A GAME INTERPRETATION OF THE NEUMANN PROBLEM FOR FULLY NONLINEAR PARABOLIC AND ELLIPTIC EQUATIONS

作者：Daniel Jean Paul^*

来源：ESAIM: Control, Optimisation and Calculus of Variations , 2013, 19(4): 1109-1165.

DOI：10.1051/cocv/2013047

• Fully nonlinear elliptic equations
• viscosity solutions
• Neumann problem
• deterministic control
• optimal control
• dynamic programming principle
• oblique problem
• mixed-type Dirichlet-Neumann boundary conditions

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 e 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 e tends to zero. Moreover, our construction allows us to treat both the oblique and the mixed type Dirichlet-Neumann boundary conditions.

• 出版日期2013-10