The two-phase problem for harmonic measure in VMO.

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, and that $\omega^+$ and $\omega^-$ are mutually absolutely continuous. In this paper we show that $\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)$ if and only if $\Omega^+$ is vanishing Reifenberg flat, the inner unit normal of $\Omega^+$ belongs to $VMO(\omega^+)$, and the density $\frac{d\omega^-}{d\omega^+}$ satisfies a reverse $3/2$-H\"older inequality with respect to $\omega^+$. This characterization can be considered as a two-phase counterpart of a well known related one-phase problem for harmonic measure solved by Kenig and Toro about 20 years ago.