Inserted: 2 mar 2017
Last Updated: 2 mar 2017
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 $\R^N$ 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 $k \ge 3$) comes from the fact that the eigenvalues are critical points.