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Journal of Geometric Analysis

, Volume 23, Issue 4, pp 2063–2080 | Cite as

The Complete Dirichlet-to-Neumann Map for Differential Forms

  • Vladimir Sharafutdinov
  • Clayton Shonkwiler
Article

Abstract

The Dirichlet-to-Neumann map for differential forms on a Riemannian manifold with boundary is a generalization of the classical Dirichlet-to-Neumann map which arises in the problem of Electrical Impedance Tomography. We synthesize the two different approaches to defining this operator by giving an invariant definition of the complete Dirichlet-to-Neumann map for differential forms in terms of two linear operators Φ and Ψ. The pair (Φ,Ψ) is equivalent to Joshi and Lionheart’s operator Π and determines Belishev and Sharafutdinov’s operator Λ. We show that the Betti numbers of the manifold are determined by Φ and that Ψ determines a chain complex whose homologies are explicitly related to the cohomology groups of the manifold.

Keywords

Hodge theory Inverse problems Dirichlet-to-Neumann map 

Mathematics Subject Classification (2000)

58A14 58J32 57R19 

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Copyright information

© Mathematica Josephina, Inc. 2012

Authors and Affiliations

  1. 1.Sobolev Institute of MathematicsNovosibirskRussia
  2. 2.Department of MathematicsUniversity of GeorgiaAthensUSA

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