A nonlinear eigenvalue optimization problem: Optimal potential functions. (April 2018) - Antunes, Pedro R.S.; Mohammadi, Seyyed Abbas; Voss, Heinrich
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.