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A game interpretation of the Neumann problem for fully nonlinear 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. daniel@ann.jussieu.fr
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Abstract

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 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

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