The two-phase problem for harmonic measure in VMO

Let $Ω^+\subset\mathbb R^{n+1}$ be an NTA domain and let $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ be an NTA domain as well. Denote by $ω^+$ and $ω^-$ their respective harmonic measures. Assume that $Ω^+$ is a $δ$-Reifenberg flat domain for some $δ>0$ small enough. In this paper we show that $\log\frac{dω^-}{dω^+}\in VMO(ω^+)$ if and only if $Ω^+$ is vanishing Reifenberg flat, $Ω^+$ and $Ω^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $Ω^+$ 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.

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