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

Abstract

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