
Journal of Convex Analysis 13 (2006), No. 1, 051060 Copyright Heldermann Verlag 2006 A Necessary Condition for the Quasiconvexity of Polynomials of Degree Four Sergio Gutiérrez Centre de Mathématiques Appliquées (UMR 7641), Ecole Polytechnique, 91128 Palaiseau, France sergio@cmap.polytechnique.fr [Abstractpdf] Using ideas from Compensated Compactness, we derive a necessary condition for any fourth degree polynomial on $I\!\!R^{p}$ to be sequentially lower semicontinuous with respect to weakly convergent fields defined on $I\!\!R^N$. We use that result to derive a necessary condition for the quasiconvexity of fourth degree polynomials of $m\times N$ gradient matrices of vector fields defined on $I\!\!R^N$. This condition is violated by the example given by \v{S}ver\'ak for $m\geq 3$ and $N\geq 2$, of a fourth degree polynomial which is rankone convex, but it is not quasiconvex. These classes of functions are used in the approach to Nonlinear Elasticity based on the Calculus of Variations. Keywords: Compensated compactness, lower semicontinuity, quasiconvexity, rankone convexity. MSC: 15A15, 15A09, 15A23 [ Fulltextpdf (286 KB)] for subscribers only. 