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Equi-integrability results for 3D-2D dimension reduction problems

Published online by Cambridge University Press:  15 September 2002

Marian Bocea  and
Irene Fonseca
Marian Bocea
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA; [email protected].
Irene Fonseca
Affiliation:
Department of Mathematical Sciences, Carnegie Mellon University, Pittsburgh, PA 15213, USA; [email protected].
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Abstract

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left( \nabla _{\alpha}u_\varepsilon\big| \frac{1}{\varepsilon}\nabla _3 u_\varepsilon\right) $ bounded in $L^p (\Omega; \mathbb{R}^9 ), \ 1 < p < +\infty .$ Here it is shown that, up to a subsequence, $u_\varepsilon$ may be decomposed as $w_\varepsilon+ z_\varepsilon,$ where $z_\varepsilon$ carries all the concentration effects, i.e.$\left\{ \left| \left( \nabla _{\alpha }w_\varepsilon| \frac{1}{\varepsilon }\nabla _3 w_\varepsilon\right) \right| ^{p} \right\} $ is equi-integrable, and $w_\varepsilon$ captures the oscillatory behavior, i.e.$z_\varepsilon\to 0$ in measure. In addition, if $\{ u_\varepsilon\} $ is a recovering sequence then $z_\varepsilon= z_\varepsilon(x_\alpha )$ nearby $\partial \Omega.$

Keywords

Equi-integrabilitydimension reductionlower semicontinuitymaximal functionoscillationsconcentrationsquasiconvexity.
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 7 , 2002 , pp. 443 - 470
Copyright
© EDP Sciences, SMAI, 2002

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