2024/12/27 by Félix del Teso, Julio D. Rossi, del Teso, Félix +1 · 1 citation
Mathematics · #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2412.19410
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 Δp u:=∇⋅(|∇ 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 ΔpNu:=|∇ u|2-p Δp u = 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.