p-harmonic functions by way of intrinsic mean value properties

Abstract

Let Ω be a bounded domain in ℝ n {\mathbb{R}^{n}} . Under appropriate conditions on Ω, we prove existence and uniqueness of continuous functions solving the Dirichlet problem associated to certain nonlinear mean value properties in Ω with respect to balls of variable radius. We also show that, when properly normalized, such functions converge to the p-harmonic solution of the Dirichlet problem in Ω for p ⩾ 2 {p\geqslant 2} . Existence is obtained via iteration, a fundamental tool being the construction of explicit universal barriers in Ω.

Strong Liouville theorems for discrete averaging operators

“…Note that when α = 0 then we retrieve harmonic functions (see also [1,2]), whereas for α = 1 we obtain the so-called harmonious functions in Ω, see [10]. In the Euclidean setting the functional equation (1.1) is closely related to game interpretations of the p-laplacian, for a certain α = α(p, n), see [12,13,3] and references therein.…”