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Quasiconvexity at the boundary and concentration effectsgenerated by gradients∗

Published online by Cambridge University Press:  17 May 2013

Martin Kružík
Martin Kružík*
Affiliation:
Institute of Information Theory and Automation of the ASCR, 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]
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Abstract

We characterize generalized Young measures, the so-called DiPerna–Majda measures whichare generated by sequences of gradients. In particular, we precisely describe thesemeasures at the boundary of the domain in the case of the compactification of ℝm × n by the sphere. We show that this characterization is closely related to the notion of quasiconvexity at the boundary introduced by Ball and Marsden [J.M. Ball and J. Marsden, Arch. Ration. Mech.Anal. 86 (1984) 251–277]. As a consequence we get new results onweak W1,2(Ω; ℝ3) sequentialcontinuity ofu → a· [Cof∇u] ϱ,where Ω ⊂ ℝ3 has a smooth boundary and a,ϱare certain smooth mappings.

Keywords

Bounded sequences of gradientsconcentrationsoscillationsquasiconvexity at the boundaryweak lower semicontinuity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 19 , Issue 3 , July 2013 , pp. 679 - 700
Copyright
© EDP Sciences, SMAI, 2013

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