Peculiar behavior of the principal Laplacian eigenvalue 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.