Proceedings of the Royal Society of Edinburgh: Section A Mathematics

Research Article

Rank-one convexity does not imply quasiconvexity

Vladimír Šveráka1

a1 Department of Mathematics, Heriot-Watt University, Riccarton, Edinburgh EH14 4AS, U.K. (On leave from MFF UK, Charles University, Prague, Czechoslovakia)

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]).

(Received November 13 1991)