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

TL;DR

This paper investigates the asymptotic behavior and numerical approximation of the optimal eigenvalues of the Robin Laplacian, revealing how the minimizers depend on boundary parameters and deriving growth estimates contrasting Weyl asymptotics.

Contribution

It introduces a Wolf-Keller type result, analyzes the growth of optimal eigenvalues, and numerically explores the dependence of minimizers on the boundary parameter for the Robin Laplacian.

Findings

Optimal eigenvalues grow at most with n^{1/N}.

The gap between consecutive eigenvalues tends to zero as n increases.

Numerical evidence suggests existence of a boundary parameter minimizing the eigenvalues for each n.

Abstract

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 $n^{1/N}$, 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 $n^{\rm th}$ eigenvalue is minimised by $n$ disks for all $0<\alpha<\alpha_{n}$ and, combined with analytic estimates, that this value is expected to grow with $n^{1/N}$.