Quasiconvexity and Rank-one Convexity

Abstract

The motivation for this chapter is two-fold. On the one hand, since Jensen’s inequality has played a prominent role in our approach to weak lower semicontinuity, our analysis would be somehow incomplete without any reference to this inequality with respect to rank-one convex functions. Because quasiconvexity implies rank-one convexity, probability measures satisfying Jensen’s inequality with respect to the class of rank-one convex functions are indeed examples of gradient parametrized measures. It turns out that this family of probability measures can be understood. at least conceptually, in a nice constructive way. They are called laminates to emphasize its layering structure. As a matter of fact, laminates are almost the only way to produce explicitly examples of gradient parametrized measures. It is true that the Riemann-Lebesgue lemma allows one to consider gradient parametrized measures associated with periodic gradients. The problem is that we do not know how to decide whether they are laminates or not. The importance of laminates in the description of some equilibrium states for crystals has been stressed in Chapter 5. They are also important in the theory of composite materials and homogenization.