Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity
type: Item
instance of: scholarly article
title: Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity
zbMATH Open document ID: 1368.35095
author: Charles Dapogny
author: Michael S. Vogelius
published in: Chinese Annals of Mathematics. Series B
publication date: 2017-04-21T00:00:00Z
review text: Denote by \(\Omega\) a bounded domain in \(\mathbb R^2\), with smooth boundary and assume that it is formed by a conductive material, and it has a thin inhomogeneity of width depending on \(\epsilon\), \({\omega}_{\epsilon}\), with conductivity \(a_{\epsilon}\). The conductivity in the entire domain will be denoted by \({\gamma}_{\epsilon}\). One defines the electric potential \(u_{\epsilon}\) as the solution of the Dirichlet problem for the second order elliptic equation \[ -div(\gamma_\epsilon\nabla u_\epsilon)=f\quad\text{in}\quad\Omega,\quad u_{\epsilon} = \varphi\quad\text{on}\quad {\partial}{\Omega}. \] The goal of the paper is to investigate the uniform asymptotic behavior of the potential \(u_{\epsilon}\) as the thin inhomogeneity converges to zero. The authors intend to find an approximate solution uniformly close to \(u_{\epsilon}\) which is independent of the behavior of \(a_{\epsilon}\). After presenting preliminaries, the statement of the problem and notations, they introduce the main tool that will be used for studying the convergence of minimizers in variational problems. In the next section the authors construct a \(0\)-th order approximation to \(u_{\epsilon}\). Further, there an extensive study of this approximation is presented. Investigated are uniform energy and regularity estimates, asymptotic exactness and the limit behavior. Using the same strategy, the derivation of the uniform first-order approximation to \(u_{\epsilon}\) is discussed. An appendix is devoted to the proof of uniform regularity estimates for the 0-th order approximation.
reviewed by: Claudia Simionescu-Badea
Mathematics Subject Classification ID: 35J25
Mathematics Subject Classification ID: 74G10
Mathematics Subject Classification ID: 74G75
zbMATH DE Number: 6707257
zbMATH Keywords: second order elliptic equation
zbMATH Keywords: Dirichlet problem
zbMATH Keywords: conductivity problems
zbMATH Keywords: thin inhomogeneities
zbMATH Keywords: uniform asymptotic expansions
MaRDI profile type: Publication
full work available at URL: s11401-016-1072-3
OpenAlex ID: W2568760610
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DOI: 10.1007/S11401-016-1072-3
description: scientific article; zbMATH DE number 6707257