Complex geometrical optics solutions for Lipschitz conductivities

Päivärinta, Lassi; Panchenko, Alexander; Uhlmann, Günther

doi:10.4171/rmi/338

Cited by 108 publications

(104 citation statements)

References 13 publications

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“…In dimensions n ≥ 3, the first global uniqueness result for C 2 -conductivities was proven in [28]. In [25], [5] the global uniqueness result was extended to less regular conductivities. Also see [14] for the determination of more singular conormal conductivities.…”

Section: Introductionmentioning

confidence: 99%

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

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137

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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Section: Introductionmentioning

confidence: 99%

The Calderón problem with partial data in two dimensions

Imanuvilov

Uhlmann

Yamamoto

2010

J. Amer. Math. Soc.

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137

193

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“…Currently, the best general global uniqueness result known for n ≥ 3 is for γ ∈ C 3 2 , proved in [28], building on [2] and using the general argument of [32], while the best known result for n = 2 is γ ∈ W 1,p (Ω), p > 2, proved in [3] using the ∂ technique of [1,25,26]. Global uniqueness for piecewise-analytic conductivities was proven [20], and special types of jump discontinuities were treated in [17].…”

Section: Theoremmentioning

confidence: 99%

The Calderón problem for conormal potentials I: Global uniqueness and reconstruction

Greenleaf

Lassas

Uhlmann

2003

Comm Pure Appl Math

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“…Theorem 1.5 has been extended to conductivities having 3/2 derivatives in some sense in [32,152]. Uniqueness for conormal conductivies in C 1+ was shown in [63].…”

Section: Uniquenessmentioning

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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Complex geometrical optics solutions for Lipschitz conductivities

Abstract: We prove the existence of complex geometrical optics solutions for Lipschitz conductivities. Moreover we show that, in dimensions n ≥ 3 that one can uniquely recover a W 3/2,∞ conductivity from its associated Dirichlet-to-Neumann map or voltage to current map.

Cited by 108 publications

References 13 publications

The Calderón problem with partial data in two dimensions

The Calderón problem with partial data in two dimensions

The Calderón problem for conormal potentials I: Global uniqueness and reconstruction

Inverse problems: seeing the unseen

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