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