On the dimension of -harmonic measure in space

  • John L. Lewis

    University of Kentucky, Lexington, USA
  • Kaj Nyström

    Uppsala University, Sweden
  • Andrew Vogel

    Syracuse University, USA

Abstract

Let , , and let , , , be given. In this paper we study the dimension of -harmonic measures that arise from non-negative solutions to the -Laplace equation, vanishing on a portion of , in the setting of -Reifenberg flat domains. We prove, for , that there exists small such that if is a -Reifenberg flat domain with , then -harmonic measure is concentrated on a set of -finite -measure. We prove, for , that for sufficiently flat Wolff snowflakes the Hausdorff dimension of -harmonic measure is always less than . We also prove that if , then there exist Wolff snowflakes such that the Hausdorff dimension of -harmonic measure is less than , while if , then there exist Wolff snowflakes such that the Hausdorff dimension of -harmonic measure is larger than . Furthermore, perturbing off the case we derive estimates when is near 2 for the Hausdorff dimension of -harmonic measure.

Cite this article

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

DOI 10.4171/JEMS/420