Vol. 286, No. 1, 2017

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ISSN: 0030-8730
On the absolute continuity of $p$-harmonic measure and surface measure in Reifenberg flat domains

Murat Akman

Vol. 286 (2017), No. 1, 25–37
Abstract

We study the set of absolute continuity of p-harmonic measure μ associated to a positive weak solution to the p-Laplace equation with continuous zero boundary values and (n 1)-dimensional Hausdorff measure n1 on locally flat domains in space. We prove that when n 2 and 2 < p < and when n 3 and 2 η < p < 2 for some η > 0 there exist locally flat domains Ω n with locally finite perimeter and Borel sets E Ω such that μ(E) > 0 = n1(E).

Keywords
Hausdorff dimension of $p$-harmonic measure, harmonic measure, $p$-harmonic measure, nonlinear elliptic PDEs, Hausdorff measure, Hausdorff dimension, singular sets for $p$-harmonic measure, Reifenberg flat domains, NTA domains
Mathematical Subject Classification 2010
Primary: 28A75, 28A78, 35J25, 37F35, 35J92, 31A15
Milestones
Received: 5 November 2015
Revised: 16 August 2016
Accepted: 7 October 2016
Published: 9 December 2016
Authors
Murat Akman
Department of Mathematics
University of Connecticut
Storrs, CT 06268-3009
United States