Quasiconvexity, rank-one convexity, and composites

In this lecture I will discuss some connections between quasiconvexity, rank-one convexity, and the problem of determining the range which the effective tensor of a linear composite can take. It has been known since the work of Kohn that the problem of bounding the effective tensor (such as the effective elasticity tensor or effective conductivity tensor) reduces to a quasiconvexification problem. The converse is true also: subject to some minor technical points, a given quasiconvexification problem can reinterpreted as a problem of finding a bound on an effective tensor.
One interesting topic, initiated by Grabovsky, is the theory of exact relations. An exact relation holds if the set of effective tensors lies within a manifold. Grabovsky found an algebraic condition which ensures an exact relation holds for laminated composites. Together with Sage we found a sufficient algebraic condition for an exact relation to hold for all composites. Curiously, it appears there is a gap between these conditions that is in many ways parallel to the gap between quasiconvexity and rank-one convexity. However in all known examples, if one algebraic condition is satisfied then the other is too.
It is well known that rank-one convexity is equivalent to quasiconvexity if and only if the Young's measure of any gradient can be reproduced by a laminate field. Briane, Nesi and myself have recently probed an analogous question for three dimensional conducting composites. Specifically we find that the determinant of the field maintains its sign in any laminate microstructure but can change its sign in an appropriately chosen microstructure of interlinked chains.