The asymptotic expansion of the solution of a boundary-value problem with rapidly alternating type of boundary conditions
type: Item
instance of: scholarly article
title: The asymptotic expansion of the solution of a boundary-value problem with rapidly alternating type of boundary conditions
zbMATH Open document ID: 0926.35028
DOI: 10.1007/BF02355849
published in: Journal of Mathematical Sciences (New York)
publication date: 1999-03-15T00:00:00Z
review text: Considered is the Poisson equation \(\Delta u^\varepsilon(x)+ f(x)= 0\) in \(\Omega\subset\mathbb{R}^n\) between hyperplanes \(\Gamma_1\) and \(\Gamma_2\). Boundary conditions are \(u^\varepsilon(x)= 0\) on \(\Gamma^\varepsilon_D\cup \Gamma_2\) and \({\partial u^\varepsilon(x)\over \partial\nu}= 0\) on \(\Gamma^\varepsilon_N= \Gamma_1\setminus \Gamma^\varepsilon_D\) where \(\nu\) is normal to \(\partial\Omega\) and \(u^\varepsilon\), \(f\), \(\Gamma^\varepsilon_D\) are of some periodicities. \(\varepsilon>0\) effects that the boundary conditions are rapidly alternating for small \(\varepsilon\). Here it is constructed an asymptotic expansion for the solution \(u^\varepsilon\) in powers of \(\varepsilon\) with terms of some boundary layers and solutions of certain homogenized problems in \(\Omega\). For the remainder term of this asymptotic expansion for the solution \(u^\varepsilon\) it is given an estimation.
Mathematics Subject Classification ID: 35C20
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 1259436
zbMATH Keywords: Poisson equation
zbMATH Keywords: boundary layers
zbMATH Keywords: homogenized problems
author: G. A. Chechkin
reviewed by: Joachim Jaenicke
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1259436