAj R 2 j,a R ) uy ) R ua R ) ) 3 ) 3 δx) uy ) δx) C GAj 2 j,a j ) R ua j ) C uy ) 2 j R 2 j R) n ua j ) ) 3 ) 3 δx) uy ) δx) 3.39) C 2 j R ωb0, 2 j R)) C uy ) 2 j R ωb0, 2 j R)). Thus using 3.23), the fact that ω is a doubling measure, and 3.8) we have ω 2 R X) C R C R C R {R< Y <2R} Ω δx) 2 j R ) 3 GX,Y ) uy ) δy ) 2 dy ωb0, 2 j R)) ) 3 δx) I R 2 j R ωb0, 2 j R)) {R< Y <2R} Ω u 2 Y ) δy ) 2 dy e SÉRIE TOME N 3

19 3.0) REIFENBERG FLAT CHORD ARC DOMAINS 3 CRn u 2 A R ) R 2 CR n 2 ωb0,r)) R n CR n [ ωb0,r)) R n ) 3 δx) 2 j R ) 2 δx) ωb0, X )) ) 3 ωb0, X )) X ] 2 ) 3 δx) X ωb0, X )). In order to finish the proof of 3.6) we need to estimate ωr X) for X R. By 3.2), 3.39) and the computation in 3.0) we obtain 3.) ω R X) C R 2 C R δx) 2 j R ) 3 ωb0, 2 j R)) ) 3 δx) 2 j R CR n [ ωb0,r)) R n ωb0, 2 j R)) I R ] 2 δx) X Ω {R< Y <2R} ) 3 ωb0, X )). uy ) 2 δy ) dy Inequality 3.6) is proved by combining 3.0) and 3.). Case 3. Let 2 R< X < R. Note that δx) < R.Let X be such that δx)= X X, which implies that X < 8R. Note that if Y B0, 2R) then Y B X;0R). We now look at ω 2 R X) C GX,Y ) uy ) R δy ) 2 dy 3.2) C R Ω {R< Y <2R} B X,0R) Ω {R< Y <2R} BX,δX)/2) + C R + C R GX,Y ) uy ) δ 2 Y ) dy Ω {R< Y <2R} B X,2δX)\BX,δX)/2))) Ω {R< Y <2R} B X,0R)\B X,2δX))) For Y Ω {R< Y < 2R} BX,cδX)/2), GX,Y ) uy ) δ 2 Y ) dy GX,Y ) uy ) δ 2 Y ) dy. 3.3) GX,Y ) C X Y n and uy ) ux) C δy ) 2 δx) 2, by Harnack s principle. Thus 3.) Ω {R< Y <2R} BX,δX)/2) C ux) δx) 2 GX,Y ) uy ) δ 2 Y ) dy Ω {R< Y <2R} BX,δX)/2) dy CuX). X Y n

20 32 C.E. KENIG AND T. TORO Fig.. If Y Ω {R< Y < 2R} [B X,2δX))\BX,δX)/2)] 3.5) GX,Y ) C uy ) uy ) GX,Z) C uz) ux) δx) n by the Comparison Principle, for Z BX,δX)/2) see Fig. ). Thus 3.5) yields GX,Y ) uy ) δy ) 2 dy 3.6) Ω {R< Y <2R} B X,2δX))\BX,δX)/2)) C ux)δx) n Ω B X,2δX)) u 2 Y ) δy ) 2 dy. A similar argument to the one used to estimate I R see 3.23)) ensures that u 2 Y ) δ 2 Y ) dy C u2 X) 3.7) δ 2 X) δn+ X). Ω B X,2δX)) Thus combining 3.6) and 3.7) we obtain 3.8) Ω {R< Y <2R} B X,2δX))\BX,δX)/2)) GX,Y ) uy ) δ 2 dy CuX). Y ) If Y Ω {R< Y < 2R} B X,0R)\B X,2δX))) there exists j {,...,j 0 } so that 2 j δx) X Y < 2 j+ δx) where j 0 is such that 2 j0 δx) > 0R 2 j0 δx). Let Y j = A X,2 j δx)) be a non-tangential point with respect to X at radius 2 j δx). Then for Y B X,2 j+ δx))\b X,2 j δx)) by the Comparison Principle, Lemma 2. and 3.8) we have GX,Y ) CGX,Y j ) uy ) ) 3 δx) uy j ) C GYj 2 j,y j ) uy ) δx) uy j ) C 2 3j e SÉRIE TOME N 3 uy ) 2 jn ) δx) n 2 jn ) δx) n ) ωb X,2 j δx)))

