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
skroemer@math.uni-koeln.de
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
kruzik@utia.cas.cz
[Abstract-pdf]
\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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