Characteristic frequencies of bodies with thin spikes. I: Convergence and estimates
type: Item
instance of: scholarly article
title: Characteristic frequencies of bodies with thin spikes. I: Convergence and estimates
zbMATH Open document ID: 0829.35088
DOI: 10.1007/BF01209080
published in: Mathematical Notes
publication date: 1995-09-12T00:00:00Z
review text: The set \(\Omega_\varepsilon\) is an \(n\)-dimensional body \(\Omega\) with thin spike \(\kappa_\varepsilon\) having ``diameter'' \(\varepsilon\) and finite length \(h\). In this paper we consider the following two eigenvalue (EV) problems: \[ (\Delta + \lambda_\varepsilon) \Psi_\varepsilon = 0, \quad x \in \Omega_\varepsilon, \quad \partial \Psi_\varepsilon/ \partial \nu = 0, \quad x \in \partial \Omega_\varepsilon, \tag{1} \] where \(\nu\) is the outer normal to \(\Omega_\varepsilon\) and \[ (\Delta + \lambda_\varepsilon) \Psi_\varepsilon = 0, \quad x \in \Omega_\varepsilon, \quad \Psi_\varepsilon = 0, \quad x \in \partial \Omega_\varepsilon. \tag{2} \] Nontrivial solutions of (2) describe the characteristic oscillations of the fixed body \(\Omega_\varepsilon\) and the solutions of (1) correspond to the characteristic oscillations of the body with free boundary. The square roots \(k_\varepsilon\) of the EV \(\lambda_\varepsilon\) are the characteristic frequencies of the bodies under the types of boundary conditions indicated. We study the question of the convergence of the EV of the boundary problems (1) and (2) as \(\varepsilon \to 0\). It would seem that as \(\varepsilon \to 0\) the EV of (1) and (2) would converge respectively to the sets \(\Sigma^\nu\) and \(\widetilde \Sigma\) of the EV of the Neumann and Dirichlet problems in \(\Omega\). Indeed such convergence holds for (2). However in the case of boundary conditions (1) it turns out that the EV converge either to \(\Sigma^\nu\) or to the set \(\Sigma^{\text{ch}} = \{\mu_m = (\pi (2m - 1)/(2h))^2\}_{m = 1}^\infty\). Firstly, we establish a correspondence between the multiplicities of the EV of the perturbed and limit problems without any restrictions on \(\Sigma^\nu\) and \(\Sigma^{\text{ch}}\) for (1), and secondly, we get estimates for solutions of the boundary problems \[ (\Delta + \lambda) u_\varepsilon = f, \quad x \in \Omega_\varepsilon, \quad \partial u_\varepsilon/ \partial \nu = 0, \quad x \in \pa…
Mathematics Subject Classification ID: 35P20
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 795647
zbMATH Keywords: scattering frequencies for an acoustic resonator
zbMATH Keywords: characteristic oscillations
zbMATH Keywords: characteristic frequencies
zbMATH Keywords: Neumann and Dirichlet problems
zbMATH Keywords: multiplicities
zbMATH Keywords: perturbed and limit problems
author: R. R. Gadyl'shin
MaRDI profile type: Publication
cites work: Decomposition of series of resonances on a 'nonphysical' sheet
cites work: Q4118213
cites work: Q3330608
cites work: Q3704306
cites work: Q3981481
cites work: Characterization of the eigenfunctions in the singularly perturbed domain
cites work: Q5541954
cites work: A comparison of three perturbation methods for non-linear hyperbolic waves
description: scientific article; zbMATH DE number 795647