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

Abstract

The asymptotic behavior (as $\varepsilon \to 0$) 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 $5N = \mathcal{O}(\varepsilon^{-1})$ of $\varepsilon$-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 $\mathcal{O}(\varepsilon)$. The density of the junction is order $\mathcal{O}(\varepsilon^{-\alpha})$ on the rods from the second class (the concentrated masses if $\alpha >0$) and $\mathcal{O}(1)$ outside of them. In addition, we study the influence of the concentrated masses on the asymptotic behavior of these magnitudes in the case $\alpha=1$ and $\alpha \in (0,1)$.