Geodesic
On the dimension of $p$-harmonic measure in space
Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2197-2256

Voir la notice de l'article provenant de la source EMS Press

Let Ω⊂Rn, n≥3, and let p, 1∞, p=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 ∂Ω, in the setting of δ-Reifenberg flat domains. We prove, for p≥n, that there exists δ~=δ~(p,n)>0 small such that if Ω is a δ-Reifenberg flat domain with δ~, then p-harmonic measure is concentrated on a set of σ-finite Hn−1-measure. We prove, for p≥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, then there exist Wolff snowflakes such that the Hausdorff dimension of p-harmonic measure is less than n−1, while if 12, 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.
DOI : 10.4171/jems/420
Classification : 35-XX
Keywords: p-harmonic function, p-harmonic measure, Hausdorff dimension, Reifenberg flat domain, Wolff snowflake 
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     title = {On the dimension of $p$-harmonic measure in space},
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John L. Lewis; Kaj Nyström; Andrew Vogel. On the dimension of $p$-harmonic measure in space. Journal of the European Mathematical Society, Tome 15 (2013) no. 6, pp. 2197-2256. doi: 10.4171/jems/420

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