Minimizing the second eigenvalue of the Laplace operator with Dirichlet boundary conditions

In this paper, we are interested in the minimization of the second eigenvalue of the Laplacian with Dirichlet boundary conditions amongst convex plane domains with given area. The natural candidate to be the optimum was the ''stadium'', convex hull of two identical tangent disks. We refute this conjecture. Nevertheless, we prove existence of a minimzer. We also study some qualitative properties of the minimizer (regularity, geometric properties)