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.
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
@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, Tome 19 (2013) no. 4, pp. 1109-1165. doi : 10.1051/cocv/2013047. http://www.numdam.org/articles/10.1051/cocv/2013047/
[1] and , A finite difference approach to the infinity Laplace equation and tug-of-war games. Trans. Amer. Math. Soc. 364 (2012) 595-636. | MR
[2] , and , An infinity Laplace equation with gradient term and mixed boundary conditions. Proc. Amer. Math. Soc. 139 (2011) 1763-1776. | MR
[3] , Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differ. Equ. 106 (1993) 90-106. | MR
[4] , Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, Math. Appl. 17 (1994). | MR
[5] , Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154 (1999) 191-224. | MR
[6] and , Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26 (2001) 2323-2337. | MR
[7] and , Remarques sur les problèmes de réflexion oblique. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 69-74. | MR
[8] and , Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133-1148. | MR
[9] and , A strong comparison result for the Bellman equation arising in stochastic exit time control problems and its applications. Commun. Partial Differ. Equ. 23 (1998) 1995-2033. | MR
[10] and , Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271-283. | MR
[11] , , and , Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 (2007) 1081-1110. | MR
[12] , and , User's guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1-67. | MR
[13] and , SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554-580. | MR
[14] , Partial differential equations, in Graduate Studies in Mathematics, vol. 19. Amer. Math. Soc., Providence, RI, second edition (2010). | MR
[15] , Differential games. In Handbook of game theory with economic applications, Vol. II, Handbooks in Econom. North-Holland, Amsterdam (1994) 781-799. | MR
[16] , Surface evolution equations, A level set approach. In Monographs in Mathematics. Birkhäuser Verlag, Basel 99 (2006). | MR
[17] and , A billiard-based game interpretation of the Neumann problem for the curve shortening equation. Adv. Differ. Equ. 14 (2009) 201-240. | MR
[18] and , Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition. | MR
[19] , Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs. Duke Math. J. 62 (1991) 633-661. | MR
[20] and , A deterministic-control-based approach to motion by curvature. Commun. Pure Appl. Math. 59 (2006) 344-407. | MR
[21] and , A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Commun. Pure Appl. Math. 63 (2010) 1298-1350. | MR
[22] , Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52 (1985) 793-820. | MR
[23] , A.-S. Sznitman, Construction de processus de diffusion réfléchis par pénalisation du domaine. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 559-562. | MR
[24] and , Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37 (1984) 511-537. | MR
[25] , On game interpretations for the curvature flow equation and its boundary problems. University of Kyoto RIMS Kokyuroku 1633 (2009) 138-150.
[26] , , and , Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc. 22 (2009) 167-210. | MR
[27] , Interface evolution with Neumann boundary condition. Adv. Math. Sci. Appl. 4 (1994) 249-264. | MR
[28] , Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163-177. | MR
Cité par Sources :