21 REIFENBERG FLAT CHORD ARC DOMAINS ) uy ) C 2 j ) 3 ωb X,2 j δx))). Hence using 3.8), Harnack s principle and an argument similar to the one used to prove 3.23) we have that GX,Y ) uy ) δ 2 Y ) dy 3.50) Ω {R< Y <2R} B X,0R)\B X,2δX))) C C C C j 0 j= 2 j ) 3 ωb X,2 j δx))) Ω {R< Y <2R} {2 j δx) X Y <2 j+ δx)} j 0 j= j 0 j= j 0 j= 2 j ) 3 ωb X,2 j δx))) 2 j ) 3 ωb X,2 j δx))) 2 j ) 3 uy j ) 2 2 j δx)) n ωb X,2 j δx))) C B X,2 j+ δx)) Ω u 2 Y ) δ 2 Y ) dy u 2 Y ) δ 2 Y ) dy 2 j+ δx) ) n+ u 2 Y j+ ) 2 j+ δx)) 2 j 0 j= 2 j ) 3 ωb X,2 j δx))) [2 j δx)] n. Since log h VMO), by 3.2), and using the fact that ω is doubling in the case where j = j 0 we have that ωb X,2 j δx))) 2 j ) n 8 δx) 3.5) ωb X,0R)) C. R Thus combining 3.50) and 3.5) we obtain Ω {R< Y <2R} B X,0R)\B X,2δX))) Cω B X,0R) ) j0 C R n 8 j 0 j= j= 2 j ) 3 GX,Y ) uy ) δ 2 Y ) dy R n 8 δx) 7 ωb X,0R)) 8 2 j ) j δx)) n 8 2 j δx)) n C δx) 7 8 R n 8 C δx) 7 8 ω B X,0R) ) 2 j0 ) δx) 7 8 R n 8 C ω B 8 R n 8 C δx) 3 R n R ) 3.52) ω B X,0R). Since ω is a doubling measure and X < 8R then ω B X,0R) ) j0 2 j ) 8 X,0R) ) R ω B X,0R) ) ω B0, 8R) ) Cω B0,R) ). j= δx) ) 8

22 3 C.E. KENIG AND T. TORO Combining this remark, 3.2), 3.), 3.8) and 3.52) we obtain that 3.53) ω 2 R X) C R ux)+c δx) R ) 3 ωb0,r)) R n. By the Harnack principle, Corollary 2.2 and 3.8), for X B0, R) we have 3.5) δx) ux) C R ) 3 uar ) C δx) R ) 3 ωb0,r)) R n. Combining 3.53) and 3.5) we obtain 3.55) ω 2 R X) ) 3 δx) ωb0,r)) C R R n. We now look at ωr X), ω R X) C R ) C R Ω {R< Y <2R} B X,0R) Ω {R< Y <2R} B X,0R) Combining 3.2), 3.), 3.8), 3.52) and 3.5) we obtain that GX,Y ) uy ) δy ) dy GX,Y ) uy ) δy ) 2 dy. 3.57) ω R X) ) 3 δx) ωb0,r)) C R R n, which concludes the proof of 3.7), and that of Lemma 3.3. In fact note that 3.5), 3.6) and 3.7) ensure that ω R vanishes continuously at the boundary, and that ω R C α Ω) for α 0, 3 ). Proof of Theorem 3.. Recall that Ω is an unbounded δ-reifenberg flat chord arc domain, satisfying Corollaries 2.2 and 2.3 with ε =. Assume that 0. LetR>, and using the notation introduced in Lemma 3.3 define for X Ω, h R X)=ϕ R X) ux) ω R X). Note that h R is a harmonic function in Ω satisfying h R 0 on \B0, 2R). In fact 3.) ensures that ω R = [ϕ R u]. The proof of Lemma 3. ensures that Nϕ R X) ux)) L dω X ) for every X Ω. Lemma 3.3 guarantees that ω R is bounded, thus Nω R ) L dω X ) for every X Ω. Thus Nh R ) L dω X ) for every X Ω and Lemma 3.3 in Appendix A ensures that 3.58) h R X)= ϕ R Q)F Q) dω X for X Ω. e SÉRIE TOME N 3

