The inverse conductivity problem in two dimensions
type: Item
instance of: scholarly article
title: The inverse conductivity problem in two dimensions
zbMATH Open document ID: 0716.35080
DOI: 10.1016/0022-0396(90)90002-7
author: Ziqi Sun
published in: Journal of Differential Equations
publication date: 1990-01-01T00:00:00Z
review text: The author considers the problem of recovering the coefficient \(\gamma\) entering the Dirichlet problem \[ div(\gamma \nabla u)=0\text{ in } \Omega,\quad u=g\text{ on } \partial \Omega \] from the Dirichlet-to- Neumann map \(g\mapsto \partial u/\partial N\) on \(\partial \Omega\). Here \(\Omega\) is a bounded domain in \({\mathbb{R}}^ 2\) with smooth boundary and \(\gamma \in L_{\infty}(\Omega)\), \(\gamma >0\). When \(\Delta \gamma \in L_{\infty}(\Omega)\) the global uniqueness result was obtained by \textit{J. Sylvester} and \textit{G. Uhlmann} [Ann. Math., II. Ser. 125, 153-169 (1987; Zbl 0625.35078)]; see also their review paper ``The Dirichlet-to- Neumann map and applications'' [Inverse problems in partial differential equations, Proc. Conf., Pt. Summer Res. Conf. Math. Sci., Arcata/CA (US) 189, 101-139 (1990)]. In the plane case uniqueness is known only if \(\gamma\) is close to a constant. In this paper by using an expansion of almost exponential solutions to the equation and some orthogonality relations the author proves global uniqueness and stability results when \(\Delta\) log \(\gamma\) \(=0\) or \(\Delta \gamma^{\beta}=0\) for some \(\beta\).
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 35J05
zbMATH DE Number: 4179970
zbMATH Keywords: conductivity
zbMATH Keywords: Dirichlet-to-Neumann map
zbMATH Keywords: global uniqueness
zbMATH Keywords: stability
reviewed by: Victor Isakov
MaRDI profile type: Publication
full work available at URL: 0022-0396(90)90002-7
OpenAlex ID: W2062546791
cites work: Stable determination of conductivity by boundary measurements
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description: scientific article; zbMATH DE number 4179970