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# Analyse non lisse : - Fonction d'appui de la Jacobienne généralisée de Clarke et de son enveloppe plénière - Quelques applications aux équations de Hamilton-Jacobi du premier ordre (fonctions de Hopf-Lax, Hamiltoniens diff. convexes, solutions sci)

Abstract : The work we present in this manuscript is divided into two parts. The first part deals with the calculus of the support functions of Clarke's generalized jacobian and of its plenary hull, associated with a locally Lipschitz continuous mapping with range in \R^m. In 1975, Clarke established that the support function of his generalized subdifferential was a generalized directional derivative. It is therefore satisfactory to prove that the support function of the generalized jacobian is a generalized directional divergence''. The second part deals with several results concerning the application of methods from Nonsmooth analysis to first order Hamilton-Jacobi equations. Techniques such as convex duality or subdifferential calculus are used to prove that Hopf-Lax formulae provide explicit solutions of the associated Hamilton-Jacobi equation. We use neither the famous comparison principle from viscosity solution theory nor regularization procedures. The finite and the infinite dimensional cases are treated successively. These results are applied to get estimates for solutions of Hamilton-Jacobi equations whose hamiltonians are differences of convex functions. The last part is devoted to the construction of a lower semicontinuous solution of a Hamilton-Jacobi equation whose hamiltonian is the supremum of a parametric family of hamiltonians H(x,u,p) that are convex in p. We use the same techniques to prove the existence of a minimal lsc solution for Hamilton-Jacobi equations under weaker assumptions than the ones found in the traditional viscosity solution theory.
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https://tel.archives-ouvertes.fr/tel-00001203
Contributor : Cyril Imbert <>
Submitted on : Tuesday, March 12, 2002 - 2:10:31 PM
Last modification on : Friday, November 13, 2020 - 9:02:03 PM
Long-term archiving on: : Friday, April 2, 2010 - 7:52:51 PM

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• HAL Id : tel-00001203, version 1

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Cyril Imbert. Analyse non lisse : - Fonction d'appui de la Jacobienne généralisée de Clarke et de son enveloppe plénière - Quelques applications aux équations de Hamilton-Jacobi du premier ordre (fonctions de Hopf-Lax, Hamiltoniens diff. convexes, solutions sci). Mathématiques [math]. Université Paul Sabatier - Toulouse III, 2000. Français. ⟨tel-00001203⟩

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