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A nonlinear eigenvalue optimization problem: Optimal potential functions
Publikationstyp
Journal Article
Date Issued
2018-04
Sprache
English
Author(s)
Institut
TORE-URI
Journal
Volume
40
Start Page
307
End Page
327
Citation
Nonlinear Analysis: Real World Applications (40): 307-327 (2018-04)
Publisher DOI
Scopus ID
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.