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oblique problem

ESAIM: Control, Optimisation and Calculus of Variations (1)

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A game interpretation of the Neumann problem for fullynonlinear parabolic and elliptic equations
Jean-Paul Daniel
Journal:

ESAIM: Control, Optimisation and Calculus of Variations / Volume 19 / Issue 4 / October 2013

Published online by Cambridge University Press:

13 August 2013, pp. 1109-1165

Print publication:

October 2013
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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.