On the dimension of $p$-harmonic measure in space

Abstract

Let $\Omega \subset {\mathbb{R}}^n$, $n\ge 3$, and let $p$, $1<p<\infty$, $p\ne 2$, be given. In this paper we study the dimension of $p$-harmonic measures that arise from non-negative solutions to the $p$-Laplace equation, vanishing on a portion of $\partial \Omega$, in the setting of $\delta$-Reifenberg flat domains. We prove, for $p\ge n$, that there exists $\tilde{\delta }=\tilde{\delta }(p,n)>0$ small such that if $\Omega$ is a $\delta$-Reifenberg flat domain with $\delta <\tilde{\delta }$, then $p$-harmonic measure is concentrated on a set of $\sigma$-finite $H^{n-1}$-measure. We prove, for $p\ge n$, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of $p$-harmonic measure is always less than $n-1$. We also prove that if $2<p<n$, then there exist Wolff snowflakes such that the Hausdorff dimension of $p$-harmonic measure is less than $n-1$, while if $1<p<2$, then there exist Wolff snowflakes such that the Hausdorff dimension of $p$-harmonic measure is larger than $n-1$. Furthermore, perturbing off the case $p=2,$ we derive estimates when $p$ is near 2 for the Hausdorff dimension of $p$-harmonic measure.