Dirichlet to Neumann operator on differential forms

Belishev, M. I.; Sharafutdinov, V. A.

doi:10.1016/j.bulsci.2006.11.003

Cited by 48 publications

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“…The Euler characteristic in turn determines the topology of X in the latter case. The article [BSh08] generalises this result to higher dimensions. More precisely, the authors define a Dirichlet to Neumann operator on the space of differential forms of all degrees and express the Betti numbers of X in terms of this operator.…”

Section: Introductionmentioning

confidence: 89%

“…The Dirichlet to Neumann operator of [BSh08] maps differential forms of degree i on ∂X to those of degree n−i−1, i.e., it does not preserve the natural graduation of the space of differential forms. Moreover, as substitution for the Dirichlet problem, the boundary value problem for harmonic forms u on X is chosen, with prescribed data t(u) = u 0 and t(d * u) = 0 on ∂X , where t(d * u) is the tangential component of d * u on the boundary.…”

Section: Introductionmentioning

confidence: 99%

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The Dirichlet to Neumann operator for elliptic complexes

Тарханов

2011

Trans. Amer. Math. Soc.

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Section: Introductionmentioning

confidence: 89%

Section: Introductionmentioning

confidence: 99%

The Dirichlet to Neumann operator for elliptic complexes

Тарханов

2011

Trans. Amer. Math. Soc.

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“…Now ( Ω, g ) and the (unknown) original (Ω, g) are two Riemannian manifolds with common boundary Γ, connected by the (unknown) diffeomorphism j := s −1 t * ; see (1.5). 7 These are metrics in which the real and imaginary parts of functions from A ( Ω) are harmonic. 6 Functions from A ( Ω) play the role of local complex coordinates.…”

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confidence: 99%

“…Without carrying out the full recovery scheme 1-4, we can use Λ to define some topological invariants of Ω: the Betti numbers and others; see [2], [7].…”

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confidence: 99%

Noncommutative geometry and the tomography of manifolds

Belishev

Demchenko

Popov

2014

Trans. Moscow Math. Soc.

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The tomography of manifolds describes a range of inverse problems in which we seek to reconstruct a Riemannian manifold from its boundary data (the "Dirichlet-Neumann" mapping, the reaction operator, and others). Different types of data correspond to physically different situations: the manifold is probed by electric currents or by acoustic or electromagnetic waves. In our paper we suggest a unified approach to these problems, using the ideas of noncommutative geometry. Within the framework of this approach, the underlying manifold for the reconstruction is obtained as the spectrum of an adequate Banach algebra determined by the boundary data.2010 Mathematics Subject Classification. Primary 35R30, 46L60, 58B34, 93B28, 35Q61. Key words and phrases. Restoration of a Riemannian manifold from boundary data, impedance, acoustic and electromagnetic tomography, connections of tomography problems with noncommutative geometry.License or copyright restrictions may apply to redistribution; see http://www.ams.org/journal-terms-of-use 134 M. I. BELISHEV, M. N. DEMCHENKO, AND A. N. POPOVmeasurements are formalised by the assignment of a reaction operator which transforms displacements of the boundary (functions of the boundary points and time) into forces on the boundary (functions of the same variables). The inverse problem consists of reconstructing the shell by means of the reaction operator. This situation also makes sense in the multidimensional case and has significant applications in three dimensions (defectoscopy, geophysics, ultrasound diagnostics in medicine and others).Electromagnetic tomography (EMT). A region in curved space (a three-dimensional Riemannian manifold with boundary) is illuminated by electromagnetic waves initiated by boundary sources. The waves interact with the internal structure and carry information on it to the boundary. The observer seeks to recover the structure: the topology and the metric.The principal difference between these similar situations and the traditional coefficient inverse problem consists of the following. In the latter, the support of the required coefficients (the domain space or the manifold; in IT, the shell) is assumed to be known: we have the ability to fix a point of the support and consider how to recover the unknown function at it (the conductivity of the shell, the density of the medium, the speed of the wave, etc.). In our settings, the support itself is subject to definition. An additional complication is that the correspondence "manifold → reaction operator" is not injective: two isometric manifolds with common boundary have identical reaction operators. For the observer, such manifolds respond identically to external stimuli; in principle, they are indistinguishable and it is unclear which of them to recover.

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“…The classical Dirichlet-to-Neumann map was generalized to an operator on differential forms independently by Joshi and Lionheart [JL05] and Belishev and Sharafutdinov [BS08]. Joshi and Lionheart called their operator Π and showed that the data (∂M, Π) determines the C ∞ -jet of the Riemannian metric at the boundary.…”

Section: Introductionmentioning

confidence: 99%

The Complete Dirichlet-to-Neumann Map for Differential Forms

Sharafutdinov

Shonkwiler²

2012

J Geom Anal

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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.

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Dirichlet to Neumann operator on differential forms

Cited by 48 publications

References 7 publications

The Dirichlet to Neumann operator for elliptic complexes

The Dirichlet to Neumann operator for elliptic complexes

Noncommutative geometry and the tomography of manifolds

The Complete Dirichlet-to-Neumann Map for Differential Forms

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