0000000000497026

AUTHOR

Yaroslav Kurylev

showing 2 related works from this author

The Calderon problem in transversally anisotropic geometries

2016

We consider the anisotropic Calderon problem of recovering a conductivity matrix or a Riemannian metric from electrical boundary measurements in three and higher dimensions. In the earlier work \cite{DKSaU}, it was shown that a metric in a fixed conformal class is uniquely determined by boundary measurements under two conditions: (1) the metric is conformally transversally anisotropic (CTA), and (2) the transversal manifold is simple. In this paper we will consider geometries satisfying (1) but not (2). The first main result states that the boundary measurements uniquely determine a mixed Fourier transform / attenuated geodesic ray transform (or integral against a more general semiclassical…

Mathematics - Differential GeometryGeodesicGeneral MathematicsBoundary (topology)Conformal map01 natural sciencessymbols.namesakeMathematics - Analysis of PDEsFOS: Mathematics[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]0101 mathematicsMathematicsCalderón problemRiemannian manifoldApplied Mathematicsta111010102 general mathematicsMathematical analysiscomplex geometrical optics solutionInverse problemRiemannian manifold010101 applied mathematicsboundary control methodFourier transformDifferential Geometry (math.DG)Transversal (combinatorics)Metric (mathematics)symbolsinverse boundary value problemAnalysis of PDEs (math.AP)
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The Linearized Calderón Problem in Transversally Anisotropic Geometries

2017

In this article we study the linearized anisotropic Calderon problem. In a compact manifold with boundary, this problem amounts to showing that products of harmonic functions form a complete set. Assuming that the manifold is transversally anisotropic, we show that the boundary measurements determine an FBI type transform at certain points in the transversal manifold. This leads to recovery of transversal singularities in the linearized problem. The method requires a geometric condition on the transversal manifold related to pairs of intersecting geodesics, but it does not involve the geodesic X-ray transform which has limited earlier results on this problem.

Mathematics - Differential GeometryGeodesicGeneral MathematicsNEUMANN MAPBoundary (topology)Type (model theory)01 natural scienceslaw.inventionMathematics - Analysis of PDEslinearized anisotropic Calderón problemlaw35R30 35J25111 MathematicsFOS: Mathematics0101 mathematicsMathematics010102 general mathematicsMathematical analysisInverse problem010101 applied mathematicsHarmonic functionDifferential Geometry (math.DG)Transversal (combinatorics)Gravitational singularityMathematics::Differential GeometryINVERSE PROBLEMManifold (fluid mechanics)Analysis of PDEs (math.AP)
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