Asymptotic Expansions of Eigenfunctions and Eigenvalues of the Steklov Spectral Problem in Thin Perforated Domains with Rapidly Varying Thickness and Different Limit Dimensions

The paper deals with the asymptotic behaviour as ε → 0 of the spectrum of the Laplace–Beltrami operator Δε on the Riemannian manifold Mε (dim Mε = N ≥ 2) depending on a small parameter ε > 0. Mε consists of two perforated domains, which are connected by an array of tubes of length qε. Each perforated domain is obtained by removing from the fixed domain Ω ⊂ ℝN the system of ε-periodically distributed balls of radius dε = ō(ε). We obtain a variety of homogenized spectral problems in Ω; their type depends on some relations between ε, dε and qε. In particular, if the limitsare positive, then the homogenized spectral problem contains the spectral parameter in a nonlinear manner, and its spectrum has a sequence of accumulation points.

Download Full-text

Asymptotic expansions and numerical algorithms of eigenvalues and eigenfunctions of the Dirichlet problem for second order elliptic equations in perforated domains

Numerische Mathematik ◽

10.1007/s00211-003-0468-7 ◽

2004 ◽

Vol 96 (3) ◽

pp. 525-581 ◽

Cited By ~ 47

Author(s):

Jun-Zhi Cui ◽

Li-Qun Cao

Keyword(s):

Dirichlet Problem ◽

Asymptotic Expansions ◽

Elliptic Equations ◽

Numerical Algorithms ◽

Second Order ◽

Eigenvalues And Eigenfunctions ◽

Perforated Domains ◽

Second Order Elliptic Equations

Download Full-text

Asymptotic approximations for eigenvalues and eigenfunctions of a spectral problem in a thin graph-like junction with a concentrated mass in the node

Analysis and Applications ◽

10.1142/s0219530520500219 ◽

2020 ◽

pp. 1-65

Author(s):

Taras Mel’nyk

Keyword(s):

Asymptotic Behavior ◽

Small Parameter ◽

Spectral Problem ◽

Asymptotic Expansions ◽

Multiscale Analysis ◽

Asymptotic Approximations ◽

Concentrated Mass ◽

Eigenvalues And Eigenfunctions

A spectral problem is considered in a thin 3D graph-like junction that consists of three thin curvilinear cylinders that are joined through a domain (node) of the diameter [Formula: see text] where [Formula: see text] is a small parameter. A concentrated mass with the density [Formula: see text] [Formula: see text] is located in the node. The asymptotic behavior of the eigenvalues and eigenfunctions is studied as [Formula: see text] i.e. when the thin junction is shrunk into a graph. There are five qualitatively different cases in the asymptotic behavior [Formula: see text] of the eigenelements depending on the value of the parameter [Formula: see text] In this paper three cases are considered, namely, [Formula: see text] [Formula: see text] and [Formula: see text] Using multiscale analysis, asymptotic approximations for eigenvalues and eigenfunctions are constructed and justified with a predetermined accuracy with respect to the degree of [Formula: see text] For irrational [Formula: see text] a new kind of asymptotic expansions is introduced. These approximations show how to account the influence of local geometric and physical inhomogeneity of the node in the corresponding limit spectral problem on the graph for different values of the parameter [Formula: see text]

Download Full-text