The one-phase problem for harmonic measure in two-sided NTA domains
The one-phase problem for harmonic measure in two-sided NTA domains
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DOI：
10.2140/apde.2017.10.559
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发表时间：
2016-04
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影响因子：
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通讯作者：
Jonas Azzam;Mihalis Mourgoglou;X. Tolsa
中科院分区：
文献类型：
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作者：
Jonas Azzam;Mihalis Mourgoglou;X. Tolsa
我们证明了：如果$\Omega\subset\mathbb R^3$是一个具有AD-正则边界的双边NTA域，使得Poisson核的对数属于$\textrm{VMO}（\sigma）$，其中$\sigma$是$\Omega$的曲面测度，那么$\partial\Omega$的外单位法向也属于$\textrm{VMO}（\sigma）$.对于大于3美元的尺寸，类似的结果失败。这回答了凯尼格和托罗以及博尔茨和霍夫曼提出的一个问题。
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.