A nonlinear eigenvalue optimization problem: Optimal potential functions

作者：Antunes Pedro R S; Mohammadi Seyyed Abbas^*; Voss Heinrich

来源：Nonlinear Analysis: Real World Applications , 2018, 40: 307-327.

DOI：10.1016/j.nonrwa.2017.09.003

• Nonlin eareigenvalue problem
• Shape optimization
• Ultra-high contrast regime
• Quantum dots

In this paper we study the following optimal shape design problem: Given an open connected set Omega subset of R-N and a positive number A is an element of (0, vertical bar Omega vertical bar), find a measurable subset D subset of Omega with vertical bar D vertical bar = A such that the minimal eigenvalue of -div(zeta(lambda, x)del u)+alpha chi(D)u = lambda u in Omega, u = 0 on partial derivative Omega, 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.

• 出版日期2018-4