Potential Analysis meets Geometric Measure Theory T. Toro Abstract A central question in Potential Theory is the extend to which the geometry of a domain influences the boundary regularity of the solution of a divergence form elliptic operator. To answer this question one studies the properties of the corresponding elliptic measure. On the other hand one of the central questions in Geometric Measure Theory GMT) is the extend to which the regularity of a measure determines the geometry of its port. The goal of this paper is to present a few instances in which techniques from GMT and Harmonic Analysis come together to produce new results in both areas. AMS classification: 28A33, 3A5 Keywords: Elliptic measure, Harmonic measure, Ahlfors regular. Introduction A central theme in potential theory is understanding the properties of solutions of divergence form elliptic operators. In this paper we focus our attention on the solvability of the Dirichlet problem for such operators on rough domains. To address this question one studies the properties of the corresponding elliptic measure. The introduction of techniques from geometric measure theory has enabled us to deepen our understanding of the subject in two different directions. On the one hand we have begun to understand the properties of the elliptic measure on rough domains. On the other hand we have studied to extent to which the regularity of the harmonic measure determines the geometry of the boundary of the domain. While these two problems are very different in nature they have common features. The key in both cases is the interplay between harmonic analysis and geometric measure theory. In the first case, the geometry of the domain allows us to develop the right tools from harmonic analysis to study the properties of the elliptic measure. In the second case, the properties of the Green function and the corresponding harmonic measure ensure that the tangent measure machinery from geometric measure theory can be effectively used to yield information about the boundary of the domain. The author was partially ported by NSF grants DMS-060095 and DMS-0856687
2 Elliptic measure on rough domains The Dirichlet problem addresses the following question: given a bounded domain Ω R n and a function f C Ω) does there exists u satisfying { Lu = div AX) u) = 0 in Ω 2.) u = f on Ω? Here AX) = a ij X)) is a symmetric measurable matrix satisfying λ ξ 2 n i,j= a ijx)ξ i ξ j Λ ξ 2 for all x Ω and ξ R n,i.e. L is strongly elliptic. The domain Ω is regular for L, if f C Ω), there exists u f = u CΩ) satisfying 2.). Moreover Ω is regular for L if and only if it is regular for the Laplacian i.e. when A is the identity matrix). If Ω is regular by the maximum principle and the Riesz representation theorem there is a family probability measures {ωl X} X Ω such that ux) = fq)dωl X Q), Ω ωl X is called the L elliptic measure of Ω with pole X. In [JK] Jerison and Kenig introduced a new class of regular domains the non-tangentially accessible NTA) domains. In particular they proved that on NTA domains the elliptic measure is doubling and that the non-tangential limit of the solution of 2.) at the boundary exists and coincides with f ω L a.e..quasispheres and Lipschitz domains are both examples of NTA domains. In the case that Ω is a Lipschitz domain i.e. locally the domain above the graph of a Lipschitz function) a natural question arises: what is the relationship between ω L and the surface measure of Ω, σ = H n Ω? This question also makes sense in a larger class of domains, that of chord arc domains see also SKT domains [HMT]). A chord arc domain CAD) is an NTA domain