Application of the methods of the mathematical theory of diffraction to the problem of the asymptotic behavior of the eigenvalues and eigenfunctions of the Laplace operator

1976 ◽  
pp. 54-56
Author(s):  
V. M. Babič
Keyword(s):  
Asymptotic Behavior ◽  
Laplace Operator ◽  
Mathematical Theory ◽  
Eigenvalues And Eigenfunctions ◽  
The Laplace Operator

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.

scholarly journals ON THE ASYMPTOTIC BEHAVIOR OF EIGENVALUES OF THE BOUNDARY VALUE PROBLEM WITH A PARAMETER

2017 ◽  
Vol 21 (6) ◽  
pp. 135-140
Author(s):  
A.V. Filinovskiy
Keyword(s):  
Boundary Condition ◽  
Asymptotic Behavior ◽  
Dirichlet Problem ◽  
Laplace Operator ◽  
Smooth Boundary ◽  
Robin Boundary Condition ◽  
Simple Eigenvalue ◽  
Robin Problem ◽  
The Laplace Operator ◽  
Robin Boundary

The paper presents the investigation of an eigenvalue problem for the Laplace operator with Robin boundary condition in a bounded domain with smooth boundary. The case of boundary condition containing a real parameter is con- sidered. It is proved that multiplicity of the eigenvalue to the Robin problem for all values of the parameter greater than some number does not exceed the mul- tiplicity of the corresponding eigenvalue to the Dirichlet problem for the Laplace operator. For simple eigenvalue of the Dirichlet problem the convergence of eigen- function of the Robin problem to the eigenfunction of the Dirichlet problem for unlimited increase of the parameter is proved. The formula for derivative on the parameter for eigenvalues of the Robin problem is established. This formula is used to justify the asymptotic expansions of eigenvalues of the Robin problem for large positive values of the parameter.

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