"Blinking eigenvalues" of the Steklov problem generate the continuous spectrum in a cuspidal domain

Abstract

We study the Steklov spectral problem for the Laplace operator in a bounded domain $\Omega \subset {\mathbb{R}}^d$, $d\ge 2$, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains ${\Omega }^{\epsilon }\subset \Omega$, $\epsilon >0$, obtained from $\Omega$ by blunting the cusp at the distance of $\epsilon$ from the cusp tip. While the spectrum in the blunted domain ${\Omega }^{\epsilon }$ consists for a fixed $\epsilon$ of an unbounded positive sequence $\{{\lambda }_j^{\epsilon }{\}}_{j=1}^{\infty }$ of eigenvalues, we single out different types of behavior of some eigenvalues as $\epsilon \to +0$: in particular, stable, blinking, and gliding families of eigenvalues are found. We also describe a mechanism which transforms the family of the eigenvalue sequences into the continuous spectrum of the problem in $\Omega$, when $\epsilon \to +0$.