Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
type: Item
instance of: scholarly article
title: Asymptotic profiles of eigenfunctions for some 1-dimensional linearized eigenvalue problems
zbMATH Open document ID: 1201.34027
author: Tohru Wakasa
author: Shoji Yotsutani
published in: Communications on Pure and Applied Analysis
publication date: 2010-05-25T00:00:00Z
review text: The authors investigate the asymptotic profiles of all eigenfunctions \(\varphi_j^n\) (\(j\in\mathbb{N}\cup\{0\}\), \(n\in\mathbb{N}\)) of the linearized eigenvalue problem \[ \begin{cases}\varepsilon^2\varphi_{xx}(x)+f_u(u_{n,\varepsilon}(x))\varphi(x)+\mu\varphi(x)=0,&\text{in }(0,1),\\ \varphi_x(0)=\varphi_x(1)=0,\end{cases} \] where \(\varepsilon\) is a positive parameter, \(f(u)=\sin u\) and \(u_{n,\varepsilon}\) (\(n\in\mathbb{N}\)) is the \(n\)-mode solution of the problem \[ \begin{cases}\varepsilon^2u_{xx}(x)+f(u(x))=0&\text{in }(0,1),\\ u_x(0)=u_x(1)=0.\end{cases} \] They show that two special eigenfunctions completely control the asymptotic profiles of the other eigenfunctions.
reviewed by: Rodica Luca
Mathematics Subject Classification ID: 34B08
Mathematics Subject Classification ID: 34A05
Mathematics Subject Classification ID: 34B05
Mathematics Subject Classification ID: 34B15
Mathematics Subject Classification ID: 34L10
zbMATH DE Number: 5711089
zbMATH Keywords: nonlinear eigenvalue problem
zbMATH Keywords: linearized eigenvalue problem
zbMATH Keywords: linearized stability
zbMATH Keywords: eigenvalue
zbMATH Keywords: eigenfunction
zbMATH Keywords: exact solution
zbMATH Keywords: asymptotic formula
MaRDI profile type: Publication
full work available at URL: cpaa.2010.9.539
OpenAlex ID: W2012589612
DOI: 10.3934/CPAA.2010.9.539
description: scientific article; zbMATH DE number 5711089