The Calderón problem for the fractional magnetic operator

Li, Li

doi:10.1088/1361-6420/ab8445

Cited by 20 publications

(20 citation statements)

References 25 publications

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“…The study of the fractional Calderón problem was initiated in [16] where the unknown potential in the fractional Schrödinger equation on a bounded domain in the Euclidean space was determined from exterior measurements. Following this work, inverse problems of recovering lower order terms for fractional elliptic equations have been studied extensively, see for example [15], [14], [35], [34], [1], [4], [5], [6], [7], [8], [9], [30], [31], [33] for some of the important contributions. In all of those works, it is assumed that the leading order coefficients are known.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Fractional anisotropic Calderón problem on closed Riemannian manifolds

Feizmohammadi¹,

Ghosh²,

Krupchyk³

et al. 2021

Preprint

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In this paper we solve the fractional anisotropic Calderón problem on closed Riemannian manifolds of dimensions two and higher. Specifically, we prove that the knowledge of the local source-to-solution map for the fractional Laplacian, given on an arbitrary small open nonempty a priori known subset of a smooth closed connected Riemannian manifold, determines the Riemannian manifold up to an isometry. This can be viewed as a nonlocal analog of the anisotropic Calderón problem in the setting of closed Riemannian manifolds, which is wide open in dimensions three and higher.

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Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Fractional anisotropic Calderón problem on closed Riemannian manifolds

Feizmohammadi¹,

Ghosh²,

Krupchyk³

et al. 2021

Preprint

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“…We are interested in the study of a high order fractional version of the MSE. There have been many studies in this direction (see for instance [54,53,55]). In our work, we will build upon the results from [14] and generalize them to higher order.…”

Section: 3mentioning

confidence: 99%

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Covi¹,

Mönkkönen²,

Railo³

2021

IPI

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We prove a unique continuation property for the fractional Laplacian (−∆) s when s ∈ (−n/2, ∞) \ Z where n ≥ 1. In addition, we study Poincaré-type inequalities for the operator (−∆) s when s ≥ 0. We apply the results to show that one can uniquely recover, up to a gauge, electric and magnetic potentials from the Dirichlet-to-Neumann map associated to the higher order fractional magnetic Schrödinger equation. We also study the higher order fractional Schrödinger equation with singular electric potential. In both cases, we obtain a Runge approximation property for the equation. Furthermore, we prove a uniqueness result for a partial data problem of the d-plane Radon transform in low regularity. Our work extends some recent results in inverse problems for more general operators.

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“…Moreover, the fractional Calderón problem has been solved even for a single measurement in [17]. Other settings include the fractional magnetic Schrödinger equation [9,11,34,35,36], the fractional conductivity equation [10], the fractional heat equation [29,46] and a fractional elasticity equation with constant coefficients [37]. A semilinear fractional Schrödinger equation was studied in [28,34,35,36].…”

Section: Connection To the Earlier Literaturementioning

confidence: 99%

“…Other settings include the fractional magnetic Schrödinger equation [9,11,34,35,36], the fractional conductivity equation [10], the fractional heat equation [29,46] and a fractional elasticity equation with constant coefficients [37]. A semilinear fractional Schrödinger equation was studied in [28,34,35,36]. Many more details about fractional Calderón problems can be found in the survey [48].…”

Section: Connection To the Earlier Literaturementioning

confidence: 99%

Uniqueness for the fractional Calderón problem with quasilocal perturbations

Covi¹

2021

Preprint

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We study the fractional Schrödinger equation with quasilocal perturbations and show that the qualitative unique continuation and Runge approximation properties hold in the assumption of sufficient decay. Quantitative versions of both results are also obtained via a propagation of smallness analysis for the Caffarelli-Silvestre extension. The results are then used to show uniqueness in the inverse problem of retrieving a quasilocal perturbation from DN data under suitable geometric assumptions. Our work generalizes recent results regarding the locally perturbed fractional Calderón problem.

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The Calderón problem for the fractional magnetic operator

Cited by 20 publications

References 25 publications

Fractional anisotropic Calderón problem on closed Riemannian manifolds

Fractional anisotropic Calderón problem on closed Riemannian manifolds

Unique continuation property and Poincaré inequality for higher order fractional Laplacians with applications in inverse problems

Uniqueness for the fractional Calderón problem with quasilocal perturbations

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