Minimization of the p-Laplacian first eigenvalue for a two-phase material

Abstract

We study the problem of minimizing the first eigenvalue of the p-Laplacian operator for a two-phase material in a bounded open domain Ω⊂RN, N⩾2 assuming that the amount of the best material is limited. We provide a relaxed formulation of the problem and prove some smoothness results for these solutions. As a consequence we show that if Ω is of class C1,1, simply connected with connected boundary, then the unrelaxed problem has a solution if and only if Ω is a ball. We also provide an algorithm to approximate the solutions of the relaxed problem and perform some numerical simulations.