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Some remarks concerning quasiconvexity and strong convergence

Published online by Cambridge University Press:  14 November 2011

L. C. Evans  and
R. F. Gariepy
L. C. Evans
Affiliation:
Department of Mathematics, University of Maryland, College Park, MD 20742, U.S.A.
R. F. Gariepy
Affiliation:
Department of Mathematics, University of Kentucky, Lexington, KY 40506, U.S.A.
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Synopsis

We show that the weak convergence of a sequence of functions in a Sobolev space plus the convergence of appropriately quasiconvex “energies” imply, in fact, strong convergence. This assertion makes rigorous, for example, the heuristic principle that “quasiconvexity damps out oscillations in the gradients” of minimising sequences in the calculus of variations.

Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 106 , Issue 1-2 , 1987 , pp. 53 - 61
Copyright
Copyright © Royal Society of Edinburgh 1987

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rational Mech. Anal. 86 (1984), 125–145.CrossRefGoogle Scholar
2Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rational Mech. Anal. 63 (1977), 337–403.CrossRefGoogle Scholar
3Dacorogna, B.. Weak Continuity and Weak Lower Semicontinuity of Non-linear Functionals (Berlin: Springer, 1982).CrossRefGoogle Scholar
4Giorgi, E. De. Sulla convergenza di alcune successioni di integrali del tipo dell'area. Rend. Mat. 8 (1975), 277–294.Google Scholar
5Evans, L. C.. Quasiconvexity and partial regularity in the calculus of variations. Arch. Rational Mech. Anal. 95 (1986), 227–252.CrossRefGoogle Scholar
6Evans, L. C. and Gariepy, R. F.. Blow-up, compactness, and partial regularity in the calculus of variations (to appear).Google Scholar
7Kohn, R. and Strang, G.. Optimal design and relaxation of variational problems. Comm. Pure Appl. Math, (to appear).Google Scholar
8Marcellini, P.. Approximation of quasiconvex functions and lower semicontinuity of multiple integrals. Manuscripta Math, (to appear).Google Scholar
9Marcellini, P.. On the definition and lower semicontinuity of certain quasiconvex integrals (to appear).Google Scholar
10Morrey, C. B. Jr, Quasiconvexity and the lower semicontinuity of multiple integrals. Pacific J. Math. 2 (1952), 25–33.CrossRefGoogle Scholar
11Visintin, A.. Strong convergence results related to strict convexity. Comm. Partial Differential Equations 9 (1984), 439–466.CrossRefGoogle Scholar