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

Abstract

In this paper, we study the set of absolute continuity of p-harmonic measure, $\mu$, and $(n-1)-$dimensional Hausdorff measure, ${\mathcal{H}}^{n-1}$, on locally flat domains in ${\mathbb{R}}^n$, $n\ge 2$. We prove that for fixed $p$ with $2<p<\infty$ there exists a Reifenberg flat domain $\Omega \subset {\mathbb{R}}^n$, $n\ge 2$, with ${\mathcal{H}}^{n-1}(\partial \Omega )<\infty$ and a Borel set $K\subset \partial \Omega$ such that $\mu (K)>0={\mathcal{H}}^{n-1}(K)$ where $\mu$ is the p-harmonic measure associated to a positive weak solution to p-Laplace equation in $\Omega$ with continuous boundary value zero on $\partial \Omega$. We also show that there exists such a domain for which the same result holds when $p$ is fixed with $2-\eta <p<2$ for some $\eta >0$ provided that $n\ge 3$. This work is a generalization of a recent result of Azzam, Mourgoglou, and Tolsa when the measure $\mu$ is harmonic measure at $x$, $\omega ={\omega }^x$, associated to the Laplace equation, i.e when $p=2$.

13 pages, some minor changes

References in corpus (1)

• Null sets of harmonic measure on NTA domains: Lipschitz approximation revisited