A counterexample regarding a two-phase problem for harmonic measure in VMO

Abstract

Let $Ω^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $Ω^+$ and $Ω^-=\mathbb R^{n+1}\setminus\overline {Ω^+}$ have joint big pieces of chord-arc subdomains and the outer unit normal to $Ω^+$ belongs to $VMO(ω^+)$, where $ω^\pm$ is the harmonic measure of $Ω^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$. In this paper we answer this question in the negative by constructing an appropriate counterexample in $\mathbb R^2$, with the additional property that the outer unit normal to $Ω^+$ is constant $ω^+$-a.e. in $\partialΩ^+$.