The two-phase problem for harmonic measure in VMO

Let Ω +⊂ Rn+1 be an NTA domain and let Ω -= Rn+1\ 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 logdω-dω+∈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.