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 $k\to +\mathrm{\infty }$. 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 mixed Dirichlet-Neumann eigenvalues, under an additional geometric constraint, and discuss some applications of our proof techniques.