Absolute continuity of harmonic measure for domains with lower regular boundaries

TL;DR

This paper investigates the conditions under which harmonic measure is absolutely continuous with respect to surface measure on certain domains, extending previous results and characterizing boundary structures that ensure this property.

Contribution

It generalizes known results by establishing absolute continuity for domains with large complements and characterizes boundary structures using cone points and rectifiability.

Findings

Absolute continuity holds for domains with $d$-Ahlfors regular boundaries splitting $ ^{d+1}$ into NTA domains.

Almost every boundary point on Lipschitz graphs is a cone point.

Results extend to elliptic measures for divergence form elliptic operators.

Abstract

We study absolute continuity of harmonic measure with respect to surface measure on domains $\Omega$ that have large complements. We show that if $\Gamma\subset \mathbb{R}^{d+1}$ is $d$-Ahlfors regular and splits $ \mathbb{R}^{d+1}$ into two NTA domains then $\omega_{\Omega}\ll \mathscr{H}^{d}$ on $\Gamma\cap \partial\Omega$. This result is a natural generalisation of a result of Wu in [Wu86]. We also prove that almost every point in $\Gamma\cap\partial\Omega$ is a cone point if $\Gamma$ is a Lipschitz graph. Combining these results and a result from [AHMMMTV], we characterize sets of absolute continuity with finite $\mathscr{H}^{d}$-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.