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

Lewis, John L., Nyström, Kaj, and Vogel, Andrew. "On the dimension of $p$-harmonic measure in space." Journal of the European Mathematical Society 015.6 (2013): 2197-2256. <http://eudml.org/doc/277562>.

@article{Lewis2013,
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 for the Hausdorff dimension of $p$-harmonic measure when $p$ is near 2.},
author = {Lewis, John L., Nyström, Kaj, Vogel, Andrew},
journal = {Journal of the European Mathematical Society},
keywords = {$p$-harmonic function; $p$-harmonic measure; Hausdorff dimension; Reifenberg flat domain; Wolff snowflake; -harmonic function; -harmonic measure; Hausdorff dimension; Reifenberg flat domain; Wolff snowflake},
language = {eng},
number = {6},
pages = {2197-2256},
publisher = {European Mathematical Society Publishing House},
title = {On the dimension of $p$-harmonic measure in space},
url = {http://eudml.org/doc/277562},
volume = {015},
year = {2013},
}

TY - JOUR
AU - Lewis, John L.
AU - Nyström, Kaj
AU - Vogel, Andrew
TI - On the dimension of $p$-harmonic measure in space
JO - Journal of the European Mathematical Society
PY - 2013
PB - European Mathematical Society Publishing House
VL - 015
IS - 6
SP - 2197
EP - 2256
AB - 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 for the Hausdorff dimension of $p$-harmonic measure when $p$ is near 2.
LA - eng
KW - $p$-harmonic function; $p$-harmonic measure; Hausdorff dimension; Reifenberg flat domain; Wolff snowflake; -harmonic function; -harmonic measure; Hausdorff dimension; Reifenberg flat domain; Wolff snowflake
UR - http://eudml.org/doc/277562
ER -