On uniqueness in the inverse conductivity problem with local data

Isakov, Victor

doi:10.3934/ipi.2007.1.95

Cited by 130 publications

(193 citation statements)

References 16 publications

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“…On the other hand, to the authors' knowledge, there are no uniqueness results similar to Theorem 1.1 with Dirichlet data supported and Neumann data measured on the same arbitrary open subset of the boundary, even for smooth potentials or conductivities. In dimension n ≥ 3 Isakov [17] proved global uniqueness assuming that Γ 0 is a subset of a plane or a sphere. In dimensions n ≥ 3, [8] proves global uniqueness in determining a bounded potential for the Schrödinger equation with Dirichlet data supported on the whole boundary and Neumann data measured in roughly half the boundary.…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

137

191

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We prove for a two-dimensional bounded domain that the Cauchy data for the Schrödinger equation measured on an arbitrary open subset of the boundary uniquely determines the potential. This implies, for the conductivity equation, that if we measure the current fluxes at the boundary on an arbitrary open subset of the boundary produced by voltage potentials supported in the same subset, we can uniquely determine the conductivity. We use Carleman estimates with degenerate weight functions to construct appropriate complex geometrical optics solutions to prove the results.

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Section: Introductionmentioning

confidence: 99%

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

137

191

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“…By Lemma 5.3 we have the orthogonality relation: 15) where E 1 , E 2 ∈ H(curl , Ω b ) solve the problem (3.1)-(3.3) with q replaced by q 1 and q 2 , respectively. We now look for solutions to the problem (3.1)-(3.3) in the following form:…”

Section: Lemma 52 the Operator F Has A Dense Range Inmentioning

confidence: 99%

“…The uniqueness result for the inverse problem in this paper is most closely related in term of result and method of argument to Kirsch on the determination of the refractive index in the TE polarization. Inspired by [27] and [15], we obtain an orthogonality relation for two different refractive indexes by using a D-to-N map on an artificial boundary on which the tangential electric fields are identical for an integral type of incident electric field. It should be remarked that the method for constructing geometry optical solutions in [19,15,27] for non-periodic inverse conductivity problems does not work since the solutions are required to be quasi-periodic in the periodic case.…”

Section: Introductionmentioning

confidence: 99%

“…Inspired by [27] and [15], we obtain an orthogonality relation for two different refractive indexes by using a D-to-N map on an artificial boundary on which the tangential electric fields are identical for an integral type of incident electric field. It should be remarked that the method for constructing geometry optical solutions in [19,15,27] for non-periodic inverse conductivity problems does not work since the solutions are required to be quasi-periodic in the periodic case. To reconstruct the refractive index, we follow Kirsch's idea [17] (see also [26]) by considering a kind of Sturm-Liouville eigenvalue problems.…”

Section: Introductionmentioning

confidence: 99%

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An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

Yang

Zhang

2011

Applicable Analysis

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This paper is concerned with uniqueness for reconstructing a periodic inhomogeneous medium covered on a perfectly conducting plate. We deal with the problem in the frame of time-harmonic Maxwell systems without T E or T M polarization. An orthogonal relation for two refractive index is obtained, and then inspired by Kirsch's idea, the refractive index can be identified by utilizing the eigenvalues and eigenfunctions to a quasi-periodic SturmLiouville eigenvalue problem.

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“…Isakov [95] proved a uniqueness result in dimension three or higher when the DN map is given on an arbitrary part of the boundary assuming that the remaining part is an open subset of a plane or a sphere and the DN map is measured on the plane or sphere. The case of partial data on a slab was studied in [122].…”

Section: The Uniqueness Proofmentioning

confidence: 99%

Inverse problems: seeing the unseen

2014

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This survey article deals mainly with two inverse problems and the relation between them. The first inverse problem we consider is whether one can determine the electrical conductivity of a medium by making voltage and current measurements at the boundary. This is called electrical impedance tomography and also Calderón's problem since the famous analyst proposed it in the mathematical literature (Calderón in On an inverse boundary value problem. Seminar on numerical analysis and its applications to continuum physics (Rio de Janeiro, 1980), Soc Brasil Mat. Rio de Janeiro, pp. [65][66][67][68][69][70][71][72][73] 1980). The second is on travel time tomography. The question is whether one can determine the anisotropic index of refraction of a medium by measuring the travel times of waves going through the medium. This can be recast as a geometry problem, the boundary rigidity problem. Can we determine a Riemannian metric of a compact Riemannian manifold with boundary by measuring the distance function between boundary points? These two inverse problems concern visibility, that is whether we can determine the internal properties of a medium by making measurements at the boundary. The last topic of this paper considers the opposite issue: invisibility: Can one make objects invisible to different types of waves, including light?Communicated by Ari Laptev.

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On uniqueness in the inverse conductivity problem with local data

Cited by 130 publications

References 16 publications

The Calderón problem with partial data in two dimensions

The Calderón problem with partial data in two dimensions

An inverse electromagnetic scattering problem for a bi-periodic inhomogeneous layer on a perfectly conducting plate

Inverse problems: seeing the unseen

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