The two-phase problem for harmonic measure in VMO (Q2189548)

Let \(\Omega ^{+},\Omega ^{-}\) be disjoint domains in \(\mathbb{R}^{n+1}\) with intersecting boundaries, and let \(\omega ^{+},\omega ^{-}\) be the corresponding harmonic measures. There has been much recent interest in investigating how the analytic properties of \(\omega ^{+}\) and \(\omega ^{-}\) on \(\partial \Omega ^{+}\cap \partial \Omega ^{-}\) are related to the geometric properties of \(\partial \Omega ^{+}\cap \partial \Omega ^{-}\). The present paper assumes that a bounded NTA domain \(\Omega ^{+}\) is \(\delta \)-Reifenberg flat for some small \(\delta >0\), and that \(\Omega ^{-}:=\mathbb{R}^{n+1}\backslash \overline{\Omega ^{+}}\) is also an NTA domain. It then gives a geometric characterization of the condition that \(\omega ^{+}\) and \(\omega ^{-}\) are mutually absolutely continuous and \(\log (d\omega^{-}/d\omega ^{+})\in \mathrm{VMO}(\omega ^{+})\). The characterization involves, in particular, the oscillation of the unit normal to the boundary.