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Compensated compactness for nonlinear homogenization and reduction of dimension
Authors:P. Courilleau  J. Mossino
Affiliation:(1) Département de mathématiques, Université de Cergy-Pontoise, 2, Avenue Adolphe Chauvin, 95302 Cergy Pontoise, France;(2) Centre de Mathématiques et Leurs Applications, Ecole Normale Supérieure de Cachan, 61, Avenue du Président Wilson, 94235 Cachan Cedex, France
Abstract:
We study the limit behaviour of some nonlinear monotone equations, such as: $-div(A^epsilon varphi (B^epsilon nabla U^epsilon)) = F^epsilon$, in a domain $Omega^epsilon$ which is thin in some directions (e.g. $Omega^epsilon$ is a plate or a thin cylinder). After rescaling to a fixed domain $Omega$, the above equation is transformed into: $-div^epsilon(a^epsilon varphi (b^epsilon nabla^epsilon u^epsilon)) = f^epsilon$, with convenient operators $div^epsilon$ and $nabla^epsilon$. Assuming that $a^epsilon$ and the inverse of $b^epsilon$ have particular forms and satisfy suitable compensated compactness assumptions, we prove a closure result, that is we prove that the limit problem has the same form. This applies in particular to the limit behaviour of nonlinear monotone equations in laminated plates.Received: 16 October 2002, Accepted: 12 June 2003, Published online: 22 September 2003Mathematics Subject Classification (2000): 35B27, 35B40, 74Q15
Keywords:
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