Rectifiability and elliptic measures on 1-sided NTA domains with Ahlfors-David regular boundaries

TL;DR

This paper characterizes the rectifiability of boundaries of 1-sided NTA domains with Ahlfors-David regularity through harmonic measure and elliptic measures, linking geometric and measure-theoretic properties.

Contribution

It provides a new characterization of boundary rectifiability in terms of harmonic and elliptic measures, extending previous results to more general elliptic operators.

Findings

Rectifiability is equivalent to absolute continuity of surface measure with harmonic measure.

Boundaries can be covered by portions of chord-arc subdomains.

Existence of exterior corkscrew points is linked to rectifiability.

Abstract

Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (aka uniform domain), i.e. a domain which satisfies interior Corkscrew and Harnack Chain conditions, and assume that $\partial\Omega$ is $n$-dimensional Ahlfors-David regular. We characterize the rectifiability of $\partial\Omega$ in terms of the absolute continuity of surface measure with respect to harmonic measure. We also show that these are equivalent to the fact that $\partial\Omega$ can be covered $\mathcal{H}^n$-a.e. by a countable union of portions of boundaries of bounded chord-arc subdomains of $\Omega$ and to the fact that $\partial\Omega$ possesses exterior corkscrew points in a qualitative way $\mathcal{H}^n$-a.e. Our methods apply to harmonic measure and also to elliptic measures associated with real symmetric second order divergence form elliptic operators with locally Lipschitz coefficients whose derivatives satisfy a natural qualitative Carleson condition.