Memory effects in homogenisation: Linear second-order equations
type: Item
instance of: scholarly article
title: Memory effects in homogenisation: Linear second-order equations
zbMATH Open document ID: 0791.35009
DOI: 10.1007/BF00411476
author: N. Antonić
published in: Archive for Rational Mechanics and Analysis
publication date: 1994-07-18T00:00:00Z
review text: The author studies homogenisation problems. The problems \(A_ \varepsilon\) employ linear differential equations, but the limit problem \(A_ 0\) is given by an integro-differential operator: The expression of the integral term is deduced under two different sets of assumptions: bounds in \(L^ 1\) or \(L^ 2\). The \(L^ 2\) setting uses the Fourier transform and natural assumptions of boundedness in \(L^ \infty\) and uniform ellipticity of the coefficients \(a^ \varepsilon\), \(b^ \varepsilon\) and \(c^ \varepsilon\). The \(L^ 1\) settings are obtained under additional assumptions. The description of the memory term is given for a problem on a bounded interval, by using the eigenfunction expansion and a representation theorem for Nevanlinna functions. An explicit study is given for one space dimensional problem. Generalization is given in several space dimensions where the operator \(-\partial^ 2_ x\) is replaced by the Laplace operator \(-\Delta\). Several examples are computed.
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 74E05
zbMATH DE Number: 488201
zbMATH Keywords: memory effects
zbMATH Keywords: homogenisation
zbMATH Keywords: integro-differential operators
zbMATH Keywords: eigenfunction expansion
zbMATH Keywords: representation theorem for Nevanlinna functions
Wikidata QID: Q58624238
reviewed by: Marco Codegone
MaRDI profile type: Publication
cites work: Q3999241
cites work: Q4405419
cites work: Q5530492
cites work: Q3878033
cites work: Q3797568
cites work: Memory effects and homogenization
description: scientific article; zbMATH DE number 488201