On singularity of harmonic measure in space

Wu, Jun

doi:10.2140/pjm.1986.121.485

Cited by 23 publications

(21 citation statements)

References 7 publications

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“…A quantitative version of this theorem was proved by Lavrentiev in [30]. Due to examples of Bishop and Jones in [8] in the planar case, and of Ziemer in [39] and Wu in [38] in higher dimensions, neither H n | ∂Ω ≪ ω nor ω ≪ H n are true for arbitrary simply connected domains Ω ⊂ R n+1 with H n (∂Ω) < ∞ without imposing additional topological and/or non-topological conditions on ∂Ω. Quantitative mutual absolute continuity of harmonic measure and surface measure in higher dimensions was proven when Ω is a Lipschitz domain by Dahlberg in [12], and when Ω is non-tangentially accessible (NTA) (see Definition 1.9) and ∂Ω is Ahlfors-David regular (ADR, see Definition 1.5) independently by David and Jerison in [14] and by Semmes in [36].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 96%

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Akman¹,

Badger²,

Hofmann³

et al. 2017

Trans. Amer. Math. Soc.

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Abstract. Let Ω ⊂ R n+1 , n ≥ 2, be 1-sided NTA domain also known as uniform domain), i.e., a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that ∂Ω is n-dimensional Ahlfors-David regular. We characterize the rectifiability of ∂Ω in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that ∂Ω can be covered H n -a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of Ω and to the fact that ∂Ω possesses exterior corkscrew points in a qualitative way H n -a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.

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Section: Introduction and Statement Of Main Resultsmentioning

confidence: 96%

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Akman¹,

Badger²,

Hofmann³

et al. 2017

Trans. Amer. Math. Soc.

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“…They also showed that the result of [58] may fail in the absence of some topological hypothesis (e.g., simple connectedness). Examples constructed in [62] and [63] show that, in higher dimensions, some topological restrictions, even stronger than those needed in the planar case, are required for the absolute continuity of with respect to surface measure to the boundary.…”

Section: One Phase Casementioning

confidence: 99%

Geometric Measure Theory–Recent Applications

Toro¹

2019

Notices Amer. Math. Soc.

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Geometric Measure Theory (GMT) provides a framework to address questions in very different areas of mathematics, including calculus of variations, geometric analysis, potential theory, free boundary regularity, harmonic analysis, and theoretical computer science. Progress in different branches of GMT has led to the emergence of new challenges, making it a very vibrant area of research. In this note we will provide a historic background to some of the

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“…Bishop and Jones [10] found a uniformly rectifiable set E on the plane and some subset of E with zero arc-length which carries positive harmonic measure relative to the domain R 2 \ E. • Example 3. Wu proved in [44] that there exists a topological ball ⊂ R 3 and a set E ⊂ ∂ lying on a 2-dimensional hyperplane so that Hausdorff dimension of E is 1 (which implies σ (E) = 0) but ω(E) > 0.…”

Section: (B)mentioning

confidence: 99%

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

2022

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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.

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On singularity of harmonic measure in space

Cited by 23 publications

References 7 publications

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

Geometric Measure Theory–Recent Applications

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

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