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

UPMC Univ. Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, CNRS, LJLL, 75005 Paris, France
daniel@ann.jussieu.fr

Received: 22 February 2012
Revised: 29 October 2012

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 ε which extend those proposed by Kohn and Serfaty [21]. 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 ε tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet–Neumann boundary conditions.