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Absolute continuity of harmonic measure for domains with lower regular boundaries

Akman, Murat and Azzam, Jonas and Mourgoglou, Mihalis (2019) 'Absolute continuity of harmonic measure for domains with lower regular boundaries.' Advances in Mathematics, 345. 1206 - 1252. ISSN 0001-8708

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Abstract

We study absolute continuity of harmonic measure with respect to surface measure on domains Ω that have large complements. We show that if Γ⊂Rd+1 is Ahlfors d-regular and splits Rd+1 into two NTA domains, then ωΩ«Hd on Γ∩∂Ω. This result is a natural generalization of a result of Wu in [49]. We also prove that almost every point in Γ∩∂Ω is a cone point if Γ is a Lipschitz graph. Combining these results and a result from [8], we characterize sets of absolute continuity (with finite Hd-measure if d>1) for domains with large complements both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. Even in the plane, this extends the results of McMillan in [38] and Pommerenke in [43], which were only known for simply connected planar domains. Finally, we also show our first result holds for elliptic measure associated with real second order divergence form elliptic operators with a mild assumption on the gradient of the matrix.

Item Type: Article
Uncontrolled Keywords: Harmonic measure, Absolute continuity, NTA domains, Chord-arc domains, Chord-arc surfaces, Elliptic measure
Divisions: Faculty of Science and Health > Mathematical Sciences, Department of
Depositing User: Elements
Date Deposited: 12 Sep 2019 15:40
Last Modified: 12 Sep 2019 16:15
URI: http://repository.essex.ac.uk/id/eprint/25311

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