Hostname: page-component-cd9895bd7-fscjk Total loading time: 0 Render date: 2024-12-24T16:28:45.819Z Has data issue: false hasContentIssue false
  • English
  • Français

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

Published online by Cambridge University Press:  13 August 2013

Jean-Paul Daniel
Jean-Paul Daniel*
Affiliation:
UPMC Univ. Paris 06, UMR 7598 Laboratoire Jacques-Louis Lions, CNRS, LJLL, 75005 Paris, France. [email protected]
Get access

Abstract

We provide a deterministic-control-based interpretation for a broad class of fullynonlinear parabolic and elliptic PDEs with continuous Neumann boundary conditions in asmooth domain. We construct families of two-person games depending on a small parameterε which extend those proposed by Kohn and Serfaty [21]. These new games treat a Neumann boundary conditionby introducing some specific rules near the boundary. We show that the value functionconverges, in the viscosity sense, to the solution of the PDE as ε tendsto zero. Moreover, our construction allows us to treat both the oblique and the mixed typeDirichlet–Neumann boundary conditions.

Keywords

Fully nonlinear elliptic equationsviscosity solutionsNeumann problemdeterministic controloptimal controldynamic programming principleoblique problemmixed-type Dirichlet–Neumann boundary conditions
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 4 , October 2013 , pp. 1109 - 1165
Copyright
© EDP Sciences, SMAI, 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Armstrong, S.N. and Smart, C.K., A finite difference approach to the infinity Laplace equation and tug-of-war games. Trans. Amer. Math. Soc. 364 (2012) 595–636. Google Scholar
Armstrong, S.N., Smart, C.K. and Somersille, S.J., An infinity Laplace equation with gradient term and mixed boundary conditions. Proc. Amer. Math. Soc. 139 (2011) 1763–1776. Google Scholar
Barles, G., Fully nonlinear Neumann type boundary conditions for second-order elliptic and parabolic equations. J. Differ. Equ. 106 (1993) 90–106. Google Scholar
G. Barles, Solutions de viscosité des équations de Hamilton-Jacobi. Springer-Verlag, Paris, Math. Appl. 17 (1994).
Barles, G., Nonlinear Neumann boundary conditions for quasilinear degenerate elliptic equations and applications. J. Differ. Equ. 154 (1999) 191–224. Google Scholar
Barles, G. and Busca, J., Existence and comparison results for fully nonlinear degenerate elliptic equations without zeroth-order term. Commun. Partial Differ. Equ. 26 (2001) 2323–2337. Google Scholar
Barles, G. and Lions, P.-L., Remarques sur les problèmes de réflexion oblique. C. R. Acad. Sci. Paris Sér. I Math. 320 (1995) 69–74. Google Scholar
Barles, G. and Perthame, B., Exit time problems in optimal control and vanishing viscosity method. SIAM J. Control Optim. 26 (1988) 1133–1148. Google Scholar
Barles, G. and Rouy, E., 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. Google Scholar
Barles, G. and Souganidis, P.E., Convergence of approximation schemes for fully nonlinear second order equations. Asymptotic Anal. 4 (1991) 271–283. Google Scholar
Cheridito, P., Soner, H.M., Touzi, N. and Victoir, N., Second-order backward stochastic differential equations and fully nonlinear parabolic PDEs. Comm. Pure Appl. Math. 60 (2007) 1081–1110. Google Scholar
Crandall, M.G., Ishii, H. and Lions, P.-L., User’s guide to viscosity solutions of second order partial differential equations. Bull. Amer. Math. Soc. (N.S.) 27 (1992) 1–67. Google Scholar
Dupuis, P. and Ishii, H., SDEs with oblique reflection on nonsmooth domains. Ann. Probab. 21 (1993) 554–580. Google Scholar
L.C. Evans, Partial differential equations, in Graduate Studies in Mathematics, vol. 19. Amer. Math. Soc., Providence, RI, second edition (2010).
A. Friedman, Differential games. In Handbook of game theory with economic applications, Vol. II, Handbooks in Econom. North-Holland, Amsterdam (1994) 781–799.
Y. Giga, Surface evolution equations, A level set approach. In Monographs in Mathematics. Birkhäuser Verlag, Basel 99 (2006).
Giga, Y. and Liu, Q., A billiard-based game interpretation of the Neumann problem for the curve shortening equation. Adv. Differ. Equ. 14 (2009) 201–240. Google Scholar
D. Gilbarg and N.S. Trudinger, Elliptic partial differential equations of second order. Classics in Mathematics. Springer-Verlag, Berlin (2001). Reprint of the 1998 edition.
Ishii, H., Fully nonlinear oblique derivative problems for nonlinear second-order elliptic PDEs. Duke Math. J. 62 (1991) 633–661. Google Scholar
Kohn, R.V. and Serfaty, S., A deterministic-control-based approach to motion by curvature. Commun. Pure Appl. Math. 59 (2006) 344–407. Google Scholar
Kohn, R. V. and Serfaty, S., A deterministic-control-based approach to fully nonlinear parabolic and elliptic equations. Commun. Pure Appl. Math. 63 (2010) 1298–1350. Google Scholar
Lions, P.-L., Neumann type boundary conditions for Hamilton-Jacobi equations. Duke Math. J. 52 (1985) 793–820. Google Scholar
Lions, P.-L., Menaldi, J.-L. Sznitman, A.-S., Construction de processus de diffusion réfléchis par pénalisation du domaine. C. R. Acad. Sci. Paris Sér. I Math. 292 (1981) 559–562. Google Scholar
Lions, P.-L. and Sznitman, A.-S., Stochastic differential equations with reflecting boundary conditions. Commun. Pure Appl. Math. 37 (1984) 511–537. Google Scholar
Liu, Q., On game interpretations for the curvature flow equation and its boundary problems. University of Kyoto RIMS Kokyuroku 1633 (2009) 138–150. Google Scholar
Peres, Y., Schramm, O., Sheffield, S. and Wilson, D.B., Tug-of-war and the infinity Laplacian. J. Amer. Math. Soc. 22 (2009) 167–210. Google Scholar
Sato, M.-H., Interface evolution with Neumann boundary condition. Adv. Math. Sci. Appl. 4 (1994) 249–264. Google Scholar
Tanaka, H., Stochastic differential equations with reflecting boundary condition in convex regions. Hiroshima Math. J. 9 (1979) 163–177. Google Scholar