The maximization of the p-Laplacian energy for a two-phase material

Abstract

We consider the optimal arrangement of two diffusion materials in a bounded openset \Omega \subset \BbbR Nin order to maximize the energy. The diffusion problem is modeled by thep-Laplacianoperator. It is well known that this type of problem has no solution in general and then that it is nec-essary to work with a relaxed formulation. In the present paper, we obtain such relaxed formulationusing the homogenization theory; i.e., we replace both materials by microscopic mixtures of them.Then we get some uniqueness results and a system of optimality conditions. As a consequence, weprove some regularity properties for the optimal solutions of the relaxed problem. Namely, we showthat the flux is in the Sobolev spaceH1(\Omega )Nand that the optimal proportion of the materials isderivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has nosolution in general. Our results extend those obtained by the first author for the Laplace operator.