A surprising application of the theory of exact relations for composite materials to Calculus of Variations

Yury Grabovsky
Department of Mathematics, Temple University, Philadelphia

Many important questions about the effective behavior of composite materials can be restated as problems in Calculus of Variations, where the unknown is a vector field. One such question is whether every composite can be mimicked by a laminate made with the same constituent materials. The parallel question in the context of Calculus of Variations is whether every rank-one convex function, (i.e. convex along rank-one lines) is quasiconvex. Both questions had been open for a long time before they were settled, and both have been answered in the negative. However, examples settling these questions are unwieldy and hard to construct. In this talk I will produce an aesthetically beautiful example of a rotationally-invariant rank-one convex, nonquasiconvex function. This example comes from the theory of exact relations - formulas that hold for effective tensors of all composites made with a given set of materials, regardless of microstructure.