The two-phase problem for harmonic measure in VMO

Let Ω ^+⊂ℝ^n+1 be an NTA domain and let Ω ^-= ℝ^n+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.