23 REIFENBERG FLAT CHORD ARC DOMAINS 35 Therefore for X Ω B0, R 2 ) using 3.5) and Lemma 3.2 we have 3.59) ux) hr X) + ωr X) hq) dω X Q)+C δx)3/ R /2. Letting R we obtain that for X Ω 3.60) ux) hq) dω X Q), which proves Theorem 3.. Proof of Theorem 3.2. Let Q 0.Letϕ C c BQ 0,δA)/)), ϕ for X Q 0 <δa)/8, 0 ϕ, ϕ C/δA) and ϕ C/δA) 2. In particular ϕ 0 in BA, δa)/). For X Ω define 3.6) ω 0 X)= GX,Y ) [ ϕy ) GA, Y ) ] dy. AsinLemma3.3wehavethatω 0 0, ω 0 C α Ω), and Ω 3.62) ω0 X) C δa) n ) 3 δx) δa) for X Ω B Q 0, δa) ). In fact ω 0 X)=ω0X)+ω 0X) 2 where 3.63) ω0x)= GX, Y ) ϕ GA, Y ) dy Ω and 3.6) Note that ω 2 0 X)= Ω GX,Y ) ϕ GA, Y ) ) dy. 3.65) GA, Y ) C GA, Y ) δy ) and 2 GA, Y ) C GA, Y ) δy ) 2, also spt ϕ, spt ϕ BQ 0, 2R)\BQ 0,R) where R = δa)/8.for Y Ω BQ 0, 2R)\BQ 0,R) Corollary 2.3 and the comparison principle ensure that

24 36 C.E. KENIG AND T. TORO ) 3 ) 3 δx) δx) GA, Y ) GX,Y ) C GA2R,Y) CGA 2R,A R ) R R GA, A 2R ) C ) 3 δx) 3.66) GA, Y ) R n R GA, A 2R ), where A 2R = AQ 0,R); i.e.r/m < A 2R Q 0 < 2R and δa 2R ) R/M and similarly for A R. Therefore by Harnack s principle and the fact that δ is chosen so that Ω satisfies Corollary 2.3 with ε = we have ω 2 0 X) C GA, Y ) 2 ) ) 3 δx) R n δy ) 2 dy R GA, A 2R ) Ω {R< Y Q 0 <2R} C ) 3 δx) GA, A2R R n ) dy 3.67). R R 3 2 δy ) 2 R 3 2 Ω {R< Y Q 0 <2R} Ω {R< Y Q 0 <2R} The computation done to prove 3.23) shows that dy 3.68) CR n. δy ) 2 Combining 3.67), 3.68) and 3.7) we have ) 3 ω0 2 X) δx) GA, A 2R ) C R R 3.69) C R n δx) R δx) C R ) 3 ω A BQ 0, R) ). ) 3 ω A BQ 0, 2R)) As similar computation shows that the same inequality holds for ω0 X), and hence for X Ω BQ 0,δA)/) R n 3.70) ω0 X) C R n δx) R ) 3 ω A BQ 0, R) ), which yields 3.62). A similar argument as the one presented in the proof of Theorem 3. shows that for any Q 0 and every X Ω BQ 0,δA)/8) 3.7) GA, X) k A Q) dω X Q)+ C ) 3 δx) ω A δa) n B Q 0,δA) )) δa) which proves 3.).. Blow up argument In this section, which is the core of the paper, we describe a general construction of blow-up sequences for Reifenberg flat chord arc domains whose Poisson kernels have logarithm in VMO. e SÉRIE TOME N 3

25 REIFENBERG FLAT CHORD ARC DOMAINS 37 The main result is that any such sequence has a subsequence whose limit satisfies the hypothesis of Theorem 2.2. Let Ω R n+ be a δ-reifenberg flat chord arc domain, with δ>0 small enough so that the conclusion of Corollary 2.3 holds and that of Corollary 2.2 in the unbounded case) for ε =. Here again u denotes either the Green function with pole at A or with pole at infinity, h denotes the corresponding Poisson kernel see 2.27)) and dω = hdσ. We assume that log h VMO). LetQ i, and assume Q i Q as i. Without loss of generality we may assume that Q =0.Let{r i } i be a sequence of positive numbers so that lim i r i =0. Consider the domains.) Ω i = r i Ω Q i ) with i = r i Q i ). Consider also the functions u i on Ω i defined by.2) u i Z)= ur iz + Q i ) r i BQi,r i)hdσ. Let Ω i =Ω i if u is the Green s function with pole at infinity and Ω =Ω i \{ A Qi r i } if u is the Green s function with pole at A.Then.3) u i =0 on Ω i Ω i, u i i =0 and.) dω i Q)=h i Q) dσ i Q) for H n -a.e. Q i. Here σ i = H n i, ω i denotes the harmonic measure of Ω i either with pole at infinity or at A Q i r i, depending on whether u is the Green s function with pole at infinity or with pole at A. Furthermore the corresponding Poisson kernel h i satisfies.5) h i Q)= hr iq + Q i ). BQi,r i)hdσ Since log h VMO, by including the term BQi,r i)hdσ in the denominator of the function u i defined in.2) we remove the singularity of the Poisson kernel of the limit domain. This is the correct type of blow up in the sense that it allows us to connect the geometry of the limit domain to the analytic properties of its Green s function with pole at infinity. THEOREM.. There exists a subsequence which we relabel) satisfying.6) Ω i Ω in the Hausdorff distance sense, uniformly on compact sets,.7) i in the Hausdorff distance sense uniformly on compact sets, where Ω is an unbounded δ-reifenberg flat chord arc domain. Moreover there exists u CΩ) such that.8) u i u uniformly on compact sets

