Game theoretical asymptotic mean value properties for non-homogeneous p-Laplace problems

Abstract

We extend the classical mean value property for the Laplacian operator to address a nonlinear and non-homogeneous problem related to the p-Laplacian operator for p > 2. Specifically, we characterize viscosity solutions to the p-Laplace equation pu := ∇· (|∇u|p−2∇u) = f with a nontrivial right-hand side f , through novel asymptotic mean value formulas. While asymptotic mean value formulas for the homogeneous case ( f = 0) have been previously established, leveraging the normalization Np u := |∇u|2−p pu = 0, which yields the 1- homogeneous normalized p-Laplacian, such normalization is not applicable when f = 0. Furthermore, the mean value formulas introduced here motivate, for the first time in the literature, a game-theoretical approach for non-homogeneous p-Laplace equations. We also analyze the existence, uniqueness, and convergence of the game values, which are solutions to a dynamic programming principle derived from the mean value property.