caa a22 4500 388108975 CHVBK 20180307125347.0 cr unu---uuuuu 161130s1999 xx s 000 0 eng 10.1017/S0308210500021466 doi S0308210500021466 pii (NATIONALLICENCE)cambridge-10.1017/S0308210500021466 Balder Erik J. Mathematical Institute, University of Utrecht, Utrecht, The Netherlands On compactness results for multi-scale convergence [Elektronische Daten] [Erik J. Balder] 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. Copyright © Royal Society of Edinburgh 1999 Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/3(1999), 467-476 0308-2105 129:3<467 1999 129 PRM https://doi.org/10.1017/S0308210500021466 text/html Onlinezugriff via DOI 1 research-article jats NATIONALLICENCE 856 40 https://doi.org/10.1017/S0308210500021466 text/html Onlinezugriff via DOI NATIONALLICENCE 100 1- Balder Erik J. Mathematical Institute, University of Utrecht, Utrecht, The Netherlands NATIONALLICENCE 773 0- Proceedings of the Royal Society of Edinburgh: Section A Mathematics Royal Society of Edinburgh Scotland Foundation 129/3(1999), 467-476 0308-2105 129:3<467 1999 129 PRM CC0 http://creativecommons.org/publicdomain/zero/1.0 nationallicence BK010053 XK010053 XK010000 NATIONALLICENCE NATIONALLICENCE NL-cambridge