26 38 C.E. KENIG AND T. TORO and.9) Furthermore u =0 in Ω, u =0 in, u > 0 in Ω..0) ω i ω, weakly as Radon measures. Moreover ω is the harmonic measure of Ω with pole at infinity corresponding to u ). Proof. Since for each i, B0, ) Ω i and 0 i, given a compact set K R n+, there exists a subsequence {i } such that Ω i K and i K converge in the Hausdorff distance sense. Taking an exhaustion of R n+ by compact sets, we can insure that there exists another subsequence {i k } such that Ω ik and ik converge in the Hausdorff distance sense, uniformly on compact sets. Hence modulo relabeling the subsequence we have that.) Ω i Ω in the Hausdorff distance sense uniformly on compact sets, and.2) i Σ in the Hausdorff distance sense uniformly on compact sets. Note that if E R n+ is a Borel set E ω i E)= h i Q) dσ i Q)= hr iq + Q i ) dσ i Q) BQi,r i)hdσ.3) E = r n i r ie+q i hq) dσq) BQi,r i)hq) dσq) Since is Ahlfors regular, there exists C> so that = r n i σ BQ i,r i ) ) ωr i E + Q i ) ω BQ i,r i ) )..) C ωr ie + Q i ) ωbq i,r i )) ω ie) C ωr ie + Q i ) ωbq i,r i )). Since ω is a doubling measure for each compact set K R n+, sup i ω i K) C K. Hence there exists a subsequence which we relabel again) so that ω i ω and µ i µ where µ i E) = ωrie+qi) ωbq. Note that i,r i)) C µ ω Cµ which ensures that spt µ =sptω, where spt denotes the support of a measure. Our immediate goal is to show that Σ =,to do this we first need to prove that Σ =sptω. It is straightforward to show that spt µ Σ see proof of Lemma 2. in [8]). Now assume that X Σ,thereexistX i = r i Z i Q i ) i with Z i so that X i X. Forr 0, ) there exists i 0 so that for i i 0 X X i < r 2 and Z i Q i Mr i,wherem = X +. Then for i i 0.5) µ i BX,r) ) = ωbr i X + Q i ; rr i )) ωbq i,r i )) ωbz i, r 2 r i)) ωbq i,r i )) ωbz i, r 2 r i)) Cr, M ), ωbz i,r i M + ))) e SÉRIE TOME N 3

27 REIFENBERG FLAT CHORD ARC DOMAINS 39 because ω is doubling. From.5) we deduce that X spt µ, which combined with the remarks above ensures that Σ =sptω. In order to prove that =Σ,let X = Ω Ω c. Given ε>0 there exist Y Ω BX,ε) and Y Ω c BX,ε). By definition Y = lim Y i Q i ) i r i for some Y i Ωc. Moreover there exists a sequence Y i R n+ such that Y = lim Y i Q i ). i r i Modulo taking a subsequence we may assume that Y i Ω. A simple connectivity argument shows that for each i there exists P i [Y i,y i ],where[y i,y i ] denotes the segment joining Y i to Y i.letp i = t i ) r i Y i + t i r i Y i for some t i 0, ) then the sequence P i Q i )= t i ) Y i Q i )+t i Y i r i r i r Q i) i is bounded, thus there exists a subsequence {i ε } such that r iε P iε Q iε ) Z ε Σ.Moreover since P iε Q iε ) Y iε Q iε ) r iε r iε Y i ε Y i ε, r iε letting i ε we have that Y Z ε Y Y and X Z ε X Y + Y Y 3ε. Summarizing we have proved that given X and given ε>0 there exists Z Σ such that X Z <ε. Hence X Σ = spt ω =sptω =Σ,i.e., Σ.Inorderto prove the other inclusion we use the fact that since Ω is a δ-reifenberg flat domain then Ω is an NTA domain. Let X Σ there exists a sequence X i such that r i X i Q i ) X.Givenρ>0 since both Ω and Ω c satisfy the corkscrew condition for i large enough so that r i ρ<r)thereexist A i Ω and A i Ωc such that B A i, r ) iρ Ω, and A i X i ρr i, M B A i, r iρ M ) Ω c, and A i X i ρr i, which implies that Ai Q i B, ρ ) Ω i, A i Q i r i M r i.6) ) Ai Q i dist, Ω i r i ρ M ; X i Q i r i ρ,

