Harmonic measure and quantitative connectivity: geometric characterization of the $L^p$-solvability of the Dirichlet problem

Abstract

It is well-known that quantitative, scale invariant absolute continuity (more precisely, the weak-$A_{\infty }$ property) of harmonic measure with respect to surface measure, on the boundary of an open set $\Omega \subset {\mathbb{R}}^{n+1}$ with Ahlfors-David regular boundary, is equivalent to the solvability of the Dirichlet problem in $\Omega$, with data in $L^p(\partial \Omega )$ for some $p<\infty$. In this paper, we give a geometric characterization of the weak-$A_{\infty }$ property, of harmonic measure, and hence of solvability of the $L^p$ Dirichlet problem for some finite $p$. This characterization is obtained under background hypotheses (an interior corkscrew condition, along with Ahlfors-David regularity of the boundary) that are natural, and in a certain sense optimal: we provide counter-examples in the absence of either of them (or even one of the two, upper or lower, Ahlfors-David bounds); moreover, the examples show that the upper and lower Ahlfors-David bounds are each quantitatively sharp.

This paper is a combination of arXiv:1712.03696 and arXiv:1803.07975 To appear in Invent. Math