The Maximization of the First Eigenvalue for a Two-Phase Material

Abstract

We study the optimal arrangement of two conductive materials in order to maximize the first eigenvalue of the corresponding diffusion operator with Dirichlet conditions. The amount of the highest conductive composite is assumed to be limited. Since this type of problems has no solution in general, we work with a relaxed formulation. We show that the problem has some good concavity properties which allow us to get some uniqueness results. By proving that it is related to the minimization of the energy for a two-phase material studied in [4] we also obtain some smoothness results. As a consequence, we show that the unrelaxed problem has never solution. The paper is completed with some numerical results corresponding to the convergence of the gradient algorithm.