Monge solutions for discontinuous Hamiltonians

Ariela Briani; Andrea Davini

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • Volume: 11, Issue: 2, page 229-251
  • ISSN: 1292-8119

Abstract

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We consider an Hamilton-Jacobi equation of the form

 H ( x , D u ) = 0 x Ω N , ( 1 ) 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.

How to cite

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Briani, Ariela, and Davini, Andrea. "Monge solutions for discontinuous Hamiltonians." ESAIM: Control, Optimisation and Calculus of Variations 11.2 (2010): 229-251. <http://eudml.org/doc/90763>.

@article{Briani2010,
abstract = { We consider an Hamilton-Jacobi equation of the form

 $$ H(x,Du)=0\quad x\in\Omega\subset\mathbb R^N,\qquad\qquad (1) $$ 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed. },
author = {Briani, Ariela, Davini, Andrea},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Viscosity solution; lax formula; Finsler metric.; Lax formula; Finsler metric; fronts in nonhomogeneous media; Monge subsolutions; Monge supersolutions; Dirichlet boundary conditions},
language = {eng},
month = {3},
number = {2},
pages = {229-251},
publisher = {EDP Sciences},
title = {Monge solutions for discontinuous Hamiltonians},
url = {http://eudml.org/doc/90763},
volume = {11},
year = {2010},
}

TY - JOUR
AU - Briani, Ariela
AU - Davini, Andrea
TI - Monge solutions for discontinuous Hamiltonians
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/3//
PB - EDP Sciences
VL - 11
IS - 2
SP - 229
EP - 251
AB - We consider an Hamilton-Jacobi equation of the form

 $$ H(x,Du)=0\quad x\in\Omega\subset\mathbb R^N,\qquad\qquad (1) $$ 
 where H(x,p) is assumed Borel measurable and quasi-convex in p. The notion of Monge solution, introduced by Newcomb and Su, is adapted to this setting making use of suitable metric devices. We establish the comparison principle for Monge sub and supersolution, existence and uniqueness for equation ([see full text]) coupled with Dirichlet boundary conditions, and a stability result. The relation among Monge and Lipschitz subsolutions is also discussed.
LA - eng
KW - Viscosity solution; lax formula; Finsler metric.; Lax formula; Finsler metric; fronts in nonhomogeneous media; Monge subsolutions; Monge supersolutions; Dirichlet boundary conditions
UR - http://eudml.org/doc/90763
ER -

References

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