Minimization of the k-th eigenvalue of the Robin-Laplacian

The paper is concerned with the minimization of the k-th eigenvalue of the Laplace operator with Robin boundary conditions, among all open sets of the space satisfying a volume constraint. We prove the existence of a solution in a relaxed framework and find some qualitative properties of the optimal sets. The main idea is to see these spectral shape optimization questions as free discontinuity problems in the framework of special functions of bounded variation. One of the key difficulties (for k greater or equal than 3) comes from the fact that the eigenvalues are critical points. The research that led to the present paper was partially supported by a grant of the group GNAMPA of INdAM.