whose boundary surface measure is Ahlfors regular, i.e. there exists C > such that for all Q Ω and r 0, diam Ω) C r n σbq, r)) Cr n. 2.2) More precisely we would like to address the following question. Let Ω be a CAD, L an operator as above and u the solution of 2.) in Ω with boundary data f. Given < p < is the non-tangential maximal function of u bounded in L p σ)? Namely we are interested in the D) p problem, i.e. given f C Ω) and u satisfying { Lu = 0 in Ω u = f on Ω does there exist C > 0 depending only on the geometry of the domain, the Ahlfors regularity constant of σ and the ellipticity constants of L such that following bound hold? Nu) L p σ) C f L p σ), 2.3) where the non-tangential maximal function of u is defined for Q Ω by Nu)Q) = ux) X ΓQ) 2
ΓQ) = {X Ω : X Q < 2δX)} and δx) = dist X, Ω). 2.4) This PDE question has an equivalent formulation in harmonic analysis, by means of the theory of weights. In fact 2.3) holds if and only if ω L elliptic measure of L and σ and mutually absolutely continuous and the Radon-Nikodym derivative k L = dω L dσ satisfies the reverse Hölder inequality: ) k q q L σ Q, r)) dσ C k L dσ, 2.5) σ Q, r)) Q,r) Q,r) where p + q ω L B q σ). =, Q, r) = BQ, r) Ω, Q Ω and r 0, diam Ω). If 2.5) holds we say that We briefly recall the history of the D) p problem. Dahlberg s pioneering work established that for the Laplacian, i.e. L = the harmonic measure ω = ω satisfies ω B 2 σ), thus D) 2 holds for all Lipschitz domains, [D]. By contrast if Ω is a CAD there exists q, ) so that ω B q σ), but such q is not uniform across the class of chord arc domains see [DJ] and [Se]). It was shown in [CFK], and independently in [MM], that there are operators L for which ω L and σ are mutually singular, hence neither 2.3) nor 2.5) hold. Thus one of the main questions in this area is to find sharp conditions that ensure that 2.3) and 2.5) are satisfied. Suppose that we have two operators L 0 and L, whose respective coefficient matrices A 0 and A coincide on a neighborhood of Ω. Then if D) p holds for L 0 it also holds for L. This is a consequence of the properties of non-negative harmonic functions on NTA domains. Thus D) p is a property that only depends on the behavior of the coefficients of L near Ω. We are lead to consider the following notion: we say that L is a perturbation of L 0 if there exists a constant C > 0 such that the deviation function satisfies 0<r<diam Ω Q Ω i.e. a2 X) δx) ax) = { A Y ) A 0 Y ) : Y BX, δx)/2)} 2.6) hq, r) C where hq, r) = σ Q, r)) T Q,r)) a 2 ) X) /2 δx) dx, 2.7) dx is a Carleson measure. Here T Q, r)) = BQ, r) Ω is the Carleson region associated to the surface ball Q, r). Note that in this case L = L 0 on Ω. We include below some of the most remarkable results in this direction. Theorem 2.. [D2] Let Ω = B0, ). If L 0 =, and lim r 0 Q = hq, r) = 0, then ω L B q σ) for all q >. In [F], Fefferman removed the smallness condition of hq, r) above by defining an appropriate quantity Aa)Q). Recall that A σ) = q> B q σ). a Theorem 2.2. [F] Let Ω = B0, ), L 0 =, ΓQ) be as in 2.4), and Aa)Q) = 2 X) ΓQ) δx) dx) /2. n If Aa) L σ) then ω A σ). 3
