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

TL;DR

This paper proves that in three-dimensional two-sided NTA domains with AD-regular boundary, if the Poisson kernel's logarithm is in VMO, then the outer unit normal also belongs to VMO, answering a question by Kenig and Toro.

Contribution

It establishes a new regularity result linking the Poisson kernel and boundary normals in 3D NTA domains, resolving an open question.

Findings

The outer unit normal belongs to VMO if the Poisson kernel's log is in VMO.

The result does not extend to dimensions higher than 3.

Answers a question posed by Kenig and Toro.

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 $\textrm{VMO}(\sigma)$, where $\sigma$ is the surface measure of $\Omega$, then the outer unit normal to $\partial\Omega$ belongs to $\textrm{VMO}(\sigma)$ 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.