On a two-phase problem for harmonic measure in general domains
  We show that, for disjoint domains in the Euclidean space, mutual absolute continuity of their harmonic measures implies absolute continuity with respect to surface measure and rectifiability in the intersection of their boundaries. This improves on our previous result which assumed that the boundaries satisfied the capacity density condition. 
Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa, and Alexander Volberg
math.CA math.AP