The Linearized Calderón Problem in Transversally Anisotropic Geometries

Ferreira, David Dos Santos; Kurylev, Yaroslav; Lassas, Matti; Liimatainen, Tony; Salo, Mikko

doi:10.1093/imrn/rny234

Cited by 11 publications

(16 citation statements)

References 22 publications

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“…This uniformity of constants is essential when proving stability estimates. We mention here a similar consideration in the Riemannian setting [23,Section 4.1].…”

Section: Gaussian Beamsmentioning

confidence: 99%

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Lassas

Uhlmann

Wang

2018

Commun. Math. Phys.

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116

140

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We consider inverse problems in space-time (M, g), a 4-dimensional Lorentzian manifold. For semilinear wave equations g u + H(x, u) = f , where g denotes the usual Laplace-Beltrami operator, we prove that the source-to-solution map L : f → u|V , where V is a neighborhood of a time-like geodesic µ, determines the topological, differentiable structure and the conformal class of the metric of the space-time in the maximal set where waves can propagate from µ and return back. Moreover, on a given space-time (M, g), the source-to-solution map determines some coefficients of the Taylor expansion of H in u .

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“…This uniformity of constants is essential when proving stability estimates. We mention here a similar consideration in the Riemannian setting [23,Section 4.1].…”

Section: Gaussian Beamsmentioning

confidence: 99%

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Lassas

Uhlmann

Wang

2018

Commun. Math. Phys.

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116

140

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“…In this article we extend the result of [11] and show that if the transversal manifold is additionally real-analytic, Question 1 has a positive answer (i.e. one can recover f ∈ L ∞ (M ) completely, not just some of its singularities).…”

mentioning

confidence: 55%

“…In [17], it is proved that Question 1 has a positive answer when (M, g) is a complex Kähler manifold with sufficiently many holomorphic functions. The article [11] establishes a recovery of singularities result: if (M, g) is transversally anisotropic and the transversal manifold satisfies a certain geometric condition, one can recover transversal singularities of f . In a related work [26], it is proved that on a general transversally anisotropic manifold products of sets of four (instead of pairs) of harmonic functions form a complete set in L 1 (M ).…”

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confidence: 99%

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Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

Krupchyk¹,

Liimatainen²,

Salo³

2020

Preprint

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In this article we study the linearized anisotropic Calderón problem on a compact Riemannian manifold with boundary. This problem amounts to showing that products of pairs of harmonic functions of the manifold form a complete set. We assume that the manifold is transversally anisotropic and that the transversal manifold is real analytic and satisfies a geometric condition related to the geometry of pairs of intersecting geodesics. In this case, we solve the linearized anisotropic Calderón problem. The geometric condition does not involve the injectivity of the geodesic X-ray transform. Crucial ingredients in the proof of our result are the construction of Gaussian beam quasimodes on the transversal manifold, with exponentially small errors, as well as the FBI transform characterization of the analytic wave front set.

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“…In this case, a global WKB approach no longer seems possible, and the complex geometric optics solutions are obtained via a Gaussian beams quasimode construction. We refer to [10], [23] for the study of the linearized anisotropic Calderón problem on transversally anisotropic manifolds.…”

Section: Introduction and Statement Of Resultsmentioning

confidence: 99%

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

Feizmohammadi,

Krupchyk,

Oksanen

et al. 2020

Preprint

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We show that a continuous potential q can be constructively determined from the knowledge of the Dirichlet-to-Neumann map for the Schrödinger operator −∆ g +q on a conformally transversally anisotropic manifold of dimension ≥ 3, provided that the geodesic ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of the uniqueness result of [11]. A crucial role in our reconstruction procedure is played by a constructive determination of the boundary traces of suitable complex geometric optics solutions based on Gaussian beams quasimodes concentrated along non-tangential geodesics on the transversal manifold, which enjoy uniqueness properties. This is achieved by applying the simplified version of the approach of [33] to our setting. We also identify the main space introduced in [33] with a standard Sobolev space on the boundary of the manifold. Another ingredient in the proof of our result is a reconstruction formula for the boundary trace of a continuous potential from the knowledge of the Dirichlet-to-Neumann map.

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The Linearized Calderón Problem in Transversally Anisotropic Geometries

Cited by 11 publications

References 22 publications

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

Linearized Calderón problem and exponentially accurate quasimodes for analytic manifolds

Reconstruction in the Calderón problem on conformally transversally anisotropic manifolds

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