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Let $\Omega \subset \mathbb{R}^{n+1}$, $n\geq 2$, be 1-sided NTA domain (akauniform domain), i.e. a domain which satisfies interior Corkscrew and HarnackChain conditions, and assume that $\partial\Omega$ is $n$-dimensionalAhlfors-David regular. We characterize the rectifiability of $\partial\Omega$in terms of the absolute continuity of surface measure with respect to harmonicmeasure. 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 ofportions of boundaries of bounded chord-arc subdomains of $\Omega$ and to thefact that $\partial\Omega$ possesses exterior corkscrew points in a qualitativeway $\mathcal{H}^n$-a.e. Our methods apply to harmonic measure and also toelliptic measures associated with real symmetric second order divergence formelliptic operators with locally Lipschitz coefficients whose derivativessatisfy a natural qualitative Carleson condition.

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