  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.

Analysis of PDEs