A Two-Phase Free Boundary Problem for Harmonic Measure

TL;DR

This paper investigates a two-phase free boundary problem related to harmonic measure, establishing sharp regularity results despite initial non-degeneracy challenges by employing monotonicity formulas.

Contribution

It introduces a novel approach to prove sharp regularity for a two-phase harmonic measure problem without assuming non-degeneracy a priori.

Findings

Proved sharp Hölder regularity for the free boundary.

Developed monotonicity formulae to establish non-degeneracy.

Extended understanding of free boundary regularity in harmonic measure problems.

Abstract

We study a 2-phase free boundary problem for harmonic measure first considered by Kenig and Toro and prove a sharp H\"older 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.