A mean value formula for the p-Laplacian

Abstract

In this paper we introduce an asymptotic mean value formula of the form $u(x) - \B_ε^p (u) (x) =ε^2 Δ_p u(x) + o(ε^2)$ for the $p-$Laplacian, $\plap u := \nabla \cdot \big( |\nabla u |^{p-2} \nabla u \big)$. Here $\B_ε^p (u)$ is a nonlinear expression that involve the mean value and the surmemum/infimum of $u$ in suitable balls. This asymptotic mean value formula allows us to characterize viscosity solutions to equations with different homogeneities in different sets of variables, like $\plap^{x'} u(x)+\qlap^{x''}u(x)=f(x)$ (for $x=(x',x'') \in\mathbb{R}^d$, $x'=(x_1,...,x_n)$ and $x''=(x_{n+1,...,x_d})$), and the parabolic $p$-Laplace heat equation $\partial_t u (x,t)-Δ_p u(x,t)=0$. Moreover, the resulting asymptotic mean value formulas are related to two-player zero sum games whose value functions, characterized as the unique solution to the corresponding dynamic programming principle, converge as the parameter that controls the size of the steps in the game goes to zero, to the unique viscosity solution to the corresponding boundary value problem for the operator under consideration. Finally, we also include a Holder regularity result for $\plap^{x'} u(x)+\qlap^{x''}u(x)=0$ proved using the game theoretical interpretation of the problem.