Absolute continuity of harmonic measure for domains with lower regular boundaries

Akman, Murat; Azzam, Jonas; Mourgoglou, Mihalis

doi:10.1016/j.aim.2019.01.021

Cited by 9 publications

(7 citation statements)

References 50 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…We showed that, for such domains, ω Ω ≪ H n on the subset of any ndimensional Lipschitz graph [AAM16], and hence, for these domains, we know that absolute continuity is equivalent to rectifiability of harmonic measure (versus rectifiability of the boundary). There are fewer positive results concerning absolute continuity and rectifiability of elliptic harmonic measures.…”

Section: 3mentioning

confidence: 92%

Tangent measures of elliptic measure and applications

Azzam

Mourgoglou

2019

Analysis & PDE

Self Cite

View full text Add to dashboard Cite

Tangent measure and blow-up methods, are powerful tools for understanding the relationship between the infinitesimal structure of the boundary of a domain and the behavior of its harmonic measure. We introduce a method for studying tangent measures of elliptic measures in arbitrary domains associated with (possibly non-symmetric) elliptic operators in divergence form whose coefficients have vanishing mean oscillation at the boundary. In this setting, we show the following for domains Ω ⊂ R n+1 :(1) We extend the results of Kenig, Preiss, and Toro [KPT09] by showing mutual absolute continuity of interior and exterior elliptic measures for any domains implies the tangent measures are a.e. flat and the elliptic measures have dimension n. (2) We generalize the work of Kenig and Toro [KT06] and show that VMO equivalence of doubling interior and exterior elliptic measures for general domains implies the tangent measures are always elliptic polynomials. (3) In a uniform domain that satisfies the capacity density condition and whose boundary is locally finite and has a.e. positive lower n-Hausdorff density, we show that if the elliptic measure is absolutely continuous with respect to n-Hausdorff measure then the boundary is rectifiable. This generalizes the work of Akman, Badger, Hofmann, and Martell [ABHM17]. Finally, we generalize one of the main results of [Bad11] by showing that if ω is a Radon measure for which all tangent measures at a point are harmonic polynomials vanishing at the origin, then they are all homogeneous harmonic polynomials. 2010 Mathematics Subject Classification. 31A15,28A75,28A78,28A33.

show abstract

Section: 3mentioning

confidence: 92%

Tangent measures of elliptic measure and applications

Azzam

Mourgoglou

2019

Analysis & PDE

Self Cite

View full text Add to dashboard Cite

show abstract

“…Beyond that, in a Wiener regular domain with large complement (cf. [1, Definition 1.5]), Akman et al [1] gave a characterization of sets of absolute continuity in terms of the cone point condition and the rectifiable structure of elliptic measure. Let us point out that in all of the just mentioned results, the absolute continuity happens locally.…”

Section: (B)mentioning

confidence: 99%

“…x ∈ ∂ for every weak solution u ∈ W 1,2 loc ( ) ∩ L ∞ ( ) of Lu = 0 in and for all (or for some) r > 0. (e) For every weak solution u ∈ W 1,2 loc ( ) ∩ L ∞ ( ) of Lu = 0 in and for σa.e. x ∈ ∂ there exists r x > 0 such that S r x α u(x) < ∞.…”

Section: (B)mentioning

confidence: 99%

Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

2022

View full text Add to dashboard Cite

In nice environments, such as Lipschitz or chord-arc domains, it is well-known that the solvability of the Dirichlet problem for an elliptic operator in $$L^p$$ L p , for some finite p, is equivalent to the fact that the associated elliptic measure belongs to the Muckenhoupt class $$A_\infty $$ A ∞ . In turn, any of these conditions occurs if and only if the gradient of every bounded null solution satisfies a Carleson measure estimate. This has been recently extended to much rougher settings such as those of 1-sided chord-arc domains, that is, sets which are quantitatively open and connected with a boundary which is Ahlfors–David regular. In this paper, we work in the same environment and consider a qualitative analog of the latter equivalence showing that one can characterize the absolute continuity of the surface measure with respect to the elliptic measure in terms of the finiteness almost everywhere of the truncated conical square function for any bounded null solution. As a consequence of our main result particularized to the Laplace operator and some previous results, we show that the boundary of the domain is rectifiable if and only if the truncated conical square function is finite almost everywhere for any bounded harmonic function. In addition, we obtain that for two given elliptic operators $$L_1$$ L 1 and $$L_2$$ L 2 , the absolute continuity of the surface measure with respect to the elliptic measure of $$L_1$$ L 1 is equivalent to the same property for $$L_2$$ L 2 provided the disagreement of the coefficients satisfy some quadratic estimate in truncated cones for almost everywhere vertex. Finally, for the case on which $$L_2$$ L 2 is either the transpose of $$L_1$$ L 1 or its symmetric part we show the equivalence of the corresponding absolute continuity upon assuming that the antisymmetric part of the coefficients has some controlled oscillation in truncated cones for almost every vertex.

show abstract

“…If H d | ∂Ω were σ-finite, then it would have tangents on a set K ⊆ ∂Ω of positive H d -measure by Theorem I . Theorem III in [AAM16] says that H d ω H d on the set of interior cone points for Ω, and since tangent points are also cone points, it follows from that H d ω H d on K. Since dim ω < d, we can find E so that H d (E) = 0 and ω(E ∩ K) = ω(K) > 0, but the mutual absolute continuity would imply 0 = H d (E) ≥ H d (E ∩ K) > 0, a contradiction, and thus proves the corollary.…”

Section: Introductionmentioning

confidence: 99%

Tangents, rectifiability, and corkscrew domains

Azzam¹

2018

Publicacions Matemàtiques

Self Cite

View full text Add to dashboard Cite

In a recent paper, Csörnyei and Wilson prove that curves in Euclidean space of σ-finite length have tangents on a set of positive H 1 -measure. They also show that a higher dimensional analogue of this result is not possible without some additional assumptions. In this note, we show that if Σ ⊆ R d+1 has the property that each ball centered on Σ contains two large balls in different components of Σ c and Σ has σfinite H d -measure, then it has d-dimensional tangent points in a set of positive H d -measure. As an application, we show that if the dimension of harmonic measure for an NTA domain in R d+1 is less than d, then the boundary domain does not have σ-finite H d -measure.We also give shorter proofs that Semmes surfaces are uniformly rectifiable and, if Ω ⊆ R d+1 is an exterior corkscrew domain whose boundary has locally finite H d -measure, one can find a Lipschitz subdomain intersecting a large portion of the boundary. CONTENTS 2010 Mathematics Subject Classification. 31A15,28A75,28A78.

show abstract

Absolute continuity of harmonic measure for domains with lower regular boundaries

Cited by 9 publications

References 50 publications

Tangent measures of elliptic measure and applications

Tangent measures of elliptic measure and applications

Elliptic Measures and Square Function Estimates on 1-Sided Chord-Arc Domains

Tangents, rectifiability, and corkscrew domains

Contact Info

Product

Resources

About