Homogenization of partial differential equations. Translated from the original Russian by M. Goncharenko and D. Shepelsky
type: Item
instance of: scholarly article
title: Homogenization of partial differential equations. Translated from the original Russian by M. Goncharenko and D. Shepelsky
zbMATH Open document ID: 1113.35003
published in: Progress in Mathematical Physics
publication date: 2006-03-28T00:00:00Z
review text: This book is devoted to present many not classical homogenization problems, i.e. problems in which coefficients are supposed to satisfy a strict periodicity and the standard boundedness assumptions. Usually these problems depend on a small parameter and the interest is to characterize, if this exists, the limit problem when this parameter goes to zero (this means to obtain some pointwise convergence for the Green's operators). For elliptic or parabolic equations like (\(b_{\varepsilon}\) respectively null or positive) \[ b_{\varepsilon}{{\partial u^{\varepsilon}}\over{\partial t}} - \sum_{i,j=1}^n {{\partial}\over{\partial x_i}} \Big(a_{ij,\varepsilon}{{\partial u^{\varepsilon}}\over{\partial x_j}}\Big) = f, \qquad x\in \Omega_{\varepsilon}, \;t > 0 \] as most of the equations considered in the present book are, the classical results often, but not always, lead to equations which have the same form as the original. Differing from the classical ones, the problems considered in the present book may depend on two (or more) small parameters and the structures may be not strictly periodic or quasi periodic, and the standard boundedness conditions may be violated. In many examples considered media have different characteristics, i.e. these parameters have different order of smallness. This may lead to have limit problem which are not of the same form of the original ones. To make some examples, the authors analyse situations in which the quadratic form associated to \((a_{ij,\varepsilon})_{i,j=1}^n\) is not bounded from above, situations in which the quadratic form associated to \((a_{ij,\varepsilon})_{i,j=1}^n\) is not bounded from below by a positive constant but may go to zero, situations in the nonstationary case in which \(b_{\varepsilon}\) are not bounded from above. In some cases these situations may lead to a nonlocal model, to a multicomponent model, i.e. to a system of two or more equations, or also to a model with memory. It is to be stressed that in general si…
reviewed by: Fabio Paronetto
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 35B27
zbMATH DE Number: 5015850
zbMATH Keywords: homogenization
zbMATH Keywords: two small parameters
zbMATH Keywords: nonlocal model
zbMATH Keywords: multicomponent model
author: Vladimir Aleksandrovich Marchenko
author: E. Ya. Khruslov
MaRDI profile type: Publication
description: scientific article; zbMATH DE number 5015850