Probing the boundary of quasiconvexity
Yury Grabovsky

Abstract: Computing a quasiconvex envelope is a very difficult problem. However, it is equally, if not more important to know the boundary of the set where the function and its quasiconvex envelope coincide. In particular, the knowledge of this boundary is sufficient to determine whether or not a given configuration is a strong local minimizer or not. In models of phase-transitions it is generally understood that crossing this boundary manifests itself through nucleation of precipitates of new phase. Generalizing this understanding we associate the points on the boundary of quasiconvexity with the solutions of "nucleation PDE" in all-space. Our methods can handle not only the localized, or decaying at infinity solutions, but also such common infinite energy precipitates as infinite slabs and cylinders. An example of "cooperative nucleation", where the minimizer is an infinite row of interacting inclusions will be given.