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

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.The first and last authors have been supported in part by the Spanish Ministry of Economy and Competitiveness, through the “Severo Ochoa Programme for Centres of Excellence in R&D” (SEV-2015-0554), and they acknowledge that the research leading to these results has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC agreement no. 615112 HAPDEGMT. The second author was partially supported by an NSF postdoctoral fellowship, DMS 1203497, and by NSF grant DMS 1500382. The third author was partially supported by NSF grant DMS 1361701.Peer reviewe