Theorem 2.3. [FKP] Let Ω be a Lipschitz domain. Let L be such that 2.7) holds then ω A dσ) whenever ω 0 A dσ). Theorem 2.4. [FKP] Let Ω be a Lipschitz domain. For i = 0, let ω i denote the L i elliptic measure in Ω with pole 0 Ω. Let G 0 denote the Green function for L 0 in Ω with pole at 0 Ω. There exists an ɛ 0 > 0, such that if then ω B 2 ω 0 ). Ω ω 0 ) T ) ) /2 a 2 X) G 0X) δ 2 X) dx ɛ 0, 2.8) Theorems 2. and 2.2 are proved using Dahlberg s idea of introducing a differential inequality to estimate the B q norm for a family of elliptic measures. Theorem 2.3 was proved by a direct method which used Theorem 2.4. Theorem 2.4 relied on techniques from harmonic analysis. The basic approach was to look at the solution of 2.) for L as a perturbation of the solution of 2.) for L 0 and estimate the error term using the duality properties of the tent spaces introduced in [CMS]. Tent spaces were initially defined as special subspaces of functions defined on the half space. The geometry of a Lipschitz domain and the properties of the surface measure of its boundary allow to extend the notion of tent space to this class of domains. Let Ω R n be a Lipschitz domain, for p the tent space T p is defined by T p = { f L 2 Ω) : Af) L p σ) } f 2 ) X) /2 Af)Q) = δx) n dx. 2.9) T = { f L 2 Ω) : Cf) L σ) } ΓQ) Cf)Q) = Q σ ) T ) f 2 ) X) /2 δx) dx. 2.0) The proof of Theorem 2.4 relies on the duality among tent spaces, i.e. for p, T q is the dual of T p where p + q = ; the Lp σ) equivalence of Cf) and Af), i.e. for 2 < p <, Cf) L p σ) Af) L p σ); and the properties of non-negative solutions to 2.) on Lipschitz domains. A careful look at the proof shows that, from the PDE point of view, only the properties of non-negative solutions to 2.) on NTA domains are used. The proof of the two other properties appears to depend heavily on the geometry of the domain, in particular, on the fact that locally, cones of a given direction and a given aperture with vertex at the boundary lie inside the domain. On the other hand the definitions in 2.9) and 2.0) as well as condition 2.8) make sense when Ω is a CAD. In recent work with Milakis and Pipher we prove that both the duality statement and the equivalence Cf) and Af) in L p σ). We also showed that Theorem 2.4 holds on chord arc domains. Theorem 2.5. [MPT] Let Ω be a CAD, there exists ɛ 0 > 0 such that if 2.8) holds then ω B 2 ω 0 ). Corollary 2.. [MPT] Let Ω be a CAD, there exists ɛ > 0 such that if Q Ω r>0 hq, r) < ɛ see 2.7)) then ω A σ) whenever ω 0 A σ). The following three questions were motivated by the results above. investigation. They are currently under 4
Question 2.. Let Ω be a CAD. Suppose that Q Ω r>0 hq, r) <. Does ω 0 A σ) imply that ω A σ)? Question 2.2. Let Ω be a CAD. Suppose that lim r 0 Q Ω hq, r) = 0. If log k 0 V MOσ) does log k V MOσ) where k j = dω j dσ? This question is motivated by the corresponding result on Lipschitz domains proved in [E] using the Dalhberg s differential inequality idea. In particular it is known that if Ω is a CAD whose unit normal is in V MOσ) then log k 0 V MOσ) for L 0 = see [KT]). A desire to give a proof of Question 2.2 on these domains for perturbations of the Laplacian satisfying the hypothesis motivated the work in [MPT]. Question 2.3. Let Ω be a CAD. Is the solvability of an endpoint BMO Dirichlet problem for a strongly elliptic operator L equivalent to ω L A σ)? This question is motivated by recent work in [], where the corresponding result for Lipschitz domains was established. The proof of theorems 2. and 2.2 rely to some extent on the ability to approximate a Lipschitz domain Ω by smooth interior domains in such a way that the surface measure and in unit normal to Ω are the limit of those of the approximating sequence. Such an approximation is not known to exist to chord arc domains. Question 2.4. Let Ω be a CAD. Does there exist a family of smooth domains {Ω m } m such that Ω m Ω is CAD with constants that only depend on the NTA and Ahlfors regularity constants of Ω, and χ Ωm χ Ω in BV loc? We turn our attention to the Neumann and regularity problems on CAD see [KP] for the corresponding results on Lipschitz domains). We say that the regularity problem for L with data in W,p σ) is solvable i.e. R) p holds) if given f C Ω) W,p σ) the corresponding u satisfying { Lu = 0 in Ω verifies u = f on Ω Ñ u) L p σ) C f W,p σ), 2.) where C > 0 depends only on the geometry of the domain, the Ahlfors regularity constant of σ and the ellipticity constants of L. Ñ is a modified non-tangential maximal function, introduced to overcome the fact that u L loc. ÑF )Q) = X ΓQ) /2 F Z)dZ) 2. 2.2) BX,δX)/2) 5
