The two-phase problem for harmonic measure in VMO

Let ${\mathrm{\Omega }}^{+}\subset {\mathbb{R}}^{n+1}$ 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 ${\mathrm{\Omega }}^{+}$ is a $\delta$-Reifenberg flat domain for some $\delta >0$ small enough. In this paper we show that $\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)$ifandonlyif$\Omega^+$isvanishingReifenbergflat,$\Omega^+$and$\Omega^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of ${\mathrm{\Omega }}^{+}$ 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.