A Necessary Condition for the Quasiconvexity of Polynomials of Degree Four

Using ideas from Compensated Compactness, we derive a necessary condition for any fourth degree polynomial on

to be sequentially lower semicontinuous with respect to weakly convergent fields defined on

\mathbb{R}^N R N

. We use that result to derive a necessary condition for the quasiconvexity of fourth degree polynomials of

m\times N m × N

gradient matrices of vector fields defined on

\mathbb{R}^N R N

. This condition is violated by the example given by Šverák for

m\geq 3 m ≥ 3

and

N\geq 2 N ≥ 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.