论文标题
VMO和和弦弧条件中的谐波测量的两相问题
The two-phase problem for harmonic measure in VMO and the chord-arc condition
论文作者
论文摘要
令$ω^+\ subset \ mathbb r^{n+1} $为一个有限的$δ$ -DEIFENBERG平面域，$δ> 0 $小，可能具有本地无限的表面度量。 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 $p^\pm\inΩ^\pm$.在本文中，我们表明，$ \ log \ dfrac {dΩ^ - } {dΩ^+} \在vmo（ω^+）$中等同于$ω^+$是一个和弦 - arc域，内部正常属于$ vmo（H^n | _ ____ {\partialΩ^+}）$。
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 $VMO(H^n|_{\partialΩ^+})$.
