A mean value formula for the variational p-Laplacian

Abstract

We prove a new asymptotic mean value formula for the p-Laplace operator, $$\begin{aligned} \Delta _pu=\text{ div }(|\nabla u|^{p-2}\nabla u), \quad 1<p<\infty \end{aligned}$$ Δ p u = div ( | ∇ u | p - 2 ∇ u ) , 1 < p < ∞ valid in the viscosity sense. In the plane, and for a certain range of p, the mean value formula holds in the pointwise sense. We also study the existence, uniqueness and convergence of the related dynamic programming principle.

Finite difference schemes for the parabolic p-Laplace equation

“…A proof of consistency (A c ) can be found in Theorem 2.1 in [8]. Assumption (A ω ) is trivially true for r = h since…”

“…Recently, a new monotone finite difference discretization of the p-Laplacian was introduced by the authors in [9]. It is based on the mean value property presented in [4,8]. The aim of this paper is to propose an explicit-in-time finite difference numerical scheme for the following Cauchy problem…”

Asymptotic mean value formulas for parabolic nonlinear equations

“…This is weaker than requiring the asymptotic formula to hold in the classical sense, yet enough to characterize p-harmonic functions. For mean value properties for the p-Laplacian in the Heisenberg group see [16], and for the standard variational p-Laplacian see [14]. See also [2] and the recently published book [5] for historical references and more general equations.…”