Existence and Regularity of Optimal Convex Shapes for Functionals Involving the Robin Eigenvalues

We study the existence and the regularity of optimal convex domains for a large class of shape optimization problems involving functions of the eigenvalues of the Robin-Laplacian on convex sets. We will prove that convex solutions exist and that, under some additional hypotheses on the functional, these optimal sets have $C^1$ boundary.