A Necessary Condition for the Quasiconvexity of Polynomials of Degree Four

S. Gutiérrez

Journal of convex analysis, Tome 13 (2006) no. 1, pp. 51-6.

Résumé

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.

Détail

Classification : 15A15, 15A09, 15A23
Mots-clés : Compensated compactness, lower semicontinuity, quasiconvexity, rank-one convexity