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

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 $$p>2$$ p > 2 . Specifically, we characterize viscosity solutions to the p -Laplace equation Δ ₚu :=∇ · (|∇ u|ᵖ⁻² ∇ u) = f Δ p 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 ( $$f = 0$$ f = 0 ) have been previously established, leveraging the normalization Δ ₚNu:=|∇ u|²⁻ᵖ Δ ₚu = 0 Δ p N u : = | ∇ u | 2 - p Δ p u = 0 , which yields the 1-homogeneous normalized p -Laplacian, such normalization is not applicable when f ≠ 0 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.