A mean value formula for the variational $p$-Laplacian

Abstract

We prove a new asymptotic mean value formula for the $p$-Laplace operator, $${\Delta }_{p}u=\text{div}(|\nabla u{|}^{p-2}\nabla u),$$ 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.