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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. daniel@ann.jussieu.fr
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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

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