Smoothness properties for the optimal mixture of two isotropic materials the compliance and eigenvalue problems

Abstract

The present paper is devoted to obtaining some smoothness results for the solution of two classical control problems relative to the optimal mixture of two isotropic materials. In the first one, the goal is to maximize the energy. In the second one, we want to minimize the first eigenvalue of the corresponding elliptic operator. At least for the first problem it is well known that it does not have a solution in general. Thus, we deal with a relaxed formulation. One of the applications of our results is in fact the nonexistence of a solution for the unrelaxed problem. In this sense, we improve a classical nonexistence result by Murat and Tartar for the maximization of the energy which was obtained assuming the solution smooth. We also get a counterexample to the existence of a solution for the eigenvalue problem which, to our knowledge, was an open problem.