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Lower semicontinuity of multiple integrals and the Biting Lemma

Published online by Cambridge University Press:  14 November 2011

J. M. Ball  and
K.-W. Zhang
J. M. Ball
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.
K.-W. Zhang
Affiliation:
Department of Mathematics, Heriot-Watt University, Edinburgh EH14 4AS, Scotland, U.K.; Department of Mathematics, Peking University, Beijing 100 871, China
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Synopsis

Weak lower semicontinuity theorems in the sense of Chacon's Biting Lemma are proved for multiple integrals of the calculus of variations. A general weak lower semicontinuity result is deduced for integrands which are acomposition of convex and quasiconvex functions. The “biting”weak limit of the corresponding integrands is characterised via the Young measure, and related to the weak* limit in the sense of measures. Finally, an example is given which shows that the Young measure corresponding to a general sequence of gradients may not have an integral representation of the type valid in the periodic case.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 114 , Issue 3-4 , 1990 , pp. 367 - 379
Copyright
Copyright © Royal Society of Edinburgh 1990

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