On the dimension of -harmonic measure in space
Kaj Nyström
Uppsala University, SwedenJohn L. Lewis
University of Kentucky, Lexington, USAAndrew Vogel
Syracuse University, USA
![On the dimension of $p$-harmonic measure in space cover](/_next/image?url=https%3A%2F%2Fcontent.ems.press%2Fassets%2Fpublic%2Fimages%2Fserial-issues%2Fcover-jems-volume-15-issue-6.png&w=3840&q=90)
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
Kaj Nyström, John L. Lewis, 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