A nonlinear eigenvalue optimization problem: Optimal potential functions

In this paper we study the following optimal shape design problem: Given an open
connected set Ω ⊂ RN 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)+αχDu =
λu 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