# The two-phase problem for harmonic measure in VMO

## TL;DR

This paper characterizes when the logarithm of the Radon-Nikodym derivative of harmonic measures in two NTA domains with VMO conditions is itself in VMO, linking geometric flatness and oscillation properties.

## Contribution

It establishes a two-phase analogue of the one-phase harmonic measure problem, connecting VMO regularity with geometric flatness and normal oscillation.

## Key findings

- VMO condition equivalent to vanishing Reifenberg flatness
- Domains have joint big pieces of chord-arc subdomains
- Inner normal oscillation vanishes under conditions

## Abstract

Let $\Omega^+\subset\mathbb R^{n+1}$ be an NTA domain and let $\Omega^-= \mathbb R^{n+1}\setminus \overline{\Omega^+}$ be an NTA domain as well. Denote by $\omega^+$ and $\omega^-$ their respective harmonic measures. Assume that $\Omega^+$ is a $\delta$-Reifenberg flat domain for some $\delta>0$ small enough. In this paper we show that $\log\frac{d\omega^-}{d\omega^+}\in VMO(\omega^+)$ if and only if $\Omega^+$ is vanishing Reifenberg flat, $\Omega^+$ and $\Omega^-$ have joint big pieces of chord-arc subdomains, and the inner unit normal of $\Omega^+$ has vanishing oscillation with respect to the approximate normal. This result can be considered as a two-phase counterpart of a more well known related one-phase problem for harmonic measure solved by Kenig and Toro.