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 ${\Delta }_{p}u:=\nabla \cdot (|\nabla u{|}^{p-2}\nabla u)=f$ 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 ${\Delta }_p^{\text{N}}u:=|\nabla u{|}^{2-p}{\Delta }_{p}u=0$, which yields the 1-homogeneous normalized $p$-Laplacian, such normalization is not applicable when $f\ne 0$. 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.

31 pages