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Multi-parameter homogenization by localization and blow-up

Published online by Cambridge University Press:  12 July 2007

Felipe Alvarez
Affiliation:
Centro de Modelamiento Matemático (CNRS UMR 2071), Departamento de Ingeniería Matemática, Universidad de Chile, Av. Blanco Encalada 2120, Santiago, Chile ([email protected])
Jean-Philippe Mandallena
Affiliation:
Equipe de Mathématiques, d'Informatique et Applications de Nîmes (EMIAN), Centre Universitaire de Formation et de Recherche de Nîmes, Site des Carmes, Place Gabriel Péri, Cedex 01, 30021 Nîmes, FranceInstitut de Mathématiques et Modélisation de Montpellier (I3M) (CNRS UMR 5149), Université Montpellier II, Place Eugène Bataillon, 34090 Montpellier, France ([email protected])

Abstract

We give an alternative self-contained proof of the homogenization theorem for periodic multi-parameter integrals that was established by the authors. The proof in that paper relies on the so-called compactness method for Γ-convergence, while the one presented here is by direct verification: the candidate to be the limit homogenized functional is first exhibited and the definition of Γ-convergence is then verified. This is done by an extension of bounded gradient sequences using the Acerbi et al. extension theorem from connected sets, and by the adaptation of some localization and blow-up techniques developed by Fonseca and Müller, together with De Giorgi's slicing method.

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 2004

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