Oscillations and concentrations generated by 𝒜 -free mappings and weak lower semicontinuity of integral functionals

Irene Fonseca; Martin Kružík

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

  • Volume: 16, Issue: 2, page 472-502
  • ISSN: 1292-8119

Abstract

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DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps { u k } k L p ( Ω ; m ) satisfying a linear differential constraint 𝒜 u k = 0 . Applications to sequential weak lower semicontinuity of integral functionals on 𝒜 -free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det ϕ k * det ϕ in measures on the closure of Ω n if ϕ k ϕ in W 1 , n ( Ω ; n ) . This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets Ω j Ω must be removed to obtain weak lower semicontinuity of u Ω Ω j v ( u ( x ) ) d x along { u k } L p ( Ω ; m ) ker 𝒜 . Specifically, Ω j are arbitrarily thin “boundary layers”.

How to cite

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Fonseca, Irene, and Kružík, Martin. "Oscillations and concentrations generated by ${\mathcal A}$-free mappings and weak lower semicontinuity of integral functionals." ESAIM: Control, Optimisation and Calculus of Variations 16.2 (2010): 472-502. <http://eudml.org/doc/250738>.

@article{Fonseca2010,
abstract = { DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\\{u_k\\}_\{k\in\{\mathbb N\}\} \subset L^p(\Omega;\{\mathbb R\}^m)$ satisfying a linear differential constraint $\{\mathcal A\}u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on $\{\mathcal A\}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla\varphi_k\stackrel\{*\}\{\rightharpoonup\}\{\rm det\}\nabla\varphi$ in measures on the closure of $\Omega\subset\{\mathbb R\}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^\{1,n\}(\Omega;\{\mathbb R\}^n)$. This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_\{\Omega\setminus\Omega_j\} v(u(x))\,\{\rm d\}x$ along $\\{u_k\\}\subset L^p(\Omega;\{\mathbb R\}^m)\cap\{\rm ker\}\ \{\mathcal A\}$. Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”. },
author = {Fonseca, Irene, Kružík, Martin},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Concentrations; oscillations; Young measures; concentrations},
language = {eng},
month = {4},
number = {2},
pages = {472-502},
publisher = {EDP Sciences},
title = {Oscillations and concentrations generated by $\{\mathcal A\}$-free mappings and weak lower semicontinuity of integral functionals},
url = {http://eudml.org/doc/250738},
volume = {16},
year = {2010},
}

TY - JOUR
AU - Fonseca, Irene
AU - Kružík, Martin
TI - Oscillations and concentrations generated by ${\mathcal A}$-free mappings and weak lower semicontinuity of integral functionals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2010/4//
PB - EDP Sciences
VL - 16
IS - 2
SP - 472
EP - 502
AB - DiPerna's and Majda's generalization of Young measures is used to describe oscillations and concentrations in sequences of maps $\{u_k\}_{k\in{\mathbb N}} \subset L^p(\Omega;{\mathbb R}^m)$ satisfying a linear differential constraint ${\mathcal A}u_k=0$. Applications to sequential weak lower semicontinuity of integral functionals on ${\mathcal A}$-free sequences and to weak continuity of determinants are given. In particular, we state necessary and sufficient conditions for weak* convergence of det$\nabla\varphi_k\stackrel{*}{\rightharpoonup}{\rm det}\nabla\varphi$ in measures on the closure of $\Omega\subset{\mathbb R}^n$ if $\varphi_k\rightharpoonup\varphi$ in $W^{1,n}(\Omega;{\mathbb R}^n)$. This convergence holds, for example, under Dirichlet boundary conditions. Further, we formulate a Biting-like lemma precisely stating which subsets $\Omega_j\subset \Omega$ must be removed to obtain weak lower semicontinuity of $u\mapsto\int_{\Omega\setminus\Omega_j} v(u(x))\,{\rm d}x$ along $\{u_k\}\subset L^p(\Omega;{\mathbb R}^m)\cap{\rm ker}\ {\mathcal A}$. Specifically, $\Omega_j$ are arbitrarily thin “boundary layers”.
LA - eng
KW - Concentrations; oscillations; Young measures; concentrations
UR - http://eudml.org/doc/250738
ER -

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