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

Abstract

Let $\Omega^+\subset\mathbb R^{n+1}$ be a vanishing Reifenberg flat domain such that $\Omega^+$ and $\Omega^-=\mathbb R^{n+1}\setminus\overline {\Omega^+}$ have joint big pieces of chord-arc subdomains and the outer unit normal to $\Omega^+$ belongs to $VMO(\omega^+)$, where $\omega^\pm$ is the harmonic measure of $\Omega^\pm$. Up to now it was an open question if these conditions imply that $\log\dfrac{d\omega^-}{d\omega^+} \in VMO(\omega^+)$. 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 $\Omega^+$ is constant $\omega^+$-a.e. in $\partial\Omega^+$.