Inverse Quasiconvexification

In the context of the Calculus of Variations for non-convex, vector variational problems, the natural process of going from a function \(\phi \) to its quasiconvexification \(Q\phi \) is quite involved, and, most of the time, an impossible task. We propose to look at the reverse process, what might be called inverse quasiconvexification: start from a function \(\phi _0\), and find functions \(\phi \) for which \(\phi _0=Q\phi \). In addition to establishing a few general principles, we show several explicit examples motivated by their application to inverse problems in conductivity.