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.