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

Abstract

We show that if $\Omega \subset {ℝ}^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $VMO\left(\sigma \right)$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial \Omega$ belongs to $VMO\left(\sigma \right)$ 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.