Reconstruction of small inhomogeneities from boundary measurements
type: Item
instance of: scholarly article
title: Reconstruction of small inhomogeneities from boundary measurements
zbMATH Open document ID: 1113.35148
author: Hyeonbae Kang
author: Habib Ammari
published in: Lecture Notes in Mathematics
publication date: 2004-11-25T00:00:00Z
review text: The book, according to the authors, describes a ``fresh and promising techiques for the reconstruction of small inclusions from boundary measurements'' and the presentation is ``intended to be self-contained''. The book consists of three parts.The first part deals with conductivity inclusions, the second part deals with the elastic ones, and the third part deals with electromagnetic inclusions. Besides the smallness of the inclusions, the authors require the inclusions to be weak. The weakness of the inclusions means that \(|a-1|\) is sufficiently small, where \(a\) is defined below. In the first part the governing equation is the static equation \(\Delta u + \text{div} (a\operatorname{grad} u)=0 \) in \(\Omega\), \(u_N|_S=g\), where \(a= \text{const}>0, a\neq 1,\) in \(z+\varepsilon D\subset \Omega\) and \(a=0\) in the complement of the region \(z+\varepsilon D\), \(\varepsilon>0\) is a small parameter, \(D\) is a ball (in many cases discussed in the book) or, more generally, \(D\) is a Lipschitz star-shaped domain, \(S\) is the boundary of \(\Omega\), \(N\) is the unit outer normal to \(S\), and \(\int_Sgds=0\). The solution \(u\) has to satisfy the requirement \(\int_Suds=0\), and the transmission boundary conditions on \(\partial D\). The method allows one to treat the Dirichlet condition also. Given \(g\) and \(a\), one finds uniquely this solution \(u\) and, therefore, \(f:=u|_S\). The problems in parts 2 and 3 are modelled by the problems for scalar equations with scalar boundary conditions. The authors derive asymptotics of \(u\) as \(\varepsilon \to 0\). The asymptotics obtained depend on the parameter \(\varepsilon\), which is not dimensionless, while in electromagnetic problems the usual asymptotics are obtained with respect to a dimensionless parameter \(k\varepsilon\gg1\), where \(k\) is the wavenumber and \(\varepsilon\) is the charactersistic dimension of a small body. Given \(g\) and \(f:=u|_S\), the authors want to recover the location of \(z\), the…
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35-02
Mathematics Subject Classification ID: 31B20
Mathematics Subject Classification ID: 74G75
Mathematics Subject Classification ID: 78A70
Mathematics Subject Classification ID: 92C55
zbMATH DE Number: 2119069
zbMATH Keywords: small inhomogeneities
zbMATH Keywords: impedance tomography
reviewed by: A. G. Ramm
MaRDI profile type: Publication
full work available at URL: b98245
OpenAlex ID: W1543038508
DOI: 10.1007/B98245
description: scientific article; zbMATH DE number 2119069