We say that the Neumann problem for L with data in L p σ) is solvable i.e. N) p holds) if given f L 2 σ) L p σ) with Ω f dσ = 0 the corresponding u satisfying verifies { Lu = 0 in Ω A u n = f on Ω Ñ u) L p σ) C f L p σ), 2.3) where C > 0 depends only on the geometry of the domain, the Ahlfors regularity constant of σ and the ellipticity constants of L. Here n Q) denotes the inward unit normal to Ω. Question 2.5. Let Ω be a CAD. Does R) p for the Laplacian for some p >? Does N) p for the Laplacian for some p >? In [HMT] it was proved that for the Laplacian, given p > there exits ɛ > 0 such that if Ω is a CAD and n BMOσ) with n BMOσ) < ɛ then R) p and N) p hold. The boundary of this type of domain satisfies very specific geometric conditions which are not shared by general CAD. Question 2.6. Let Ω be a CAD. Suppose that Q Ω r>0 hq, r) <. If R) q0 holds for L 0 for some q 0 > 0, does R) q holds for L for some q > 0? Question 2.7. Let Ω be a CAD. Suppose that lim r 0 Q Ω hq, r) = 0. If R) q0 hold for L 0 for some q 0 > 0 do R) q and N) q hold for L 0? and N) q0 References [CFK] [CMS] L. Caffarelli, E. Fabes & C. Kenig, Completely singular elliptic-harmonic measures, Indiana Univ. Math. J. 30 98), 97-924. R. Coifman, Y. Meyer & E. Stein, Some new function spaces and their applications to harmonic analysis, J. Funct. Anal. 62 985), 304 335. [D] B. Dahlberg, Estimates of harmonic measure, Arch. Rational Mech. Anal. 65 977), no. 3, 275 288. [D2] [DJ] B. Dahlberg, On the absolute continuity of elliptic measure, American Journal of Mathematics 08 986), 9-38. G. David & D. Jerison, Lipschitz Approximation to Hypersurfaces, Harmonic Measure, and Singular Integrals, Indiana Univ. Math. J. 39 990), 83 845. [] [DKP]M. Dindos, C. Kenig & J. Pipher, BMO solvability and the A condition for elliptic operators, to appear J. Geometric Analysis. 6
[E] [F] [FKP] [HMT] [JK] [KP] L. Escauriaza, The L p Dirichlet problem for small perturbations of the Laplacian, Israel J. Math. 94 996), 353 366 R. Fefferman, A criterion for the absolute continuity of the harmonic measure associated with an elliptic operator, J. Amer. Math. Soc. 2 989), no., 27 35. R. Fefferman, C. Kenig & J. Pipher, The theory of weights and the Dirichlet problem for elliptic equations, Ann. of Math. 34 99), 65 24. S. Hofmann, M. Mitrea & M. Taylor, Singular integrals and elliptic boundary problems on regular Semmes-Kenig-Toro domains, to appear in International Mathematics Research Notices, Oxford University Press. D. Jerison & C. Kenig, Boundary behavior of harmonic functions in nontangentially accessible domains, Adv. in Math. 46 982), 80 47. C. Kenig & J. Pipher, The Neumann problem for elliptic equations with non-smooth coefficients, Invent. math. 3 993), 447-509. [KT] C. Kenig,& T. Toro, Harmonic measure on locally flat domains, Duke Math. J. 87 997), no. 3, 509 55. [MPT] [MM] [Se] E. Milakis, J. Pipher & T. Toro, Harmonic analysis on chord arc domains, in preparation. L. Modica & S. Mortola, Construction of a singular elliptic-harmonic measure, Manuscripta Math. 33 980), 8-98. S. Semmes, Analysis vs. Geometry on a Class of Rectifiable Hypersurfaces, Indiana Univ. J. 39 990), 005 035. 7