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Monge solutions for discontinuous Hamiltonians

Published online by Cambridge University Press:  15 March 2005

Ariela Briani
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; [email protected]; [email protected]
Andrea Davini
Affiliation:
Dipartimento di Matematica, Università di Pisa Lago B. Pontecorvo 5, 56127 Pisa, Italy; [email protected]; [email protected]
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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 inp. 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 andsupersolution, existence and uniqueness for equation ([see full text])coupled with Dirichlet boundary conditions, and a stability result. Therelation among Monge and Lipschitz subsolutions is also discussed.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2005

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