Homogenization and correctors for the wave equation in non periodic perforated domains
type: Item
instance of: scholarly article
title: Homogenization and correctors for the wave equation in non periodic perforated domains
zbMATH Open document ID: 1166.35003
author: P. Donato
author: Florian Gaveau
published in: Networks and Heterogeneous Media
publication date: 2008-08-22T00:00:00Z
review text: The authors consider the homogenization of the wave equation in a (not necessarily periodically) perforated domain. They consider the wave equation \(\rho ^{\varepsilon }u_{\varepsilon }^{\prime \prime }-\text{div} (A^{\varepsilon }\nabla u_{\varepsilon })=f_{\varepsilon }\) in \(\Omega _{\varepsilon }\times (0,T)\), with the homogeneous Dirichlet boundary condition \(u_{\varepsilon }=0\) on \(\partial \Omega \times (0,T)\) and the homogeneous Neumann boundary condition \(A^{\varepsilon }\nabla u_{\varepsilon }\cdot \nu =0\) on \(\partial S_{\varepsilon }\times (0,T)\). Here \(\Omega \) is a bounded domain of \(\mathbb{R}^{n}\), \( S_{\varepsilon }\) is a collection of holes and \(\Omega _{\varepsilon }=\Omega \backslash S_{\varepsilon }\). The authors suppose that \( (S_{\varepsilon })_{\varepsilon }\) satisfies an ``admissible'' condition, in terms of the existence of linear (and uniformly continuous) extension operators belonging to \(\mathcal{L}(V_{\varepsilon },H_{0}^{1}(\Omega ))\) where \(V_{\varepsilon }=\{u\in H^{1}(\Omega _{\varepsilon }):u=0\) on \( \partial \Omega \}\). In the first part of their paper, the authors recall the definition of the \(H^{0}\)-convergence for sequences \((A^{\varepsilon },S_{\varepsilon })_{\varepsilon }\), previously defined by [\textit{M. Briane} et. al. Nonlinear partial differential equations and their applications. Collège de France seminar. Vol. XIII. Lectures held at the J. L. Lions seminar on applied mathematics, Paris, France, 1994--1996. Harlow: Longman. Pitman Res. Notes Math. Ser. 391, 62--100 (1998; Zbl 0943.35005)], and its compactness result. They then define the notion of global corrector and recall the existence of such a corrector in the present context, with its properties. As a first result, the authors prove the existence of an extension operator for every sequence \((S_{\varepsilon })_{\varepsilon }\) of admissible holes, in a stronger definition of this admissibility. They then build the convergence resul…
reviewed by: Alain Brillard
Mathematics Subject Classification ID: 35B27
Mathematics Subject Classification ID: 35L05
Mathematics Subject Classification ID: 35L20
zbMATH DE Number: 5315160
zbMATH Keywords: extension operator
zbMATH Keywords: \(H^{0}\)-convergence
zbMATH Keywords: homogeneous Dirichlet boundary condition
zbMATH Keywords: homogeneous Neumann boundary condition
MaRDI profile type: Publication
full work available at URL: nhm.2008.3.97
OpenAlex ID: W2053894927
DOI: 10.3934/NHM.2008.3.97
description: scientific article; zbMATH DE number 5315160