"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 $\Omega \subset \mathbb{R}^d$, $d \geq 2$, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains $\Omega^\varepsilon \subset \Omega$, $\varepsilon > 0$, obtained from $\Omega$ by blunting the cusp at the distance of $\varepsilon$ from the cusp tip. While the spectrum in the blunted domain $\Omega^\varepsilon$ consists for a fixed $\varepsilon$ of an unbounded positive sequence $\{ \lambda_j^\varepsilon \}_{j=1}^\infty$ of eigenvalues, we single out different types of behavior of some eigenvalues as $\varepsilon \to +0$: 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 $\varepsilon \to +0$.