On a two-phase problem for harmonic measure in general domains

TL;DR

This paper proves that for disjoint Euclidean domains, mutual absolute continuity of harmonic measures ensures boundary rectifiability and surface measure absolute continuity, extending previous results that required capacity density conditions.

Contribution

It establishes a new link between harmonic measure mutual absolute continuity and boundary rectifiability without capacity density assumptions.

Findings

Mutual absolute continuity implies boundary rectifiability.

Harmonic measures are absolutely continuous with respect to surface measure.

Results extend previous capacity density condition requirements.

Abstract

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.