Absolute continuity of harmonic measure for domains with lower regular boundaries

2016/05/24 by Akman, Murat, Azzam, Jonas, Mourgoglou, Mihalis · 1 citation
#28A75 #28A78 #31A15 #31B05 #35J25 #Analysis of PDEs (math.AP) #Classical Analysis and ODEs (math.CA) #FOS: Mathematics

paper · doi:10.48550/arxiv.1605.07291

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⊂ ℝ^d+1 is d-Ahlfors regular and splits ℝ^d+1 into two NTA domains then ω_Ω≪ \mathscrH^d on Γ∩ ∂Ω. This result is a natural generalisation of a result of Wu in [Wu86]. 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 [AHMMMTV], we characterize sets of absolute continuity with finite \mathscrH^d-measure both in terms of the cone point condition and in terms of the rectifiable structure of the boundary. This generalizes the results of McMillan in [McM69] and Pommerenke in [Pom86]. 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.

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• On the dimension drop for harmonic measure on uniformly non-flat Ahlfors-David regular boundaries

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