Asymptotic behavior of the eigenvalues and eigenfunctions to a spectral problem in a thick cascade junction with concentrated masses

Abstract

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a boundaryvalue problem for the Laplace operator in a thick cascade junction with concentrated masses is investigated. This cascade junction consists of the junction's body and great number 5N=O(ε−1) of ε-alternating thin rods belonging to two classes. One class consists of rods of finite length and the second one consists of rods of small length of order O(ε). The density of the junction is order O(ε−α) on the rods from the second class (the concentrated masses if α>0) and O(1) outside of them. In addition, we study the influence of the concentrated masses on the asymptotic behavior of these magnitudes in the case α=1 and α∈(0,1).