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Correctors for the homogenization of almost periodic monotone operators

Correctors for the homogenization of almost periodic monotone operators In a previous paper (Braides et al., 1990) it has been proven, under very mild almost periodicity conditions, that we have weak convergence in H 1,p (Ω) of the solutions uε of boundary problems in an open set Ω related to the quasi-linear monotone operator −div(a(x/ε,Duε)), to a function u, which solves an analogous problem related to a homogenized operator −div(b(Du)). In general we do not have strong convergence of Duε, to Du in (L p (Ω)) n , even in the linear periodic case. It is possible however (Theorems 2.1 and 4.2) to express Duε in terms of Du, up to a rest converging strongly to 0 in (L p (Ω)) n , applying correctors built up exploiting only the geometric properties of a. In the last section, we use the correctors result to obtain a homogenization theorem for quasi-linear equations with natural growth terms. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Asymptotic Analysis IOS Press

Correctors for the homogenization of almost periodic monotone operators

Asymptotic Analysis , Volume 5 (1) – Jan 1, 1991

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Publisher
IOS Press
Copyright
Copyright © 1991 by IOS Press, Inc
ISSN
0921-7134
eISSN
1875-8576
DOI
10.3233/ASY-1991-5103
Publisher site
See Article on Publisher Site

Abstract

In a previous paper (Braides et al., 1990) it has been proven, under very mild almost periodicity conditions, that we have weak convergence in H 1,p (Ω) of the solutions uε of boundary problems in an open set Ω related to the quasi-linear monotone operator −div(a(x/ε,Duε)), to a function u, which solves an analogous problem related to a homogenized operator −div(b(Du)). In general we do not have strong convergence of Duε, to Du in (L p (Ω)) n , even in the linear periodic case. It is possible however (Theorems 2.1 and 4.2) to express Duε in terms of Du, up to a rest converging strongly to 0 in (L p (Ω)) n , applying correctors built up exploiting only the geometric properties of a. In the last section, we use the correctors result to obtain a homogenization theorem for quasi-linear equations with natural growth terms.

Journal

Asymptotic AnalysisIOS Press

Published: Jan 1, 1991

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