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

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.

Locations

• KTH Publication Database DiVA (KTH Royal Institute of Technology) - View - PDF
• Journal of the European Mathematical Society - View - PDF