Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

Alessandrini, Giovanni

doi:10.1016/0022-0396(90)90078-4

Cited by 155 publications

(179 citation statements)

References 14 publications

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“…Salo [10] proves the coefficient W is uniquely determined by the Dirichlet to Neumann map in dimensions three and higher. Salo's argument at the boundary uses the singular solutions of Alessandrini [1]. It would be interesting to adapt Alessandrini's techniques to give a reconstruction of W at the boundary, rather than just showing uniqueness.…”

Section: Corollary 14mentioning

confidence: 99%

Identifiability at the boundary for first-order terms

Brown

Salo

2006

Applicable Analysis

716

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Abstract.Let Ω be a domain in R n whose boundary is C 1 if n ≥ 3 or C 1,β if n = 2. We consider a magnetic Schrödinger operator L W,q in Ω and show how to recover the boundary values of the tangential component of the vector potential W from the Dirichlet to Neumann map for L W,q . We also consider a steady state heat equation with convection term ∆+2W ·∇ and recover the boundary values of the convection term W from the Dirichlet to Neumann map. Our method is constructive and gives a stability result at the boundary.

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Section: Corollary 14mentioning

confidence: 99%

Identifiability at the boundary for first-order terms

Brown

Salo

2006

Applicable Analysis

716

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“…We note that this gives also a reconstruction procedure. We first can reconstruct γ at the boundary since γ ∂ |ξ | is the principal symbol of γ [see (5)]. In other words in coordinates (x , x n ) so that ∂ is locally given by x n = 0 we have…”

Section: Boundary Determinationmentioning

confidence: 99%

“…In a similar fashion, using (5), one can find ∂γ ∂ν ∂ by computing the principal symbol of ( γ − γ ∂ 1 ) where 1 denotes the Dirichlet to Neumann map associated to the conductivity 1. The other terms can be reconstructed recursively in a similar fashion.…”

Section: Boundary Determinationmentioning

confidence: 99%

“…In the case that γ ∈ C 1 ( ) we can determine, knowing the DN map, γ and its normal derivative at the boundary using the estimate (7) above and an approximation argument. For other results and approaches to boundary determination of the conductivity see [5,31,135,141]. In one way or another the boundary determination involves testing the DN map against highly oscillatory functions at the boundary.…”

Section: Theorem 12 Suppose That γmentioning

confidence: 99%

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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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“…In the case that γ ∈ C 1 (Ω) we can determine, knowing the DN map, γ and its normal derivative at the boundary using the estimate (1.7) above and an approximation argument. For other results and approaches to boundary determination of the conductivity see [5], [32], [136], [142]. In one way or another the boundary determination involves testing the DN map against highly oscillatory functions at the boundary.…”

Section: Boundary Determinationmentioning

confidence: 99%

Inverse Problems: Visibility and Invisibility

Uhlmann¹

2013

Journées Équations Aux Dérivées Partielles

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Singular solutions of elliptic equations and the determination of conductivity by boundary measurements

Cited by 155 publications

References 14 publications

Identifiability at the boundary for first-order terms

Identifiability at the boundary for first-order terms

Inverse problems: seeing the unseen

Inverse Problems: Visibility and Invisibility

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