Homogenization of multiparameter integrals. (English) Zbl 1005.49008

Alvarez, Felipe; Mandallena, Jean-Philippe

Nonlinear Anal., Theory Methods Appl., Ser. A, Theory Methods 50, No. 6, 839-870 (2002).

The paper is concerned with some multiparameter integral functionals of the form \[ \int_\Omega W_\lambda\bigg({x\over\varepsilon},\nabla u(x)\bigg) dx, \] where \(\Omega\) is a bounded domain in \({\mathbb R}^N\), \(u\colon\Omega\to{\mathbb R}^m\), \(\varepsilon>0\), and \(W_\lambda\colon {\mathbb R}^N\times{\mathbb R}^{mN}\to[0,+\infty[\) is \(]0,1[^N\) periodic with respect to the first variable. The novelty of the paper is that the integrand is permitted to depend on a vector of parameters \(\lambda=(\lambda_1,\dots,\lambda_k)\in{\mathbb R}^k\), with \(k\geq 1\). The asymptotic behaviour of the above functionals is studied as \(\varepsilon\to 0\) and \(\lambda\to 0\), possibly following particular paths linking them and forcing them to vanish simultaneously.
Homogenization formulas are deduced, as well as applications to convex and non-convex integrands.

Reviewer: Riccardo De Arcangelis (Napoli)

MSC:

49J45	Methods involving semicontinuity and convergence; relaxation
49N20	Periodic optimal control problems
35B27	Homogenization in context of PDEs; PDEs in media with periodic structure

Keywords:

constrained variational problems; \(\Gamma\)-convergence; interface problems; homogenization; multiparameter integral functionals

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