Two-scale convergence of Stekloff eigenvalue problems in perforated domains
type: Item
instance of: scholarly article
title: Two-scale convergence of Stekloff eigenvalue problems in perforated domains
zbMATH Open document ID: 1214.47020
DOI: 10.1155/2010/853717
author: Hermann Douanla
published in: Boundary Value Problems
publication date: 2011-02-08T00:00:00Z
full work available at URL: https://eudml.org/doc/226372
review text: The author deals with the spectral asymptotics of Stekloff eigenvalue problems for an elliptic linear differential operator \[ \begin{aligned} -\sum_{i,j=1}^{N}\frac{\partial}{\partial x_i}\left(a_{ij}\left(x,\frac{x}{\varepsilon}\right)\frac{\partial u_{\varepsilon}}{\partial x_{j}}\right)=0, &\quad \text{in } \Omega^{\varepsilon},\\ \sum_{i,j=1}^{N} a_{ij}\left(x,\frac{x}{\varepsilon}\right)\frac{\partial u_{\varepsilon}}{\partial x_{j}}\nu_i=\lambda_{\varepsilon}u_{\varepsilon}, &\quad \text{on } \partial T^{\varepsilon},\\ u_{\varepsilon}=0, &\quad \text{on } \partial \Omega,\\ \varepsilon\int_{S^{\varepsilon}}|u_{\varepsilon}|^2\,d\sigma_{\varepsilon}(x)=1, &{}\end{aligned} \] where \(\Omega\) is a bounded open set and \(\Omega^{\varepsilon}\) a perforated domain. By using the two-scale convergence method, a very accurate, precise, and concise homogenization result is obtained. Also, a convergence result for gradients of eigenfunctions is provided, which improves the understanding of the asymptotic behavior of eigenfunctions. Finally, it is justified that the natural local problem is not an eigenvalue problem.
reviewed by: Sotiris K. Ntouyas
Mathematics Subject Classification ID: 47A75
Mathematics Subject Classification ID: 47F05
zbMATH DE Number: 5847805
zbMATH Keywords: concise homogenization
zbMATH Keywords: eigenfunctions
zbMATH Keywords: two-scale convergence method
Wikidata QID: Q59251850
MaRDI profile type: Publication
OpenAlex ID: W2017593015
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description: scientific article; zbMATH DE number 5847805