Inserted: 27 may 2005
We consider a general formulation for optimization problems involving the eigenvalues of the Laplace operator. Both the cases of Dirichlet and Neumann conditions on the free boundary are studied. We list several results concerning the existence of optimal domains, together with some conjectures and open problems. The last section contains some numerical computations.
Keywords: shape optimization, eigenvalues, Laplace operator, Finite elements method