# Oscillations and concentrations generated by $\mathcal{A}$-free mappings and weak lower semicontinuity of integral functionals

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

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

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topFonseca, 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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