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

Azzam, Jonas; Mourgoglou, Mihalis; Tolsa, Xavier

doi:10.2140/apde.2017.10.559

Download this article
	For screen
	For printing

Recent Issues

The Journal

Submission Guidelines

Submission Form

Policies for Authors

Ethics Statement

ISSN: 1948-206X (e-only)

ISSN: 2157-5045 (print)

Author Index

To Appear

Other MSP Journals

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

DOI: 10.2140/apde.2017.10.559

Abstract

We show that if $Ω \subset ℝ^{3}$ 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 $\partial Ω$ 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

Vol. 10, No. 3, 2017