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Fermeture des fonctionnelles de diffusion et de l'élasticité linéaire pour la topologie de la Mosco-convergence

Abstract : The purpose of this thesis is to characterize all possible Mosco-limits of sequences of diffusion functionals or isotropic elasticity ones. It is a well-known fact that, when the diffusion coefficients in the scalar case, or the elasticity coefficients in the vectorial one, are not uniformly bounded, non local terms and killing terms can appear in the limit functional, despite the strong local nature of any element of those sequences. In the vectorial case, the limit functional can even involve some second derivative of the displacement. From a mechanical point of view, the effective properties of a composite material can differ fundamentally from those of its components. Umberto Mosco has shown that any limit of a sequence of diffusion functionals has to be a Dirichlet form. The contribution of the first part of this work provides a positive answer to the inverse problem. We show that any Dirichlet form is the Mosco-limit of some sequence of diffusion functionals. In a crucial step, we exhibit an explicit composite diffusive material, the effective properties of which contain an elementary non-local interaction. Then, using a step by step approach, we reach at each step a more general non-local interaction until obtaining all the Dirichlet forms. The second part of this work deals with the vectorial case. We show that the Mosco-closure of the set of isotropic elasticity functionals coincides with the set of all non-negative lower semi-continuous quadratic functionals which are objective. The proof of this result, which is far from being a simple generalisation of the scalar case, is based, at the start, on a result which is comparable to the scalar case. Then a fundamentally different approach is necessary.
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https://tel.archives-ouvertes.fr/tel-00006576
Contributor : Mohamed Camar-Eddine <>
Submitted on : Saturday, July 24, 2004 - 11:49:12 PM
Last modification on : Thursday, March 5, 2020 - 4:23:04 PM
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Mohamed Camar-Eddine. Fermeture des fonctionnelles de diffusion et de l'élasticité linéaire pour la topologie de la Mosco-convergence. Mathématiques [math]. Université du Sud Toulon Var, 2002. Français. ⟨tel-00006576⟩

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