Numerical Optimization of Eigenvalues of the Dirichlet–Laplace Operator on Domains in Surfaces

Let (M,g) be a smooth and complete surface, Ω⊂M be a domain in M, and Δg be the Laplace operator on M. The spectrum of the Dirichlet–Laplace operator on Ω is a sequence 0<λ1(Ω)≤λ2(Ω)≤⋯↗∞. A classical question is to ask what is the domain Ω∗ which minimizes λm(Ω) among all domains of a given area, and what is the value of the corresponding λm(Ω∗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. 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.