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

TL;DR

This paper establishes a geometric framework linking boundary rectifiability and connectivity conditions of domains to the absolute continuity of harmonic measure, thereby characterizing the solvability of the $L^p$-Dirichlet problem.

Contribution

It provides a new geometric characterization of domains with $L^p$-solvability of the Dirichlet problem via harmonic measure and connectivity conditions.

Findings

Harmonic measure is weak-$A_ abla$ with respect to surface measure under certain conditions.

Domains with boundary approximable by chord-arc subdomains have harmonic measure with good absolute continuity properties.

The results connect geometric boundary properties to PDE solvability in a quantitative manner.

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 $p$.