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Rank-one convexity does not imply quasiconvexity

Published online by Cambridge University Press:  14 November 2011

Vladimír Šverák
Affiliation:
Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K. (On leave from MFF UK, Charles University, Prague, Czechoslovakia)

Extract

We consider variational integrals

defined for (sufficiently regular) functions u: Ω→Rm. Here Ω is a bounded open subset of Rn, Du(x) denotes the gradient matrix of u at x and f is a continuous function on the space of all real m × n matrices Mm × n. One of the important problems in the calculus of variations is to characterise the functions f for which the integral I is lower semicontinuous. In this connection, the following notions were introduced (see [3], [9], [10]).

Type
Research Article
Copyright
Copyright © Royal Society of Edinburgh 1992

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References

1Acerbi, E. and Fusco, N.. Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1986) 125145.CrossRefGoogle Scholar
2Alibert, J. J. and Dacorogna, B.. An example of a quasiconvex function not polyconvex in dimension two (preprint).Google Scholar
3Ball, J. M.. Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal 63 (1978) 337403.CrossRefGoogle Scholar
4Ball, J. M.. Sets of gradients with no rank-one connections. J. Math, pures et appl., 69 (1990) 241–59.Google Scholar
5Ball, J. M., Currie, J. C. and Olver, P. J.. Null lagrangians, weak continuity, and variational problems of arbitrary order. J. of Func. Anal. 41 No. 2, April 1981.CrossRefGoogle Scholar
6Dacorogna, B.. Direct methods in the calculus of variations. (Berlin: Springer, 1988).Google Scholar
7Dacorogna, B. and Marcellini, P.. A counterexample in the vectorial calculus of variations, in Ball, J. M. ed. Material instabilities in continuum mechanics, pp 7783. (Oxford: Clarendon Press, 1988).Google Scholar
8Kohn, R. V.. The relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991), 192236.CrossRefGoogle Scholar
9Morrey, Ch. B.. Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 2553.CrossRefGoogle Scholar
10Morrey, Ch. B.. Multiple integrals in the calculus of variations (Berlin: Springer, 1966).CrossRefGoogle Scholar
11Serre, D.. Formes quadratiques et calcul des variations. J. Math. pures et appl., 62 (1983), 177–96.Google Scholar
12Sivaloganathan, J.. Implications of rank-one convexity, Ann. Inst. H. Poincaré, Analise Non linéaire. 5 (1988), 99118.CrossRefGoogle Scholar
13Šverák, V.. Examples of rank-one convex functions. Proc. Roy. Soc. Edinburgh 114A (1990), 237–42.CrossRefGoogle Scholar
14Šverák, V.. Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. Land. A 433 (1991) 723–5.Google Scholar
15Tartar, L.. Compensated compactness and applications to partial differential equations, in Nonlinear analysis and mechanics: Heriot-Watt Symposium IV, Pitman Research Notes in Mathematics 39, 136212. (London: Pitman, 1979.)Google Scholar
16Terpstra, F. J.. Die Darstellung der biquadratischen Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166–80.CrossRefGoogle Scholar