# A Two-Phase Free Boundary Problem for Harmonic Measure

Presenter: Max Engelstein (University of Chicago)

### Abstract

We study the regularity of the free boundary for the following two-phase problem: let $\omega^+$ be the harmonic measure of a domain $\Omega$ and $\omega^-$ be the harmonic measure of $\Omega^- = (\overline{\Omega})^c$. Assume these measures are mutually absolutely continuous and let $h$, the Radon-Nikodym derivative, satisfy $\log(h) \in C^{0,\alpha}(\partial \Omega)$. We prove, under minimal geometric assumptions, that $\Omega$ is a $C^{1,\alpha}$ domain. The situation where $\log(h)$ has higher regularity is also discussed.