Mutual absolute continuity of interior and exterior harmonic measure implies rectifiability

Abstract

We show that, for disjoint domains in the Euclidean space whose boundaries satisfy a non-degeneracy condition, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries.

arXiv commentMade suggested changes by referee. To appear in Comm. Pure Appl. Math