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

Abstract

Let $\Omega \subset {\mathbb{R}}^{n+1}$ be an open set, not necessarily connected, with an $n$-dimensional uniformly rectifiable boundary. We show that $\partial \Omega$ may be approximated in a "Big Pieces" sense by boundaries of chord-arc subdomains of $\Omega$, and hence that harmonic measure for $\Omega$ is weak-$A_{\infty }$ with respect to surface measure on $\partial \Omega$, provided that $\Omega$ satisfies a certain weak version of a local John condition. Under the further assumption that $\Omega$ satisfies an interior Corkscrew condition, and combined with our previous work, and with recent work of Azzam, Mourgoglou and Tolsa, this yields a geometric characterization of domains whose harmonic measure is quantitatively absolutely continuous with respect to surface measure and hence a haracterization of the fact that the associated $L^p$-Dirichlet problem is solvable for some finite $p$.