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

### John L. Lewis

University of Kentucky, Lexington, USA### Kaj Nyström

Uppsala University, Sweden### Andrew Vogel

Syracuse University, USA

## Abstract

Let $Ω⊂R_{n}$, $n≥3$, and let $p$, $1<p<∞$, $p=2$, 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 $∂Ω$, in the setting of $δ$-Reifenberg flat domains. We prove, for $p≥n$, that there exists $δ~=δ~(p,n)>0$ small such that if $Ω$ is a $δ$-Reifenberg flat domain with $δ<δ~$, then $p$-harmonic measure is concentrated on a set of $σ$-finite $H_{n−1}$-measure. We prove, for $p≥n$, 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.

## Cite this article

John L. Lewis, Kaj Nyström, Andrew Vogel, On the dimension of $p$-harmonic measure in space. J. Eur. Math. Soc. 15 (2013), no. 6, pp. 2197–2256

DOI 10.4171/JEMS/420