We consider the shape optimisation of the Dirichlet eigenvalues of the Laplacian $\lambda_k$, $k \in \mathbb{N}$.

In a chosen collection of open sets in $\mathbb{R}^m$, $m \geq 2$, that are subject to certain geometric constraints, the focus is on determining a set which minimises the $k$-th Dirichlet eigenvalue. If it can be shown that an optimal set exists for each $\lambda_k$, then it is interesting to investigate the asymptotic behaviour of a sequence of optimal sets as $k \to \infty$.

It was shown by Antunes and Freitas in $2012$ that among planar rectangles of unit measure, any sequence of minimising rectangles for the Dirichlet eigenvalues converges to the unit square as $k \to \infty$. The corresponding result for $m=3$ was obtained in $2016$ by van den Berg and Gittins. Some of the arguments used in these lower-dimensional cases cannot be invoked when $m \geq 4$.

For $m \geq 4$, we consider the collection of all unit measure cuboids in $\mathbb{R}^m$, that is sets of the form $\prod_{i=1}^m (0,a_i)$. Our goal is to prove that any sequence of such cuboids $(R_k^*)_k$ that minimise the Dirichlet eigenvalues converges to the unit cube as $k \rightarrow \infty$. We interpret the Dirichlet counting function as a Riesz mean of order zero and use properties of Riesz means to show that any sequence of minimising cuboids is bounded independently of $k$. This allows us to prove convergence to the unit cube, as well as to obtain a stability result for the optimal eigenvalues. This is joint work with Simon Larson (KTH).

In addition, we consider the shape optimisation of the Neumann eigenvalues of the Laplacian.

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