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Variational and viscosity operators for the evolutionary Hamilton–Jacobi equation

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VarViscOpHJ.pdf (752.6Kb)
Date
2018
Link to item file
https://hal.archives-ouvertes.fr/hal-01637617
Dewey
Analyse
Sujet
Hamilton–Jacobi equation; variational solution; viscosity solution; minmax selector; Lax-Oleinik semigroup
Journal issue
Communications in Contemporary Mathematics
Publication date
2018
Article pages
1-67
DOI
http://dx.doi.org/10.1142/S0219199718500189
URI
https://basepub.dauphine.fr/handle/123456789/18005
Collections
  • CEREMADE : Publications
Metadata
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Author
Roos, Valentine
60 CEntre de REcherches en MAthématiques de la DEcision [CEREMADE]
66 Département de Mathématiques et Applications - ENS Paris [DMA]
106 Unité de Mathématiques Pures et Appliquées [UMPA-ENSL]
Type
Article accepté pour publication ou publié
Abstract (EN)
We study the Cauchy problem for the first order evolutive Hamilton-Jacobi equation with a Lipschitz initial condition. The Hamiltonian is not necessarily convex in the momentum variable and not a priori compactly supported. We build and study an operator giving a variational solution of this problem, and get local Lipschitz estimates on this operator. Iterating this variational operator we obtain the viscosity operator and extend the estimates to the viscosity framework. We also check that the construction of the variational operator gives the Lax-Oleinik semigroup if the Hamiltonian is convex or concave in the momentum variable.

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