Inverse boundary value problems for the perturbed polyharmonic operator

Krupchyk, Katsiaryna; Lassas, Matti; Uhlmann, Günther

doi:10.1090/s0002-9947-2013-05713-3

Cited by 68 publications

(92 citation statements)

References 41 publications

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“…The proofs in [15] and [14] rely upon L 2 methods only. Inverse spectral problems for a potential perturbation of the polyharmonic operator were studied in [20], and inverse boundary value problems for a first order perturbation of the polyharmonic operator were addressed in [18,19], again using L 2 techniques.…”

Section: Introductionmentioning

confidence: 99%

Inverse boundary problems for polyharmonic operators with unbounded potentials

Krupchyk¹,

Uhlmann²

2016

J. Spectr. Theory

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Abstract. We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded open set in R n for the perturbed polyharmonic operator (−∆) m + q with q ∈ L n 2m , n > 2m, determines the potential q in the set uniquely. In the course of the proof, we construct a special Green function for the polyharmonic operator and establish its mapping properties in suitable weighted L 2 and L p spaces. The L p estimates for the special Green function are derived from L p Carleman estimates with linear weights for the polyharmonic operator.

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Section: Introductionmentioning

confidence: 99%

Inverse boundary problems for polyharmonic operators with unbounded potentials

Krupchyk¹,

Uhlmann²

2016

J. Spectr. Theory

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“…We use the following result from [11,12]. (1) such that for all 0 < h ≤ h 0 1 and u ∈ C ∞ c (Ω), we have the following interior estimate:…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…We are interested in the inverse problem of determining q from N q . The uniqueness question of determination of q from N q was answered in [9,10] and recently in [11,12,17] where they showed that unique determination of both zeroth-and first-order perturbations of the birharmonic operator is possible from boundary Neumann data. We note that the papers [11,17] also show unique determination of the first order perturbation terms from Neumann data measured on possibly small subsets of the boundary.…”

Section: Introductionmentioning

confidence: 99%

Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials

Choudhury

Krishnan

2015

Journal of Mathematical Analysis and Applications

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Please cite this article in press as: A.P. Choudhury, V.P. Krishnan, Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl. (2015), http://dx. AbstractIn this article, stability estimates are given for the determination of the zeroth-order bounded perturbations of the biharmonic operator when the boundary Neumann measurements are made on the whole boundary and on slightly more than half the boundary, respectively. For the case of measurements on the whole boundary, the stability estimates are of ln-type and for the case of measurements on slightly more than half of the boundary, we derive estimates that are of ln ln-type.

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“…The DN map with partial data for the magnetic Schrödinger operator was studied in [51,109,117,198]. The case of the polyharmonic operator was considered in [118]. The case of Helmholtz equation [139].…”

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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Inverse boundary value problems for the perturbed polyharmonic operator

Cited by 68 publications

References 41 publications

Inverse boundary problems for polyharmonic operators with unbounded potentials

Inverse boundary problems for polyharmonic operators with unbounded potentials

Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials

Inverse problems: seeing the unseen

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