Asymptotic behaviour and numerical approximation of optimal eigenvalues of the Robin Laplacian

We consider the problem of minimising the $n^{th}-$eigenvalue of the Robin Laplacian in ${\mathbb{R}}^N$. Although for $n=1,2$ and a positive boundary parameter $\alpha$ it is known that the minimisers do not depend on $\alpha$, we demonstrate numerically that this will not always be the case and illustrate how the optimiser will depend on $\alpha$. We derive a Wolf-Keller type result for this problem and show that optimal eigenvalues grow at most with $n^{1\mathrm{/}N}$, which is in sharp contrast with the Weyl asymptotics for a fixed domain. We further show that the gap between consecutive eigenvalues does go to zero as $n$ goes to infinity. Numerical results then support the conjecture that for each $n$ there exists a positive value of ${\alpha }_n$ such that the $n^{\rm th}$eigenvalueisminimisedby$n$disksforall$0<\alpha<\alpha_{n}$ and, combined with analytic estimates, that this value is expected to grow with $n^{1\mathrm{/}N}$.