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

TL;DR

This paper investigates how the Steklov spectral problem's eigenvalues behave in domains with cusps, revealing phenomena like blinking and gliding eigenvalues that lead to the emergence of a continuous spectrum as the cusp is sharpened.

Contribution

It introduces the concepts of blinking and gliding eigenvalues and explains their role in forming the continuous spectrum in cuspidal domains.

Findings

Identification of stable, blinking, and gliding eigenvalue families.

Description of the mechanism transforming eigenvalues into continuous spectrum.

Analysis of eigenvalue behavior as the cusp is sharpened.

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 $\varepsilon \to +0$.