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On compactness results for multi-scale convergence

Published online by Cambridge University Press:  14 November 2011

Erik J. Balder
Erik J. Balder
Affiliation:
Mathematical Institute, University of Utrecht, Utrecht, The Netherlands
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Abstract

Two relative compactness results for two-scale convergence in homogenization, due to G. Nguetseng, were recently extended to the multi-scale case by G. Allaire and M. Briane. Whereas their extension of Nguetseng's first result, which is in L2, is straightforward, their extension of his second result, which takes place in the Sobolev space H1, is quite complicated, even though it follows Nguetseng by using the fact that the image of H1 under the gradient mapping is the orthogonal complement of the set of divergence-free functions. Here a much simpler proof is provided by deriving the H1-type result from combining the first extension result with the fact that the above-mentioned image space is also the space of all rotation-free fields. Moreover, this approach reveals that the two results can be seen as corollaries of a fundamental relative compactness result for Young measures.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 129 , Issue 3 , 1999 , pp. 467 - 476
Copyright
Copyright © Royal Society of Edinburgh 1999

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