Calderón’s inverse conductivity problem in the plane

Astala, Kari; Päivärinta, Lassi

doi:10.4007/annals.2006.163.265

Cited by 499 publications

(818 citation statements)

References 30 publications

(33 reference statements)

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“…Later the regularity assumptions were relaxed in [6] and [2]. In particular, the paper [2] proves uniqueness for L ∞ -conductivities.…”

Section: Introductionmentioning

confidence: 99%

“…Later the regularity assumptions were relaxed in [6] and [2]. In particular, the paper [2] proves uniqueness for L ∞ -conductivities. In two dimensions a recent breakthrough result of Bukhgeim [7] gives unique identifiability of the potential from Cauchy data measured on the whole boundary for the associated Schrödinger equation.…”

Section: Introductionmentioning

confidence: 99%

“…The question of whether one can determine the conductivity up to the obstruction (1.6) in the case of full Cauchy data has been solved in two dimensions for C 2 conductivities in [24], Lipschitz conductivities in [26], and merely L ∞ conductivities in [3]. The method of proof in all these papers is the reduction to the isotropic case performed using isothermal coordinates [27].…”

Section: Introductionmentioning

confidence: 99%

“…We mention that in [3] K. Astala, M. Lassas, and L. Päivärinta have shown a partial data result in the anisotropic problem in two dimensions for bounded measurable conductivities, similar to Theorem 1.2, assuming that one knows both the Dirichlet-to-Neumann and Neumann-to-Dirichlet map on Γ. 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.…”

Section: Introductionmentioning

confidence: 99%

See 3 more Smart Citations

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

139

193

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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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“…Later the regularity assumptions were relaxed in [6] and [2]. In particular, the paper [2] proves uniqueness for L ∞ -conductivities.…”

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

139

193

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“…The relation between (26) and (20), (21) in the case of a real valued function f 2 was observed in [2] and resulted to be essential for solving the Calderón problem in the plane.…”

Section: Remark 8 Observe That the Pair Of Functionsmentioning

confidence: 99%

On a factorization of second-order elliptic operators and applications

Kravchenko¹

2006

J. Phys. A: Math. Gen.

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We show that given a nonvanishing particular solution of the equation (div p grad +q)u = 0,the corresponding differential operator can be factorized into a product of two first order operators. The factorization allows us to reduce the equation (1) to a first order equation which in a two-dimensional case is the Vekua equation of a special form. Under quite general conditions on the coefficients p and q we obtain an algorithm which allows us to construct in explicit form the positive formal powers (solutions of the Vekua equation generalizing the usual powers (z − z0) n , n = 0, 1, . . .). This result means that under quite general conditions one can construct an infinite system of exact solutions of (1) explicitly, and moreover, at least when p and q are real valued this system will be complete in ker(div p grad +q) in the sense that any solution of (1) in a simply connected domain Ω can be represented as an infinite series of obtained exact solutions which converges uniformly on any compact subset of Ω. Finally we give a similar factorization of the operator (div p grad +q) in a multidimensional case and obtain a natural generalization of the Vekua equation which is related to second order operators in a similar way as its two-dimensional prototype does.

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Electrical Impedance Tomography

Seo

Woo

2012

Nonlinear Inverse Problems in Imaging

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We i n troduce an output least-squares method for impedance tomography problems that have regions of high conductivity surrounded by regions of lower conductivity. The high conductivity is modeled on network approximation results from an asymptotic analysis and its recovery is based on this model. The smoothly varying part of the conductivity is recovered by a linearization process as is usual. We present the results of several numerical experiments that illustrate the performance of the method.

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Calderón’s inverse conductivity problem in the plane

Cited by 499 publications

References 30 publications

The Calderón problem with partial data in two dimensions

The Calderón problem with partial data in two dimensions

On a factorization of second-order elliptic operators and applications

Electrical Impedance Tomography

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