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

Abstract

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

More detailed arguments in Lemmas 3.1 and 3.2