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

Marian Bocea; Irene Fonseca

ESAIM: Control, Optimisation and Calculus of Variations (2010)

  • 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 (2010): 443-470. <http://eudml.org/doc/90631>.

@article{Bocea2010,
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 $\partial \Omega.$},
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.; equi-integrability; lower semicontinuity; quasiconvexity},
language = {eng},
month = {3},
pages = {443-470},
publisher = {EDP Sciences},
title = {Equi-integrability results for 3D-2D dimension reduction problems},
url = {http://eudml.org/doc/90631},
volume = {7},
year = {2010},
}

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
DA - 2010/3//
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 $L^p (\Omega; \mathbb{R}^9 ), \ 1 < p < +\infty .$ 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 $\partial \Omega.$
LA - eng
KW - Equi-integrability; dimension reduction; lower semicontinuity; maximal function; oscillations; concentrations; quasiconvexity.; equi-integrability; lower semicontinuity; quasiconvexity
UR - http://eudml.org/doc/90631
ER -

References

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