 
Summary: MULTIPARAMETER HOMOGENIZATION BY LOCALIZATION
AND BLOWUP
FELIPE ALVAREZ AND JEANPHILIPPE MANDALLENA
Abstract. It is given an alternative selfcontained 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 socalled 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 blowup 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 socalled com
