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

Daniel, Jean-Paul

ESAIM: Control, Optimisation and Calculus of Variations, Tome 19 (2013) no. 4, pp. 1109-1165.

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.

Classification : 49L25, 35J60, 35K55, 49L20, 35D40, 35M12, 49N90
Mots clés : fully nonlinear elliptic equations, viscosity solutions, Neumann problem, deterministic control, optimal control, dynamic programming principle, oblique problem, mixed-type Dirichlet-Neumann boundary conditions

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