JournalsjemsVol. 15, No. 6pp. 2197–2256

On the dimension of pp-harmonic measure in space

  • Kaj Nyström

    Uppsala University, Sweden
  • John L. Lewis

    University of Kentucky, Lexington, USA
  • Andrew Vogel

    Syracuse University, USA
On the dimension of $p$-harmonic measure in space cover
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Abstract

Let ΩRn\Omega\subset\mathbb {R}^{n}, n3n\geq 3, and let pp, 1<p<1 < p < \infty, p2p \not = 2, be given. In this paper we study the dimension of pp-harmonic measures that arise from non-negative solutions to the pp-Laplace equation, vanishing on a portion of Ω\partial\Omega, in the setting of δ\delta-Reifenberg flat domains. We prove, for pnp \geq n, that there exists δ~=δ~(p,n)>0\tilde\delta=\tilde\delta(p,n)>0 small such that if Ω\Omega is a δ\delta-Reifenberg flat domain with δ<δ~\delta<\tilde\delta, then pp-harmonic measure is concentrated on a set of σ\sigma-finite Hn1H^{n-1}-measure. We prove, for pnp \geq n, that for sufficiently flat Wolff snowflakes the Hausdorff dimension of pp-harmonic measure is always less than n1n-1. We also prove that if 2<p<n2<p<n, then there exist Wolff snowflakes such that the Hausdorff dimension of pp-harmonic measure is less than n1n-1, while if 1<p<21<p<2, then there exist Wolff snowflakes such that the Hausdorff dimension of pp-harmonic measure is larger than n1n-1. Furthermore, perturbing off the case p=2,p = 2, we derive estimates when pp is near 2 for the Hausdorff dimension of pp-harmonic measure.

Cite this article

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

DOI 10.4171/JEMS/420