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 $E(R_{\Omega }^{\alpha })$ 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.

arXiv comment19 pages

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Cited by in corpus (1)

• Laplacian eigenvalues for large negative Robin parameters on domains with outward peaks