| Issue |
ESAIM: COCV
Volume 19, Number 3, July-September 2013
|
|
|---|---|---|
| Page(s) | 679 - 700 | |
| DOI | https://doi.org/10.1051/cocv/2012028 | |
| Published online | 17 May 2013 | |
Quasiconvexity at the boundary and concentration effects generated by gradients∗
1
Institute of Information Theory and Automation of the
ASCR, Pod vodárenskou věží
4, 182 08
Praha 8, Czech
Republic
2
Faculty of Civil Engineering, Czech Technical
University, Thákurova
7, 166 29
Praha 6, Czech
Republic
kruzik@utia.cas.cz
Received:
12
September
2011
Revised:
2
August
2012
We characterize generalized Young measures, the so-called DiPerna–Majda measures which are generated by sequences of gradients. In particular, we precisely describe these measures 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 on weak W1,2(Ω; ℝ3) sequential continuity of u → a· [Cof∇u] ϱ, where Ω ⊂ ℝ3 has a smooth boundary and a,ϱ are certain smooth mappings.
Mathematics Subject Classification: 49J45 / 35B05
Key words: Bounded sequences of gradients / concentrations / oscillations / quasiconvexity at the boundary / weak lower semicontinuity
© EDP Sciences, SMAI, 2013
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