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

TL;DR

This paper establishes an equivalence between the vanishing mean oscillation condition of the harmonic measure ratio and the geometric property of the domain being chord-arc with a normal in VMO, in the context of Reifenberg flat and NTA domains.

Contribution

It proves that the VMO condition on the harmonic measure ratio characterizes the chord-arc domain with VMO-normal, linking harmonic measure behavior to geometric regularity.

Findings

VMO condition on harmonic measure ratio implies chord-arc domain.

Chord-arc domain with VMO-normal is characterized by harmonic measure ratio in VMO.

Establishes a geometric-analytic equivalence in Reifenberg flat and NTA domains.

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 $\log\dfrac{d\omega^-}{d\omega^+} \in VMO(\omega^+)$ is equivalent to $\Omega^+$ being a chord-arc domain with inner normal belonging to $VMO(H^n|_{\partial\Omega^+})$.