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MULTIPARAMETER HOMOGENIZATION BY LOCALIZATION AND BLOW-UP
 

Summary: MULTIPARAMETER HOMOGENIZATION BY LOCALIZATION
AND BLOW-UP
FELIPE ALVAREZ AND JEAN-PHILIPPE MANDALLENA
Abstract. It is given an alternative self-contained proof of the homogeniza-
tion theorem for periodic multiparameter integrals that was established by the
authors in [4]. The proof in that paper relies on the so-called compactness
method for -convergence while the one presented here is by direct verifica-
tion: the candidate to be the limit homogenized functional is first exhibited,
the definition of -convergence is then verified. This is done by extension of
bounded gradient sequences using the Acerbi et al. extension theorem from
connected sets [2], and by adaptation of some localization and blow-up tech-
niques developed by Fonseca and M¨uller [14] together with De Giorgi's slicing
method [11].
1. Introduction
In a recent paper we developped a framework to deal with some multiparam-
eter homogenization problems by establishing a general -convergence result for
sequences of periodic integral functionals [4, Theorem 2.2]. We also gave applica-
tions to different "degenerate" homogenization processes (soft inclusions, iterated
homogenization, thin inclusions), showing the versatility of this unified approach.
The proof of the abstract result that we gave there is based on the so-called com-

  

Source: Alvarez, Felipe - Departamento de Ingeniería Matemática, Universidad de Chile

 

Collections: Mathematics; Engineering