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
Revised: 29 October 2012
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 . 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.
Mathematics Subject Classification: 49L25 / 35J60 / 35K55 / 49L20 / 35D40 / 35M12 / 49N90
Key words: Fully nonlinear elliptic equations / viscosity solutions / Neumann problem / deterministic control / optimal control / dynamic programming principle / oblique problem / mixed-type Dirichlet–Neumann boundary conditions
© EDP Sciences, SMAI, 2013