Equi-integrability results for 3D-2D dimension reduction problems
ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 443-470.

3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients $\left({\nabla }_{\alpha }{u}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{u}_{\epsilon }\right)$ bounded in ${L}^{p}\left(\Omega ;{ℝ}^{9}\right),\phantom{\rule{4pt}{0ex}}1 Here it is shown that, up to a subsequence, ${u}_{\epsilon }$ may be decomposed as ${w}_{\epsilon }+{z}_{\epsilon },$ where ${z}_{\epsilon }$ carries all the concentration effects, i.e. $\left\{{\left|\left({\nabla }_{\alpha }{w}_{\epsilon }|\frac{1}{\epsilon }{\nabla }_{3}{w}_{\epsilon }\right)\right|}^{p}\right\}$ is equi-integrable, and ${w}_{\epsilon }$ captures the oscillatory behavior, i.e. ${z}_{\epsilon }\to 0$ in measure. In addition, if $\left\{{u}_{\epsilon }\right\}$ is a recovering sequence then ${z}_{\epsilon }={z}_{\epsilon }\left({x}_{\alpha }\right)$ nearby $\partial \Omega .$

DOI : https://doi.org/10.1051/cocv:2002063
Classification : 49J45,  74B20,  74G10,  74K15,  74K35
Mots clés : equi-integrability, dimension reduction, lower semicontinuity, maximal function, oscillations, concentrations, quasiconvexity
@article{COCV_2002__7__443_0,
author = {Bocea, Marian and Fonseca, Irene},
title = {Equi-integrability results for 3D-2D dimension reduction problems},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {443--470},
publisher = {EDP-Sciences},
volume = {7},
year = {2002},
doi = {10.1051/cocv:2002063},
zbl = {1044.49010},
mrnumber = {1925037},
language = {en},
url = {http://www.numdam.org/item/COCV_2002__7__443_0/}
}
Bocea, Marian; Fonseca, Irene. Equi-integrability results for 3D-2D dimension reduction problems. ESAIM: Control, Optimisation and Calculus of Variations, Tome 7 (2002) , pp. 443-470. doi : 10.1051/cocv:2002063. http://www.numdam.org/item/COCV_2002__7__443_0/

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