Minimising Dirichlet eigenvalues on cuboids of unit measure

We consider the minimisation of Dirichlet eigenvalues ${\lambda }_k$, $k \in \N$,oftheLaplacianoncuboidsofunitmeasurein$\R^3$. We prove that any sequence of optimal cuboids in ${\mathbb{R}}^3$ converges to a cube of unit measure in the sense of Hausdorff as $k\to \mathrm{\infty }$.