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

### Kaj Nyström

Uppsala University, Sweden### John L. Lewis

University of Kentucky, Lexington, USA### Andrew Vogel

Syracuse University, USA

## Abstract

Let $\Omega\subset\mathbb {R}^{n}$, $n\geq 3$, and let $p$, $1 < p < \infty$, $p \not = 2$, 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 $p \geq n$, 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 $p \geq n$, 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

Kaj Nyström, John L. Lewis, 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