The one-phase problem for harmonic measure in two-sided NTA domains

Abstract

We show that if $\Omega \subset {\mathbb{R}}^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\text{VMO}(\sigma )$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial \Omega$ belongs to $\text{VMO}(\sigma )$ too. The analogous result fails for dimensions larger than $3$. This answers a question posed by Kenig and Toro and also by Bortz and Hofmann.

In this version we have corrected minor typos, and also we have cited correctly some previous work by Kenig and Toro, of which we were not aware before