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arXiv · 1604.06343

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

Jonas Azzam·Mihalis Mourgoglou·Xavier Tolsa
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Abstract

We show that if $Ω\subset\mathbb R^3$ is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to $\textrm{VMO}(σ)$, where $σ$ is the surface measure of $Ω$, then the outer unit normal to $\partialΩ$ belongs to $\textrm{VMO}(σ)$ 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.

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Jonas Azzam, Mihalis Mourgoglou, Xavier Tolsa. 2016-05-02. The one-phase problem for harmonic measure in two-sided NTA domains. https://doi.org/10.2140/apde.2017.10.559

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