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

John L. Lewis; Kaj Nyström; Andrew Vogel

Journal of the European Mathematical Society (2013)

- Volume: 015, Issue: 6, page 2197-2256
- ISSN: 1435-9855

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topLewis, 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 -

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