A mean value formula for the variational p-Laplacian

Abstract

AbstractWe 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.