Absolute continuity of harmonic measure for domains with lower regular boundaries

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.