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Journal of Convex Analysis 13 (2006), No. 1, 051--060
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


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

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