scholarly journals Asymptotic behavior of the spectrum of an elliptic problem in a domain with aperiodically distributed concentrated masses

2017 ◽  
Vol 345 (10) ◽  
pp. 671-677 ◽  
Author(s):  
Gregory A. Chechkin ◽  
Tatiana P. Chechkina
Keyword(s):  
Asymptotic Behavior ◽  
Elliptic Problem ◽  
Concentrated Masses

VIBRATIONS OF A MEMBRANE WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1995 ◽  
Vol 05 (05) ◽  
pp. 565-585 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PÉREZ
Keyword(s):  
Perturbation Theory ◽  
Asymptotic Behavior ◽  
Small Parameter ◽  
Local Problem ◽  
Micro Structure ◽  
Spectral Perturbation Theory ◽  
Concentrated Masses ◽  
The Masses ◽  
Spectral Perturbation ◽  
Small Eigenvalues

We consider the asymptotic behavior of the vibrations of a membrane occupying a domain Ω ⊂ ℝ2. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε−m) with m>0. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. Depending on the value of the parameter m(m>2, m=2 or m<2) we describe the asymptotic behavior of the eigenvalues. Small eigenvalues, of order O(εm−2) for m>2, are approached via those of a local problem obtained from the micro-structure of the problem, while the eigenvalues of order O(1) are approached through those of a homogenized problem, which depend on the relation between ε and η. Techniques of boundary homogenization and spectral perturbation theory are used to study this problem.

ON VIBRATIONS OF A BODY WITH MANY CONCENTRATED MASSES NEAR THE BOUNDARY

1993 ◽  
Vol 03 (02) ◽  
pp. 249-273 ◽  
Author(s):  
MIGUEL LOBO ◽  
EUGENIA PEREZ
Keyword(s):  
Asymptotic Behavior ◽  
Small Parameter ◽  
Critical Size ◽  
Local Problem ◽  
Boundary Values ◽  
Micro Structure ◽  
Spectral Convergence ◽  
Fourier And Laplace Transforms ◽  
Concentrated Masses ◽  
The Masses

We consider the asymptotic behavior of the vibration of a body occupying a region Ω⊂ℝ3. The density, which depends on a small parameter ε, is of order O(1) out of certain regions where it is O(ε–m) with m>2. These regions, the concentrated masses with diameter O(ε), are located near the boundary, at mutual distances O(η), with η=η(ε)→0. We impose Dirichlet (respectively Neumann) conditions at the points of ∂Ω in contact with (respectively, out of) the masses. For the critical size ε=O(η2), the asymptotic behavior of the eigenvalues of order O(εm−2) is described via a Steklov problem, where the ‘mass’ is localized on the boundary, or through the eigenvalues of a local problem obtained from the micro-structure of the problem. We use the techniques of the formal asymptotic analysis in homogenization to determine both problems. We also use techniques of convergence in homogenization, Semigroups theory, Fourier and Laplace transforms and boundary values of analytic functions to prove spectral convergence. In the same framework we study the case m=2 as well as the case when other boundary conditions are imposed on ∂Ω.

scholarly journals Random Homogenization in a Domain with Light Concentrated Masses

Mathematics ◽  
2020 ◽  
Vol 8 (5) ◽  
pp. 788
Author(s):  
Gregory A. Chechkin ◽  
Tatiana P. Chechkina
Keyword(s):  
Small Parameter ◽  
Elliptic Problem ◽  
Stochastic Perturbation ◽  
Initial Problem ◽  
Compactness Theorem ◽  
Concentrated Masses ◽  
Homogenization Methods

In the paper, we consider an elliptic problem in a domain with singular stochastic perturbation of the density located near the boundary, depending on a small parameter. Using the boundary homogenization methods, we prove the compactness theorem and study the behavior of eigenelements to the initial problem as the small parameter tends to zero.

VIBRATIONS OF A THICK PERIODIC JUNCTION WITH CONCENTRATED MASSES

2001 ◽  
Vol 11 (06) ◽  
pp. 1001-1027 ◽  
Author(s):  
TARAS A. MEL'NYK
Keyword(s):  
Asymptotic Behavior ◽  
Boundary Value Problem ◽  
Laplace Operator ◽  
Mixed Boundary ◽  
Asymptotic Estimates ◽  
Mixed Boundary Value Problem ◽  
Frequency Range ◽  
Eigenvalues And Eigenfunctions ◽  
Concentrated Masses ◽  
The Laplace Operator

The asymptotic behavior (as ε→0) of eigenvalues and eigenfunctions of a mixed boundary-value problem for the Laplace operator in a plane thick periodic junction with concentrated masses is investigated. This junction consists of the junction's body and a large number N=O(ε-1) of thin rods. The density of the junction is order O(ε-α) on the rods (the concentrated masses if α>0), and O(1) outside. The results depend on the value of the parameter α(α<2, α=2, or α>2). There are three kinds of vibrations, which are present in each of these cases: vibrations, whose energy is concentrated in the junction's body; vibrations, whose energy is concentrated on the thin rods; and vibrations (pseudovibrations), in which each thin rod can have its own frequency. The frequency range, where pseudovibrations can be present, is indicated. The asymptotic estimates for the corresponding eigenfunctions and eigenvalues are proved.

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