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Weak lower semicontinuity of polyconvex integrals

Published online by Cambridge University Press:  14 November 2011

Jan Malý
Jan Malý
Affiliation:
Faculty of Mathematics and Physics–KMA, Charles University, Sokolovská 83, 18600 Praha, Czech Republic
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Synopsis

Multiple integrals with polyconvex integrands are studied on the class of all sense-preserving diffeomorphisms from W1,p(Ω, Rn) where Ω is an open subset of Rn. They are proved to be sequentially weakly lower semicontinuous if 1 < p = n –1. An example is presented showing that a similar result is not valid if p <n –1.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 123 , Issue 4 , 1993 , pp. 681 - 691
Copyright
Copyright © Royal Society of Edinburgh 1993

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References

1Acerbi, E., Butazzo, G. and Fusco, N.. Semicontinuity and relaxation for integrals depending on vector valued functions. J. Math. Pures Appl. 62 (1983), 371–387.Google Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Meek Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
3Ball, J. M. and Murat, F.. W 1, p quasiconvexity and variational problems for multiple integrals. J. Funct. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
4Dacorogna, B.. Direct Methods in the Calculus of Variations, Appl. Math. Sciences 78 (Berlin: Springer, 1989).Google Scholar
5Dacorogna, B. and Marcellini, P.. Semicontinuité pour des integrandes polyconvexes sans continuité des determinants. C.R. Acad. Sci. Paris Sér. I Math. 311 (6) (1990), 393–396.Google Scholar
6Marcellini, P.. On the definition and the lower semicontinuity of certain quasiconvex integrals. Ann. Inst. H. Poincaré Anal. Non Linéaire 3 (1986), 391–409.CrossRefGoogle Scholar
7Müller, Stefan, Tang, Q. and Yang, B. S.. On a new class of elastic deformations not allowing for cavitation (in prep).Google Scholar
8Reshetnyak, Y. G.. Stability theorems for mappings with bounded excursions. Siberian Math. J. 9 (1968), 499–512.CrossRefGoogle Scholar
9Zhang, K.. Biting theorems for Jacobians and their applications. Ann. Inst. H. Poincaré Anal. Non Linéaire 7(4) (1990), 345–365.CrossRefGoogle Scholar