Asymptotic analysis of the spectral Neumann problem in thick multi-structure of type 3:1:1

Abstract

The spectral Neumann problem is considered in a thick multi-structure, which is the union of some three-dimensional domain (the junction’s body) and a large number of ε-periodically situated thin cylinders along some curve (the joint zone) on the boundary of junction’s body. The asymptotic behaviour (as ε → 0) of the eigenvalues and eigenfunctions is investigated. Three spectral problems form asymptotics for the eigenvalues and eigenfunctions of this problem, namely, the spectral Neumann problem in junction's body; some spectral problem in a plane domain, which is filled up by the thin cylinders in the limit passage (each eigenvalue of this problem has infinite multiplicity); and the spectral problem for some singular integral operator given on the joint zone. The Hausdorff convergence of the spectrum is proved, the leading terms of asymptotics are constructed (as ε → 0) and asymptotic estimates are justified for the eigenvalues and the eigenfunctions.