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

Abstract:

Let $\Omega \subset \mathbb {R}^{n+1}$, $n\geq 2$, be a 1-sided NTA domain (also known as a 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.