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Quasiconvexity at the boundary and concentration effectsgenerated by gradients

Published online by Cambridge University Press:  17 May 2013

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∇uϱ,where Ω ⊂ ℝ3 has a smooth boundary and a,ϱare certain smooth mappings.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2013

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