The two-phase problem for harmonic measure in VMO and the chord-arc condition

Abstract

Let ${\Omega }^{+}\subset {\mathbb{R}}^{n+1}$ be a bounded $\delta$-Reifenberg flat domain, with $\delta >0$ small enough, possibly with locally infinite surface measure. Assume also that ${\Omega }^{-}={\mathbb{R}}^{n+1}\setminus \overline{{\Omega }^{+}}$ is an NTA domain as well and denote by ${\omega }^{+}$ and ${\omega }^{-}$ the respective harmonic measures of ${\Omega }^{+}$ and ${\Omega }^{-}$ with poles $p^{\pm }\in {\Omega }^{\pm }$. In this paper we show that the condition that $\log \frac{d{\omega }^{-}}{d{\omega }^{+}}\in VMO({\omega }^{+})$ is equivalent to ${\Omega }^{+}$ being a chord-arc domain with inner normal belonging to $VMO(H^n{|}_{\partial {\Omega }^{+}})$.

Theorem 1.1 has been improved by eliminating the assumption that the domain is vanishing Reifenberg flat in one of the implications. There are also other minor corrections and more precise citations. arXiv admin note: text overlap with arXiv:1904.00751