A necessary condition for the quasiconvexity of polynomials of degree four

Abstract

Using ideas from Compensated Compactness, we derive a necessary condition for any fourth degree polynomial on R^p to be sequentially lower semicontinuous with respect to weakly convergent fields defined on R^N. We use that result to derive a necessary condition for the quasiconvexity of fourth degree polynomials of m x N gradient matrices of vector fields defined on R^N. This condition is violated by the example given by Sverák for m=> 3 and 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.