TY - JOUR

T1 - “Blinking eigenvalues” of the Steklov problem generate the continuous spectrum in a cuspidal domain

AU - Nazarov, Sergei A.

AU - Taskinen, Jari

PY - 2020/8/5

Y1 - 2020/8/5

N2 - We study the Steklov spectral problem for the Laplace operator in a bounded domain Ω⊂Rd, d≥2, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains Ωε⊂Ω, ε>0, obtained from Ω by blunting the cusp at the distance of ε from the cusp tip. While the spectrum in the blunted domain Ωε consists for a fixed ε of an unbounded positive sequence {λj ε}j=1 ∞ of eigenvalues, we single out different types of behavior of some eigenvalues as ε→+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 Ω, when ε→+0.

AB - We study the Steklov spectral problem for the Laplace operator in a bounded domain Ω⊂Rd, d≥2, with a cusp such that the continuous spectrum of the problem is non-empty, and also in the family of bounded domains Ωε⊂Ω, ε>0, obtained from Ω by blunting the cusp at the distance of ε from the cusp tip. While the spectrum in the blunted domain Ωε consists for a fixed ε of an unbounded positive sequence {λj ε}j=1 ∞ of eigenvalues, we single out different types of behavior of some eigenvalues as ε→+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 Ω, when ε→+0.

KW - LAPLACIAN

UR - http://www.scopus.com/inward/record.url?scp=85079403023&partnerID=8YFLogxK

U2 - 10.1016/j.jde.2020.02.009

DO - 10.1016/j.jde.2020.02.009

M3 - Article

AN - SCOPUS:85079403023

VL - 269

SP - 2774

EP - 2797

JO - Journal of Differential Equations

JF - Journal of Differential Equations

SN - 0022-0396

IS - 4

ER -