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Journal of Convex Analysis 20 (2013), No. 3, 723--752
Copyright Heldermann Verlag 2013

Oscillations and Concentrations in Sequences of Gradients up to the Boundary

Stefan Krömer
Mathematisches Institut, Universität zu Köln, 50923 Köln, Germany

Martin Kruzík
Institute of Information Theory and Automation, Academy of Sciences of the Czech Republic, Pod vodárenskou vezí 4, 182 08 Praha 8, Czech Republic


\renewcommand{\O}{\Omega} \newcommand{\R}{{\mathbb R}} Oscillations and concentrations in sequences of gradients $\{\nabla u_k\}$, bounded in $L^p(\O; \R^{M\times N})$ if $p>1$ and $\O\subset\R^n$ is a bounded domain with the extension property in $W^{1,p}$, and their interaction with local integral functionals can be described by a generalization of Young measures due to DiPerna and Majda. We characterize such DiPerna-Majda measures, thereby extending a result by A. Ka{\l}amajska and M. Kru\v{z}{\'\i}k [``Oscillations and concentrations in sequences of gradients'', ESAIM, Control Optim. Calc. Var. 14(1) (2008) 71--104], where the full characterization was possible only for sequences subject to a fixed Dirichlet boundary condition. As an application we state a relaxation result for noncoercive multiple-integral functionals.

Keywords: Sequences of gradients, concentrations, oscillations, quasiconvexity.

MSC: 49J45, 35B05

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