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

Murat Akman

Vol. 286 (2017), No. 1, 25–37

DOI: 10.2140/pjm.2017.286.25

Abstract

We study the set of absolute continuity of $p$-harmonic measure $\mu$ associated to a positive weak solution to the $p$-Laplace equation with continuous zero boundary values and $\left(n-1\right)$-dimensional Hausdorff measure ${\mathcal{\mathscr{H}}}^{n-1}$ on locally flat domains in space. We prove that when $n\ge 2$ and $2<p<\infty$ and when $n\ge 3$ and $2-\eta <p<2$ for some $\eta >0$ there exist locally flat domains $\Omega \subset {ℝ}^n$ with locally finite perimeter and Borel sets $E\subset \partial \Omega$ such that $\mu \left(E\right)>0={\mathcal{\mathscr{H}}}^{n-1}\left(E\right)$.

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