Numerical Optimization of Dirichlet-Laplace eigenvalues on domains in surfaces

The spectrum of the Dirichlet-Laplace operator defined on a bounded domain in a smooth and complete surface consists in a strictly positive sequence, increasing to infinity. The aim of this thesis is to approximate numerically the first eigenvalues of this operator using a finite element based method, then to address the following optimization problem: what is the domain which minimizes the $k$-th eigenvalue among all domains of a given area, and what is this eigenvalue equal to? This latter has its roots in the Faber-Krahn and Krahn-Szegö theorems, which answer the question for the first and the second eigenvalue of a domain in the Euclidean space. For higher eigenvalues and other underlying surfaces like the sphere and hyperbolic space, shape optimization has been performed to provide domains which are candidates to be solutions. This gives rise to some observations about the comparison of eigenvalues of domains in various surfaces. The problem of locating a circular obstacle inside a ball to maximize the first eigenvalues is also addressed in this document.