A nonlinear eigenvalue optimization problem: Optimal potential functions

TY - JOUR

T1 - A nonlinear eigenvalue optimization problem

T2 - Optimal potential functions

AU - Antunes, Pedro R.S.

AU - Mohammadi, Seyyed Abbas

AU - Voss, Heinrich

N1 - Copyrights © 2017 Elsevier Ltd. All rights reserved.

PY - 2018/4

Y1 - 2018/4

N2 - 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

AB - 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

KW - Nonlinear eigenvalue problem

KW - Shape optimization

KW - Ultra-high contrast regime

KW - Quantum dots

UR - https://www.scopus.com/pages/publications/85031014227

U2 - 10.1016/j.nonrwa.2017.09.003

DO - 10.1016/j.nonrwa.2017.09.003

M3 - Article

SN - 1468-1218

VL - 40

SP - 307

EP - 327

JO - Nonlinear Analysis: Real World Applications

JF - Nonlinear Analysis: Real World Applications

ER -