A game-theoretical interpretation for a doubly nonlinear parabolic equation

TL;DR

This paper develops a game-theoretical framework for a doubly nonlinear parabolic PDE involving the p-Laplacian, introducing a new asymptotic mean value formula and connecting solutions to a stochastic game.

Contribution

It introduces a novel asymptotic mean value formula and a dynamic programming principle for the p-Laplacian, linking PDE solutions to a stochastic game.

Findings

New AMVF for p-Laplacian robust to gradient vanishing

Solutions to the DPP converge to viscosity solutions

Solutions coincide with value functions of a stochastic game

Abstract

We introduce a game-theoretical framework for the doubly nonlinear parabolic equation \[ |\partial_t u|^{p-2} \partial_t u - \Delta_p u = 0. \] where $\Delta_p u = \nabla \cdot ( |\nabla u |^{p-2} \nabla u)$ with $p>2$ is the standard $p-$Laplacian. A key feature to our approach is a new asymptotic mean value formula (AMVF) for the $p-$Laplacian that is robust even when the gradient vanishes and is independent of the sign of the $p-$Laplacian. This new AMVF leads naturally to a dynamic programming principle (DPP) whose solutions converge to the viscosity solution of the boundary value problem for the differential equation. In addition, solutions to the DPP coincide with value functions for a stochastic, two-players, zero-sum game that we introduce and analyze here.