Optimal shapes maximizing the Steklov eigenvalues

In this paper we consider the problem of maximizing the kth Steklov eigenvalue of the Laplacian (or a more general spectral functional), among all sets of ℝ^d of prescribed volume. We prove existence of an optimal set and get some qualitative properties of the solutions in a relaxed setting. In particular, in ℝ², we prove that the optimal set consists in the union of at most k disjoint Jordan domains with finite perimeter. A key point of our analysis is played by an isodiametric control of the Stelkov spectrum. We also perform some numerical experiments and exhibit the optimal shapes maximizing the kth eigenvalues under area constraint in ℝ² for k = 1, . . . , 10.