Optimization of sums and quotients of Dirichlet-Laplacian eigenvalues

We study some shape optimization problems related to sums and quotients of Dirichlet Laplacian eigenvalues kn for planar domains. We show how to minimize a sum ðkk þ kkþ1ÞjXj; k 1⁄4 1;2; . . . when the minimizing domain is disconnected. In particular, we prove that the optimizers in the cases k 1⁄4 1 and k 1⁄4 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 kk k1, k 1⁄4 2;3; . . .. This last problem was already studied by Osting using a different numerical method and we obtain similar results. 2012 Elsevier Inc. All rights reserved.