Vol. 10, No. 3, 2017

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ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
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 Ω 3 is a two-sided NTA domain with AD-regular boundary such that the logarithm of the Poisson kernel belongs to VMO(σ), where σ is the surface measure of Ω, then the outer unit normal to Ω belongs to 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.

Keywords
harmonic measure, nontangentially accessible (NTA) domains, VMO, Poisson kernel, free boundary problem
Mathematical Subject Classification 2010
Primary: 31B25, 31A15, 31B15, 35R35
Milestones
Received: 28 April 2016
Accepted: 13 February 2017
Published: 17 April 2017
Authors
Jonas Azzam
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Edifici C Facultat de Ciències
08193 Bellaterra
Catalonia
Mihalis Mourgoglou
Departament de Matemàtiques
Universitat Autònoma de Barcelona
Edifici C Facultat de Ciències
08193 Bellaterra
Catalonia
Xavier Tolsa
Department of Mathematics
Universitat Autònoma de Barcelona
08193 Bellaterra
Catalonia
ICREA
Passeig Lluís Companys, 23
08010 Barcelona
Catalonia