An asymptotic mean value characterization for 𝑝-harmonic functions

Abstract

Abstract. We characterize p-harmonic functions in terms of an asymptotic mean value property. A p-harmonic function u is a viscosity solution to ∆ p u = div(|∇u| p−2 ∇u) = 0 with 1 < p ≤ ∞ in a domain Ω if and only if the expansionholds as ε → 0 for x ∈ Ω in a weak sense, which we call the viscosity sense. Here the coefficients α, β are determined by α + β = 1 and α/β = (p − 2)/(N + 2).

Dynamic Programming Principle for tug-of-war games with noise

“…In the general case in [6], see also [5], the authors studied functions that satisfy (1.1) and proved that they approximate solutions to the p-Laplacian…”

Waiting time effect for motion by positive second derivatives and applications

“…This approximation gives an interesting representation of the solutions to the equation. See also [17] for the deterministic game approach to general elliptic and parabolic equations and [22][23][24][25] for a stochastic tug-of-war game approach to the p-Laplace equation with p > 1. Related extensions of this new method to the Heisenberg group are recently addressed in [9,10].…”

A homogenization process for the strong -Laplacian

“…Functions verifying (2.4) 1 are called p-harmonious. Regarding literature about p-harmonious functions we refer to [22,6]. The choices α = (p − 2)/(p + N) and β = (N + 2)/(p + N) relate the continuous value of this game, with solutions to the 1-Homogeneous p-Laplacian (1.7).…”