A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations

TL;DR

This paper introduces a new game-theoretic approach to solving fully nonlinear parabolic and elliptic PDEs with Neumann boundary conditions, extending previous models to handle boundary-specific rules and mixed conditions.

Contribution

It develops a deterministic two-person game framework that converges to the PDE solution, accommodating Neumann and mixed boundary conditions, which was not previously achieved.

Findings

Game-based solutions converge to PDE solutions in the viscosity sense.

New boundary rules effectively handle Neumann and mixed boundary conditions.

Framework extends previous models to broader classes of boundary conditions.

Abstract

We provide a deterministic-control-based interpretation for a broad class of fully nonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in a smooth domain. We construct families of two-person games depending on a small parameter which extend those proposed by Kohn and Serfaty (2010). These new games treat a Neumann boundary condition by introducing some specific rules near the boundary. We show that the value function converges, in the viscosity sense, to the solution of the PDE as the parameter tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.