Asymptotic behaviour of cuboids optimising Laplacian eigenvalues

We prove that in dimension $n \geq 2$, within the collection of unit measure cuboids in $\mathbb{R}^n$ (i.e. domains of the form $\prod_{i=1}^{n}(0, a_n)$), any sequence of minimising domains $R_k^\mathcal{D}$ for the Dirichlet eigenvalues $\lambda_k$ converges to the unit cube as $k \to \infty$. Correspondingly we also prove that any sequence of maximising domains $R_k^\mathcal{N}$ for the Neumann eigenvalues $\mu_k$ within the same collection of domains converges to the unit cube as $k\to \infty$. For $n=2$ this result was obtained by Antunes and Freitas in the case of Dirichlet eigenvalues and van den Berg, Bucur and Gittins for the Neumann eigenvalues. The Dirichlet case for $n=3$ was recently treated by van den Berg and Gittins. In addition we obtain stability results for the optimal eigenvalues as $k \to \infty$. We also obtain corresponding shape optimisation results for the Riesz means of eigenvalues in the same collection of cuboids. For the Dirichlet case this allows us to address the shape optimisation of the average of the first $k$ eigenvalues.