Optimization of sums and quotients of Dirichlet-Laplacian eigenvalues

We study some shape optimization problems related to sums and quotients of Dirichlet Laplacian eigenvalues λn for planar domains. We show how to minimize a sum (λk+λk+1)|Ω|,k=1,2,… when the minimizing domain is disconnected. In particular, we prove that the optimizers in the cases k=1 and k=2 are connected. We develop a numerical method for solving shape optimization eigenvalue problems which is applied to determine the first fourteen optimizers for sums of consecutive Dirichlet eigenvalues and quotients of type λkλ1, k=2,3,…. This last problem was already studied by Osting using a different numerical method and we obtain similar results.