A nonlinear eigenvalue optimization problem: optimal potential functions

Abstract In this paper we study the following optimal shape design problem: Given an open connected set Ω ⊂ R N and a positive number A ∈ ( 0 , | Ω | ) , find a measurable subset D ⊂ Ω with | D | = A such that the minimal eigenvalue of − div ( ζ ( λ , x ) ∇ u ) + α χ D u = λ u in Ω , u = 0 on ∂ Ω , is as small as possible. This sort of nonlinear eigenvalue problems arises in the study of some quantum dots taking into account an electron effective mass. We establish the existence of a solution and we determine some qualitative aspects of the optimal configurations. For instance, we can get a nearly optimal set which is an approximation of the minimizer in ultra-high contrast regime. A numerical algorithm is proposed to obtain an approximate description of the optimizer.