28 350 C.E. KENIG AND T. TORO.7) A B i Q i, ρ ) Ω c i r i M, A i Q i r i ) A dist i Q i, Ω i r i Modulo passing to a subsequence we may assume that ρ M. X i Q i r i ρ, A i Q i r i A ρ) Ω, and A i Q i r i A ρ). Let i in.6) and.7) we obtain B A ρ), ρ ).8) Ω, M A ρ) X ρ, and.9) A ρ) X ρ, dist A ρ), Ω ) ρ 2M..8) and.9) prove that there exists M > such that given X Σ and ρ>0 there exist A ρ) Ω and A ρ) Ωc such that.20) A ρ) X ρ, A ρ) X ρ, and.2) B A ρ), ρ ) Ω, M B A ρ), ρ ) Ω c M. Letting ρ tend to 0, and using.2) we conclude that X hence =Σ. The fact that is a δ-reifenberg flat set is a direct consequence of the fact that is a δ-reifenberg flat set and that the quantity θq, r) is scale invariant. Let K R n+ be a compact set, since is a δ-reifenberg flat set there exists R K so that for every Q { X R n+, distx,k) } and r 0,R K ), θq, r) δ, i.e., given ε>0 there exists an n-plane L containing Q so that.22) r D[ BQ, r); LQ, r) BQ, r) ] <δ+ ε. Let P K, there exists a sequence {P i } so that lim i r i P i Q i )=P, note that since by assumption lim i Q i =0then lim i P i =0. Let r 0,R K ) be fixed. Since i in the Hausdorff distance sense there exists r 0 so that for i i 0 and r ε)r, r).23) D [ i BP,r ); BP,r ) ] <εr, and if X i = r i P i Q i ), X i P <εr.fori i 0 let Λ i = LP i,r i r) P i + P then D [ BP,r); Λ i BP,r) ] D [ BP,r); i BP,r) ].2) + D [ i BP,r), Λ i BP,r) ]. e SÉRIE TOME N 3

29 REIFENBERG FLAT CHORD ARC DOMAINS 35 Note that.23) implies that.25) D [ BP,r); i BP,r) ] εr. Moreover by our choice of Λ i since r D[ i BX i,r); Λ i P + X i BX i,r) ] = r i r D[ BP i r i r); LP i,r i r) BP i,r i r) ] we have, as in the proof of Theorem 2.2 in [8], that i BP,r) i B X i,r + ε) ).26) and.27) Hence combining.26) and.27) we have Λ i B X i,r + ε) ) ;2δr + ε)+2εr ) ) Λ i BP,r); 2δr + ε)+5ε r, Λ i BP,r) i BP,r); 2δr +εr )..28) D [ i BP,r); Λ i BP,r) ] δr +0εr. Combining.2),.25) and.28) we obtain.29) r D[ BP,r); Λ i BP,r) ] δ +ε. Thus.30) θ P,r) δ. The fact that is a δ-reifenberg set combined with.2) ensures that Ω satisfies the separation property and therefore Ω is a δ-reifenberg flat domain. Since is Ahlfors regular, and the measure theoretic boundary of Ω coincides with its topological boundary, we have that for each R>0.3) ) σb0,rr i )) sup σ i B0,R) =sup i i ri n C. The compactness theorem for BV functions see [7, 5.2.3]), guarantees that modulo passing to a subsequence) χ Ωj χ E in L loc Rn+ ) where E is a set of locally finite perimeter. We claim that E =Ω. First note that since has H n+ measure zero, we may assume that E =. We can also assume that all points of E are density points for χ E.LetX int Ω c,there exists r>0 so that BX,r) Ω c.sinceω i Ω in the Hausdorff distance sense there is i 0 so that for i i 0, BX, r 2 ) Ω i =, therefore H n+ BX, r 2 ) E)=0thus X/ E. Hence E Ω.LetX int Ω there exists r>0, BX,r) Ω,since i in the Hausdorff distance there exists i 0 so that for i i 0 ; BX, r 2 ) i =. LetP i i so that ρ i = X P i =distx, i ) r 2.SinceΩ i satisfies the separation property then either BX, r ) Ω i or BX, r ) Ωc i.sincex Ω, we conclude that for i large BX, r ) Ω i and