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

Jean-Paul Daniel

ESAIM: Control, Optimisation and Calculus of Variations (2013)

  • Volume: 19, Issue: 4, page 1109-1165
  • ISSN: 1292-8119

Abstract

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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.

How to cite

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Daniel, Jean-Paul. "A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations." ESAIM: Control, Optimisation and Calculus of Variations 19.4 (2013): 1109-1165. <http://eudml.org/doc/272773>.

@article{Daniel2013,
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.},
author = {Daniel, Jean-Paul},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {fully nonlinear elliptic equations; viscosity solutions; Neumann problem; deterministic control; optimal control; dynamic programming principle; oblique problem; mixed-type Dirichlet–Neumann boundary conditions; nonlinear elliptic equations; nonliner parabolic equations},
language = {eng},
number = {4},
pages = {1109-1165},
publisher = {EDP-Sciences},
title = {A game interpretation of the Neumann problem for fully nonlinear parabolic and elliptic equations},
url = {http://eudml.org/doc/272773},
volume = {19},
year = {2013},
}

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
PB - EDP-Sciences
VL - 19
IS - 4
SP - 1109
EP - 1165
AB - 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.
LA - eng
KW - fully nonlinear elliptic equations; viscosity solutions; Neumann problem; deterministic control; optimal control; dynamic programming principle; oblique problem; mixed-type Dirichlet–Neumann boundary conditions; nonlinear elliptic equations; nonliner parabolic equations
UR - http://eudml.org/doc/272773
ER -

References

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