On the Isoperimetric and Isodiametric Inequalities and the Minimisation of Eigenvalues of the Laplacian.

We consider the problem of minimising the k-th eigenvalue of the Laplacian with some prescribed boundary condition over collections of convex domains of prescribed perimeter or diameter. It is known that these minimisation problems are well-posed for Dirichlet eigenvalues in any dimension $d\ge 2$ and any sequence of minimisers converges to the ball of unit perimeter or diameter respectively as $k\to +\infty$ . In this paper, we show that the same is true in the case of Neumann eigenvalues under diameter constraint in any dimension and under perimeter constraint in dimension $d=2$ . We also consider these problems for Robin eigenvalues and mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint.