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  • Proceedings of the Royal Society of Edinburgh: Section A Mathematics, Volume 120, Issue 1-2
  • January 1992, pp. 185-189

Rank-one convexity does not imply quasiconvexity

  • Vladimír Šverák (a1)
  • DOI:
  • Published online: 14 November 2011

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]).

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1E. Acerbi and N. Fusco . Semicontinuity problems in the calculus of variations. Arch. Rat. Mech. Anal. 86 (1986) 125145.

3J. M. Ball . Convexity conditions and existence theorems in nonlinear elasticity. Arch. Rat. Mech. Anal 63 (1978) 337403.

5J. M. Ball , J. C. Currie and P. J. Olver . Null lagrangians, weak continuity, and variational problems of arbitrary order. J. of Func. Anal. 41 No. 2, April 1981.

8R. V. Kohn . The relaxation of a double-well energy. Cont. Mech. Thermodyn. 3 (1991), 192236.

9Ch. B. Morrey . Quasi-convexity and the lower semicontinuity of multiple integrals, Pacific J. Math. 2 (1952), 2553.

14V. Šverák . Quasiconvex functions with subquadratic growth, Proc. Roy. Soc. Land. A 433 (1991) 723–5.

16F. J. Terpstra . Die Darstellung der biquadratischen Formen als Summen von Quadraten mit Anwendung auf die Variationsrechnung, Math. Ann. 116 (1938), 166–80.

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Proceedings of the Royal Society of Edinburgh Section A: Mathematics
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