Peculiar behavior of the principal Laplacian eigenvalue for large negative Robin parameters

TL;DR

This paper investigates the asymptotic behavior of the principal eigenvalue of the Laplacian with Robin boundary conditions as the Robin parameter becomes large and negative, providing a counterexample to a previously open question.

Contribution

The authors demonstrate that the limit of the ratio of the principal eigenvalue to the square of the Robin parameter does not always exist as the parameter tends to negative infinity.

Findings

Counterexample showing the limit does not exist

Negative answer to the open question from 2017

Insights into eigenvalue behavior for large negative Robin parameters

Abstract

Let $\Omega\subset\mathbb{R}^n$ with $n\ge 2$ be a bounded Lipschitz domain with outer unit normal $\nu$. For $\alpha\in\mathbb{R}$ let $R_\Omega^\alpha$ be the Laplacian in $\Omega$ with the Robin boundary condition $\partial_\nu u+\alpha u=0$, and denote by $E(R^\alpha_\Omega)$ its principal eigenvalue. In 2017 Bucur, Freitas and Kennedy stated the following open question: Does the limit of the ratio $E(R_\Omega^\alpha)/ \alpha^2$ for $\alpha\to-\infty$ always exist? We give a negative answer.