Journal Home Page Cumulative Index List of all Volumes Complete Contentsof this Volume Previous Article Journal of Convex Analysis 13 (2006), No. 1, 051--060Copyright 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 [Abstract-pdf] 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 rank-one 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, rank-one convexity. MSC: 15A15, 15A09, 15A23 [ Fulltext-pdf  (286  KB)] for subscribers only.