\(\varGamma \)-convergence of quadratic functionals with oscillating linear perturbations
type: Item
instance of: scholarly article
title: \(\varGamma \)-convergence of quadratic functionals with oscillating linear perturbations
zbMATH Open document ID: 1180.35080
author: Pablo Pedregal
author: Hélia Serrano
published in: Nonlinear Analysis. Theory, Methods \& Applications. Series A: Theory and Methods
publication date: 2009-05-07T00:00:00Z
review text: In the present paper the authors take into account the \(\Gamma\)-convergence for functionals \[ I_{\varepsilon} (u) = \int_{\Omega} [(A_{\varepsilon}(x) \cdot \nabla u(x), \nabla u (x)) + b_{\varepsilon}(x) \cdot \nabla u(x) ] dx \] where \(A_{\varepsilon}(x) = A(x, \langle \frac{x}{l_1(\varepsilon)} \rangle)\) and \(b_{\varepsilon}(x) = b(x, \langle \frac{x}{l_2(\varepsilon)} \rangle)\) (\(\langle y \rangle\) stands for the fractional part of \(y \in {\mathbb R}\)) and \(l_j(\varepsilon) \searrow 0\) when \(\varepsilon \searrow 0\), \(j=1,2\), possibly \(l_1 = l_2\). The goal is to investigate which is the interaction between \(A_{\varepsilon}\) and \(b_{\varepsilon}\) in the limit. In the case \(n=1\) they study the general case (not only the periodic one). For \(n \geq 2\) they study the periodic case in detail and in the last section they also give some examples to try to answer the question in the general case.
reviewed by: Fabio Paronetto
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35A15
zbMATH DE Number: 5551962
zbMATH Keywords: multi-scale problems
zbMATH Keywords: joint Young measures
zbMATH Keywords: periodic case
zbMATH Keywords: \(\Gamma\)-convergence
MaRDI profile type: Publication
full work available at URL: j.na.2008.09.007
OpenAlex ID: W1979688394
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DOI: 10.1016/J.NA.2008.09.007
description: scientific article; zbMATH DE number 5551962