The two-phase problem for harmonic measure in VMO

Abstract

Let ${\Omega }^{+}\subset {\mathbb{R}}^{n+1}$ be an NTA domain and let ${\Omega }^{-}={\mathbb{R}}^{n+1}\setminus \overline{{\Omega }^{+}}$ be an NTA domain as well. Denote by ${\omega }^{+}$ and ${\omega }^{-}$ their respective harmonic measures. Assume that ${\Omega }^{+}$ is a $\delta$-Reifenberg flat domain for some $\delta >0$ small enough. In this paper we show that $\log \frac{d{\omega }^{-}}{d{\omega }^{+}}\in VMO({\omega }^{+})$ if and only if ${\Omega }^{+}$ is vanishing Reifenberg flat, ${\Omega }^{+}$ and ${\Omega }^{-}$ have joint big pieces of chord-arc subdomains, and the inner unit normal of ${\Omega }^{+}$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.

Minor corrections