Optimization of sums and quotients of Dirichlet–Laplacian eigenvalues

摘要

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.