A nonlinear eigenvalue optimization problem: optimal potential functions

Antunes, Pedro R. S.; Mohammadi, Seyyed Abbas; Voss, Heinrich

http://hdl.handle.net/10400.2/7176

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Authors

Antunes, Pedro R. S.

Mohammadi, Seyyed Abbas

Voss, Heinrich

Abstract(s)

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.