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.
Keywords: fully nonlinear elliptic equations, viscosity solutions, Neumann problem, deterministic control, optimal control, dynamic programming principle, oblique problem, mixed-type Dirichlet-Neumann boundary conditions
@article{COCV_2013__19_4_1109_0, author = {Daniel, Jean-Paul}, title = {A game interpretation of the {Neumann} problem for fully nonlinear parabolic and elliptic equations}, journal = {ESAIM: Control, Optimisation and Calculus of Variations}, pages = {1109--1165}, publisher = {EDP-Sciences}, volume = {19}, number = {4}, year = {2013}, doi = {10.1051/cocv/2013047}, mrnumber = {3182683}, zbl = {1283.49028}, language = {en}, url = {http://www.numdam.org/articles/10.1051/cocv/2013047/} }
TY - JOUR AU - Daniel, Jean-Paul TI - A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations JO - ESAIM: Control, Optimisation and Calculus of Variations PY - 2013 SP - 1109 EP - 1165 VL - 19 IS - 4 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2013047/ DO - 10.1051/cocv/2013047 LA - en ID - COCV_2013__19_4_1109_0 ER -
%0 Journal Article %A Daniel, Jean-Paul %T A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations %J ESAIM: Control, Optimisation and Calculus of Variations %D 2013 %P 1109-1165 %V 19 %N 4 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2013047/ %R 10.1051/cocv/2013047 %G en %F COCV_2013__19_4_1109_0
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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