p-harmonic functions by way of intrinsic mean value properties

2020/03/02 by Arroyo, Ángel, Llorente, José G.
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2003.01210

Abstract

Let Ω⊂ℝⁿ be a bounded domain satisfying the uniform exterior cone condition. We establish existence and uniqueness of continuous solutions of the Dirichlet Problem associated to certain intrinsic nonlinear mean value properties in Ω. Furthermore we show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω, for p∈[2,∞). The proof of existence is constructive and the methods are entirely analytic, a fundamental tool being the construction of explicit, p-independent barrier functions in Ω.

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