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

Marian Bocea; Irene Fonseca

ESAIM: Control, Optimisation and Calculus of Variations (2002)

  • Volume: 7, page 443-470
  • ISSN: 1292-8119

Abstract

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3D-2D asymptotic analysis for thin structures rests on the mastery of scaled gradients ∇ α u ε | 1 ε ∇ 3 u ε bounded in L p ( Ω ; ℝ 9 ) , 1 < p < + ∞ . Here it is shown that, up to a subsequence, u ε may be decomposed as w ε + z ε , where z ε carries all the concentration effects, i.e. ∇ α w ε | 1 ε ∇ 3 w ε p is equi-integrable, and w ε captures the oscillatory behavior, i.e. z ε → 0 in measure. In addition, if { u ε } is a recovering sequence then z ε = z ε ( x α ) nearby ∂ Ω .

How to cite

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Bocea, Marian, and Fonseca, Irene. "Equi-integrability results for 3D-2D dimension reduction problems." ESAIM: Control, Optimisation and Calculus of Variations 7 (2002): 443-470. <http://eudml.org/doc/245996>.

@article{Bocea2002,
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 $L^p (\Omega ; \mathbb \{R\}^9 ), \ 1&lt;p&lt;+\infty .$ 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\lbrace \left| \left( \nabla _\{\alpha \}w_\{\varepsilon \}| \frac\{1\}\{\{\varepsilon \}\}\nabla _3 w_\{\varepsilon \}\right) \right| ^\{p\} \right\rbrace $ is equi-integrable, and $w_\{\varepsilon \}$ captures the oscillatory behavior, i.e. $z_\{\varepsilon \}\rightarrow 0$ in measure. In addition, if $\lbrace u_\{\varepsilon \}\rbrace $ is a recovering sequence then $z_\{\varepsilon \}= z_\{\varepsilon \}(x_\alpha )$ nearby $\partial \Omega .$},
author = {Bocea, Marian, Fonseca, Irene},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {equi-integrability; dimension reduction; lower semicontinuity; maximal function; oscillations; concentrations; quasiconvexity},
language = {eng},
pages = {443-470},
publisher = {EDP-Sciences},
title = {Equi-integrability results for 3D-2D dimension reduction problems},
url = {http://eudml.org/doc/245996},
volume = {7},
year = {2002},
}

TY - JOUR
AU - Bocea, Marian
AU - Fonseca, Irene
TI - Equi-integrability results for 3D-2D dimension reduction problems
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2002
PB - EDP-Sciences
VL - 7
SP - 443
EP - 470
AB - 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 $L^p (\Omega ; \mathbb {R}^9 ), \ 1&lt;p&lt;+\infty .$ 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\lbrace \left| \left( \nabla _{\alpha }w_{\varepsilon }| \frac{1}{{\varepsilon }}\nabla _3 w_{\varepsilon }\right) \right| ^{p} \right\rbrace $ is equi-integrable, and $w_{\varepsilon }$ captures the oscillatory behavior, i.e. $z_{\varepsilon }\rightarrow 0$ in measure. In addition, if $\lbrace u_{\varepsilon }\rbrace $ is a recovering sequence then $z_{\varepsilon }= z_{\varepsilon }(x_\alpha )$ nearby $\partial \Omega .$
LA - eng
KW - equi-integrability; dimension reduction; lower semicontinuity; maximal function; oscillations; concentrations; quasiconvexity
UR - http://eudml.org/doc/245996
ER -

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