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

Abstract

Let $Ω^+\subset\mathbb R^{n+1}$ be a bounded $δ$-Reifenberg flat domain, with $δ>0$ small enough, possibly with locally infinite surface measure. Assume also that $Ω^-= \mathbb R^{n+1}\setminus \overline{Ω^+}$ is an NTA domain as well and denote by $ω^+$ and $ω^-$ the respective harmonic measures of $Ω^+$ and $Ω^-$ with poles $p^\pm\inΩ^\pm$. In this paper we show that the condition that $\log\dfrac{dω^-}{dω^+} \in VMO(ω^+)$ is equivalent to $Ω^+$ being a chord-arc domain with inner normal belonging to $VMO(H^n|_{\partialΩ^+})$.

Theorems & Definitions (23)

• Theorem 1.1
• Lemma 2.1
• Theorem 2.2
• Theorem 2.3
• Lemma 3.1
• Lemma 3.2
• proof
• Lemma 3.3
• Lemma 4.1
• Lemma 4.2