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Monge solutions for discontinuous Hamiltonians

Published online by Cambridge University Press:  15 March 2005

Ariela Briani  and
Andrea Davini
Ariela Briani
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; briani@mail.dm.unipi.it; davini@dm.unipi.it
Andrea Davini
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; briani@mail.dm.unipi.it; davini@dm.unipi.it
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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.

Keywords

Viscosity solutionlax formulaFinsler metric.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 11 , Issue 2 , April 2005 , pp. 229 - 251
Copyright
© EDP Sciences, SMAI, 2005

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