On eigenfrequencies of bodies with thin branches. II: Asymptotics
type: Item
instance of: scholarly article
title: On eigenfrequencies of bodies with thin branches. II: Asymptotics
zbMATH Open document ID: 0839.35091
DOI: 10.1007/BF02110759
published in: Mathematical Notes
publication date: 1995-09-24T00:00:00Z
review text: In [Math. Notes 54, No. 6, 1192-1199 (1993); translation from Mat. Zametki 54, No. 6, 10-21 (1993; Zbl 0829.35088)] we investigated eigenfrequencies of \(n\)-dimensional bodies \(\Omega_\varepsilon\) consisting of two components: a fixed bounded volume \(\Omega\) and a thin cylindrical branch \(\kappa_\varepsilon\) adjoined to \(\Omega\) and having finite length \(h\); \(\varepsilon\ll 1\) is the ``diameter'' of the section of \(\kappa_\varepsilon\) (the precise definition of \(\Omega_\varepsilon\) will be given below). The mathematical statement of the problem is reduced to the following boundary value problems on eigenvalues: \[ \begin{alignedat}{4} (\Delta+ \lambda_\varepsilon) \psi_\varepsilon & = 0, &\quad x & \in \Omega_\varepsilon; &\quad \partial\psi_\varepsilon/\partial\nu & = 0, &\quad x & \in \partial\Omega_\varepsilon,\tag{1}\\ (\Delta+ \lambda_\varepsilon)\psi_\varepsilon & = 0, &\quad x & \in \Omega_\varepsilon; &\quad \psi_\varepsilon & = 0, &\quad x &\in \partial\Omega_\varepsilon,\tag{2}\end{alignedat} \] where \(\nu\) is an exterior normal to \(\Omega_\varepsilon\). The boundary value problem (2) corresponds to a fixed body, and the problem (1) to a body with free boundary. The eigenvalues \(\lambda_\varepsilon\) are squares of the corresponding eigenfrequencies. In [loc. cit.] it was shown that when reducing the diameter of the section of the branch all bounded eigenvalues of the problem (2) converge to the set \(\widetilde\Sigma\) of eigenvalues of the Dirichlet problem for the Laplace operator in \(\Omega\). Conversely, if \(\lambda_0\) is a \(q\)-multiple eigenvalue of the Dirichlet problem in \(\Omega\), then the total algebraic multiplicity of eigenvalues of (2) that are converging to \(\lambda_0\) is also equal to \(q\). But for the problem (1) the situation is quite different: as \(\varepsilon\to 0\) the eigenvalues of this problem converge both to the set \(\Sigma^\nu\) of eigenvalues of the Neumann problem for the Laplace operator in the …
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 796760
zbMATH Keywords: boundary value problems on eigenvalues
zbMATH Keywords: Dirichlet problem for the Laplace operator
zbMATH Keywords: Neumann problem for the Laplace operator
zbMATH Keywords: Sturm-Liouville problem
zbMATH Keywords: asymptotics of eigenvalues
zbMATH Keywords: limit values of the eigenfunctions
author: R. R. Gadyl'shin
MaRDI profile type: Publication
cites work: Q5541954
description: scientific article; zbMATH DE number 796760