Asymptotics of eigenelements to spectral problem in thick cascade junction with concentrated masses

Abstract

The asymptotic behaviour (as ϵ → 0) of eigenvalues and eigenfunctions of a boundary-value 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 behaviour of these magnitudes in that case α = 1 and α ∈ (0, 1).

Keywords:

• concentrated masses
• thick junction
• cascade junction
• asymptotics

AMS Subject Classifications::

• 35B27
• 35B25
• 47A75
• 74K30

Acknowledgements

This article was mainly written in Mathematisches Forschungsinstitut Oberwolfach during November 2009 under the support of the programm ‘Research in Pairs’. The authors wish to express deep gratitude for the hospitality and wonderful working conditions. The work of Gregory A. Chechkin was also supported, in part, by RFBR grant (no. 09-01-00353).