We study some shape optimization problems related to sums and quotients of Dirichlet Laplacian eigenvalues ?n for planar domains. We show how to minimize a sum View the MathML source 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 View the MathML source, k=2,3,…. This last problem was already studied by Osting using a different numerical method and we obtain similar results.

CEMAT - Center for Computational and Stochastic Mathematics