Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces
Régis Straubhaar · Computational Methods in Applied Mathematics · 2014
Abstract. Let (M,g) be a smooth and complete surface, Ω ⊂ M $\Omega \subset M$ be a domain in M, and Δ g $\Delta _g$ be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence 0 < λ 1 ( Ω ) ≤ λ 2 ( Ω ) ≤ ⋯ ↗ ∞ $0 < \lambda _1(\Omega ) \le \lambda _2(\Omega ) \le \cdots earrow \infty $ . A classical question is to ask what is the domain Ω * $\Omega ^*$ which minimizes λ m ( Ω ) $\lambda _m(\Omega )$ among all domains of a given area, and what is the value of the corresponding λ m ( Ω m * ) $\lambda _m(\Omega _m^*)$ . The aim of this article is to present a numerical algorithm using shape optimization and based on the finite element method to find an approximation of a candidate for Ω m * $\Omega _m^*$ . Some verifications with existing numerical results are carried out for the first eigenvalues of domains in ℝ2. Furthermore, some investigations are presented in the two-dimensional sphere to illustrate the case of the positive curvature, in hyperbolic space for the negative curvature and in a hyperboloid for a non-constant curvature.