Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 38, 2013, 351-366

DECAY OF A p-HARMONIC MEASURE IN THE PLANE

Niklas L.P. Lundström and Jonatan Vasilis

Umeå University, Department of Mathematics and Mathematical Statistics
SE-901 87 Umeå, Sweden; niklas.lundstrom 'at' math.umu.se

Chalmers University of Technology, Department of Mathematical Sciences
and University of Gothenburg, Department of Mathematical Sciences
SE-412 96 Göteborg, Sweden

Abstract. We study the asymptotic behaviour of a p-harmonic measure \omega_p, p \in (1,\infty], in a domain \Omega \subseteq R², subject to certain regularity constraints. Our main result is that \omega_p(B(w,\delta) \cap \partial\Omega, w₀) \approx \delta^q as \delta \to 0⁺, where q = q(v,p) is given explicitly as a function of v and p. Here, v is related to properties of \Omega near w. If p = \infty, this extends to some domains in Rⁿ. By a result due to Hirata, our result implies that the p-Green function for p \in (1,2) is not quasi-symmetric in plane C^1,1-domains.

2010 Mathematics Subject Classification: Primary 35J25, 35J70.

Key words: Harmonic measure, harmonic function, p-Laplace operator, generalized interior ball.

Reference to this article: N.L.P. Lundström and J. Vasilis: Decay of a p-harmonic measure in the plane. Ann. Acad. Sci. Fenn. Math. 38 (2013), 351-366.

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doi:10.5186/aasfm.2013.3808