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Weak continuity and weak lower semicontinuity for some compensation operators

Published online by Cambridge University Press:  14 November 2011

Pablo Pedregal
Pablo Pedregal
Affiliation:
School of Mathematics, University of Minnesota, Minneapolis, Minnesota 55455, U.S.A
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Synopsis

We study a special class of linear differential operators well-behaved with respect to weakconvergence. Questions related to weak lower semicontinuity, associated Young measures, weak continuity and quasi-convexity are addressed. Specifically, it is shown that the well-known necessary conditions for weak lower semicontinuity are also sufficient in this case. Some examples are given, including a discussion on how well the operator curl fits inthis context.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 113 , Issue 3-4 , 1989 , pp. 267 - 279
Copyright
Copyright © Royal Society of Edinburgh 1989

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References

1Ball, J.. A version of the fundamental theorem for Young measures, (to appear)Google Scholar
2Ball, J. and Murat, F.. Wl.p-quasiconvexity and variational problems for multiple integrals. J. Fund. Anal. 58 (1984), 225–253.CrossRefGoogle Scholar
3Ball, J. M.. Constitutive equations and existence theorems in non-linear elastostatics. In Heriot-Watt Symposium, 1, ed. Knops, R. (London: Pitman, 1976).Google Scholar
4Ball, J. and Marsden, J.. Quasiconvexity at the boundary, positivity of the second variation and elastic stability. Arch. Rational Mech. Anal. 86 (1984), 251–277.CrossRefGoogle Scholar
5Dacorogna, B.. Quasiconvexity and relaxation of non convex problems in the calculus of variations. J. Fund. Anal. 46 (1982), 102–118.CrossRefGoogle Scholar
6Dacorogna, B.. Weak Continuity and Weak Lower Semicontinuity of Nonlinear functional, Lecture Notes in Mathematics 922 (Berlin: Springer, 1982).CrossRefGoogle Scholar
7Meyer, N. G.. Quasiconvexity and lower-semicontinuity of multiple variational integrals of any order. Trans. Amer. Math. Soc. 119 (1965), 125–149.CrossRefGoogle Scholar
8Morrey, C. B.. Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math 2 (1952), 25–53.CrossRefGoogle Scholar
9Morrey, C. B.. Multiple integrals in the calculus of the variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
10Murat, F.. Compacité par compensation: condition necessaire et suffisante de continuité faible sous une hypothése de rang constant. Ann. Scuola Norm. Sup. Pisa Cl. Sci. 4 8 (1981), 69–102.Google Scholar
11Tartar, L.L., Compensated compactness and partial differential equations. In Nonlinear Analysis and Mechanics: Heriot-Watt Symposium, ed. Knops, R. J., pp. 136–212 (London: Pitman, 1979).Google Scholar
12Young, L. C.. Lectures on the calculus of variations and optimal control theory. (Philadelphia: Saunders, 1969).Google Scholar