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

Marian BOCEA and Irene FONSECA
Department of Mathematical Sciences
Carnegie Mellon University
Pittsburgh, PA 15213 U.S.A.

ABSTRACT: 3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $ \left( \nabla _{\alpha}u_\varepsilon \big\vert \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\vert \left( \nabla _{\alpha }w_\varepsilon \vert \frac{1}{\varepsilon }\nabla _3 w_\varepsilon \right) \right\vert ^{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 .$


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