Qualitative and numerical analysis of a spectral problem with perimeter constraint

We consider the problem of optimizing the kth eigenvalue of the Dirichlet Laplace operator under perimeter constraint. We provide a new method based on a Γ-convergence result for approximating the corresponding optimal shapes. We also give new optimality conditions in the case of multiple eigenvalues. We deduce from previous conditions the fact that optimal shapes never contain flat parts in their boundaries.