A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations
ESAIM: Control, Optimisation and Calculus of Variations, Volume 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.

DOI: 10.1051/cocv/2013047
Classification: 49L25, 35J60, 35K55, 49L20, 35D40, 35M12, 49N90
Keywords: 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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     title = {A game interpretation of the {Neumann} problem for fully nonlinear parabolic and elliptic equations},
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     publisher = {EDP-Sciences},
     volume = {19},
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     doi = {10.1051/cocv/2013047},
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Daniel, Jean-Paul. A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations. ESAIM: Control, Optimisation and Calculus of Variations, Volume 19 (2013) no. 4, pp. 1109-1165. doi : 10.1051/cocv/2013047. http://www.numdam.org/articles/10.1051/cocv/2013047/

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