Limiting Carleman weights and anisotropic inverse problems
type: Item
instance of: scholarly article
title: Limiting Carleman weights and anisotropic inverse problems
zbMATH Open document ID: 1181.35327
author: David Dos Santos Ferreira
author: C. E. Kenig
author: Mikko Salo
author: Gunther Uhlmann
published in: Inventiones Mathematicae
publication date: 2009-09-09T00:00:00Z
full work available at URL: https://arxiv.org/abs/0803.3508
review text: The anisotropic Calderon problem and related inverse problems are considered in the article. This inverse problem is connected with so called Electrical Impedance Tomography (EIT), which consists in determining the conductivity of a medium by making voltage and current measurments at the boundary. Applications range from geophysical prospection to medical imaging. The approach is based on limiting Carleman weights [see \textit{C. E. Kenig, J. Sjöstrand} and \textit{G. Uhlmann}, Ann. Math. (2) 165, No. 2, 567--591 (2007; Zbl 1127.35079)]. The authors characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via attenuated geodesic ray transform. The inverse problems considered are the recovery of an electric potential and a magnetic field from boundary measurements on an admissible Riemannian manifold, and determination of an admissible metric within a conformal class from the Dirichlet-to-Neumann map (DN map). Earlier results in dimension \(n \geq 3\) were restricted to real-analytic metrics.
reviewed by: Elena V. Tabarintseva
Mathematics Subject Classification ID: 35R30
Mathematics Subject Classification ID: 78A46
Mathematics Subject Classification ID: 78A05
Mathematics Subject Classification ID: 58J60
zbMATH DE Number: 5602517
zbMATH Keywords: ill-posed problem
zbMATH Keywords: backward heat equation
zbMATH Keywords: wavelets
zbMATH Keywords: quasireversibility
MaRDI profile type: Publication
OpenAlex ID: W3098329251
arXiv ID: 0803.3508
cites work: Some methods for the study of multidimensional inverse problems for differential equations
cites work: Calderón's inverse conductivity problem in the plane
cites work: Calderóns' Inverse Problem for Anisotropic Conductivity in the Plane
cites work: Q3200517
cites work: Uniqueness in the inverse conductivity problem for nonsmooth conductivities in two dimensions
cites work: Recovering a potential from Cauchy data in the two-dimensional case
cites work: Q4264393
cites work: Determining a magnetic Schrödinger operator from partial Cauchy data
cites work: Q5801822
cites work: Inverse problems for Einstein manifolds
cites work: Q3752751
cites work: Inverse spectral problems on hyperbolic manifolds and their applications to inverse boundary value problems in Euclidean space
cites work: Riemannian geometry and geometric analysis
cites work: The Calderón problem with partial data
cites work: Determining nonsmooth first order terms from partial boundary measurements
cites work: Q3692180
cites work: On determining a Riemannian manifold from the Dirichlet-to-Neumann map
cites work: The Dirichlet-to-Neumann map for complete Riemannian manifolds with boundary
cites work: Contróle Exact De Léquation De La Chaleur
cites work: Stabilization of the wave equation by the boundary
cites work: Determining anisotropic real-analytic conductivities by boundary measurements
cites work: Conformal uniqueness results in anisotropic electrical impedance imaging
cites work: Q4150304
cites work: Global uniqueness for a two-dimensional inverse boundary value problem
cites work: Global identifiability for an inverse problem for the Schrödinger equation in a magnetic field
cites work: An optimal Poincaré inequality for convex domains
cites work: Q4371081
cites work: Carleman estimates and inverse problems for Dirac operators
cites work: Integral geometry for tensor fields. Transl. from the Russian
cites work: On Emission Tomography of Inhomogeneous Media
cites work: Generic uniqueness for an inverse boundary value problem
cites work: Anisotropic inverse problems in two dimensions
cites work: An anisotropic inverse boundary value problem
cites work: A global uniqueness theorem for an inverse boundary value problem
cites work: Inverse boundary value problems at the boundary—continuous dependence
cites work: Q3900552
DOI: 10.1007/S00222-009-0196-4
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description: scientific article