Oscillations and concentrations in sequences of gradients

Martin Kružík; Agnieszka Kałamajska

ESAIM: Control, Optimisation and Calculus of Variations (2008)

  • Volume: 14, Issue: 1, page 71-104
  • ISSN: 1292-8119

Abstract

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We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, { u k } , bounded in L p ( Ø ; m × n ) if p > 1 and Ω n is a bounded domain with the extension property in W 1 , p . Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of Ω are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.

How to cite

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Kružík, Martin, and Kałamajska, Agnieszka. "Oscillations and concentrations in sequences of gradients." ESAIM: Control, Optimisation and Calculus of Variations 14.1 (2008): 71-104. <http://eudml.org/doc/246024>.

@article{Kružík2008,
abstract = {We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\lbrace \nabla u_k\rbrace $, bounded in $L^p(Ø;\mathbb \{R\}^\{m\times n\})$ if $p &gt; 1$ and $\Omega \subset \mathbb \{R\}^n$ is a bounded domain with the extension property in $W^\{1,p\}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega $ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.},
author = {Kružík, Martin, Kałamajska, Agnieszka},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {sequences of gradients; concentrations; oscillations; quasiconvexity},
language = {eng},
number = {1},
pages = {71-104},
publisher = {EDP-Sciences},
title = {Oscillations and concentrations in sequences of gradients},
url = {http://eudml.org/doc/246024},
volume = {14},
year = {2008},
}

TY - JOUR
AU - Kružík, Martin
AU - Kałamajska, Agnieszka
TI - Oscillations and concentrations in sequences of gradients
JO - ESAIM: Control, Optimisation and Calculus of Variations
PY - 2008
PB - EDP-Sciences
VL - 14
IS - 1
SP - 71
EP - 104
AB - We use DiPerna’s and Majda’s generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\lbrace \nabla u_k\rbrace $, bounded in $L^p(Ø;\mathbb {R}^{m\times n})$ if $p &gt; 1$ and $\Omega \subset \mathbb {R}^n$ is a bounded domain with the extension property in $W^{1,p}$. Our main result is a characterization of those DiPerna-Majda measures which are generated by gradients of Sobolev maps satisfying the same fixed Dirichlet boundary condition. Cases where no boundary conditions nor regularity of $\Omega $ are required and links with lower semicontinuity results by Meyers and by Acerbi and Fusco are also discussed.
LA - eng
KW - sequences of gradients; concentrations; oscillations; quasiconvexity
UR - http://eudml.org/doc/246024
ER -

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