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 $\mathbb{R}^{n}$, $n\geq 2$. We prove that for fixed $p$ with $2<p<\infty$ there exists a Reifenberg flat domain $\Omega\subset\mathbb{R}^{n}$, $n\geq 2$, 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 $n\geq 3$. 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 $p=2$.