A Two-Phase Free Boundary Problem for Harmonic Measure

Abstract

We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp Hölder regularity result. The central difficulty is that there is no a priori non-degeneracy in the free boundary condition. Thus we must establish non-degeneracy by means of monotonicity formulae.

45 pages. This version has minor revisions as suggested by the referee