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

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

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

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topDaniel, 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 -

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