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Oscillations and concentrations in sequences of gradients

Published online by Cambridge University Press:  21 September 2007

Agnieszka Kałamajska  and
Martin Kružík
Agnieszka Kałamajska
Affiliation:
Institute of Mathematics, Warsaw University, ul. Banacha 2, 02-097 Warsaw, Poland; [email protected]
Martin Kružík
Affiliation:
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic. Corresponding address Pod vodárenskou věží 4, 182 08 Praha 8, Czech Republic. Faculty of Civil Engineering, Czech Technical University, Thákurova 7, 166 29 Praha 6, Czech Republic; [email protected] (corresponding author).
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Abstract

We use DiPerna's and Majda's generalization of Young measures to describe oscillations and concentrations in sequences of gradients, $\{\nabla u_k\}$ , bounded in $L^p(\Omega;{\mathbb R}^{m\times n})$ if p > 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. Caseswhere no boundary conditions nor regularity of Ω arerequired and links with lower semicontinuity results by Meyers andby Acerbi and Fusco are also discussed.

Keywords

Sequences of gradientsconcentrationsoscillationsquasiconvexity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 14 , Issue 1 , January 2008 , pp. 71 - 104
Copyright
© EDP Sciences, SMAI, 2007

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