#### Vol. 10, No. 3, 2017

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

### Jonas Azzam, Mihalis Mourgoglou and Xavier Tolsa

Vol. 10 (2017), No. 3, 559–588
##### Abstract

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

##### Keywords
harmonic measure, nontangentially accessible (NTA) domains, VMO, Poisson kernel, free boundary problem
##### Mathematical Subject Classification 2010
Primary: 31B25, 31A15, 31B15, 35R35