Approximate tangents, harmonic measure, and domains with rectifiable boundaries

TL;DR

This paper investigates the structure of boundaries of certain domains in Euclidean space, showing they can be decomposed into Lipschitz boundaries, which implies absolute continuity of surface measure with harmonic measure under specific conditions.

Contribution

It establishes that boundaries with approximate tangents and weak regularity conditions can be covered by Lipschitz domain boundaries, extending geometric measure theory results.

Findings

Boundaries with approximate tangents are coverable by Lipschitz domain boundaries.

Domains with finite perimeter and regular boundaries have measure-theoretic boundaries covered by Lipschitz domains.

Surface measure is absolutely continuous with harmonic measure in these domains.

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$, $n \geq 1$, be an open and connected set. Set $\mathcal{T}_n$ to be the set of points $\xi \in \partial \Omega$ so that there exists an approximate tangent $n$-plane for $\partial\Omega$ at $\xi$ and $\partial\Omega$ satisfies the weak lower Ahlfors-David $n$-regularity condition at $\xi$. We first show that $\mathcal{T}_n$ can be covered by a countable union of boundaries of bounded Lipschitz domains. Then, letting $\partial^\star \Omega$ be a subset of $\mathcal{T}_n$ where $\Omega$ satisfies an appropriate thickness condition, we prove that $\partial^\star \Omega$ can be covered by a countable union of boundaries of bounded Lipschitz domains contained in $\Omega$. As a corollary we obtain that if $\Omega$ has locally finite perimeter, $\partial\Omega$ is weakly lower Ahlfors-David $n$-regular, and the measure-theoretic boundary coincides with the topological boundary of $\Omega$ up to a set of $\mathcal{H}^n$-measure zero, then $\partial \Omega$ can be covered, up to a set of $\mathcal{H}^n$-measure zero, by a countable union of boundaries of bounded Lipschitz domains that are contained in $\Omega$. This implies that in such domains, $\mathcal{H}^n|_{\partial\Omega}$ is absolutely continuous with respect to harmonic measure.