The Calderón problem in transversally anisotropic geometries

By David dos Santos Ferreira, Yaroslav Kurylev, Matti Lassas, Mikko Salo. 2013

133 citations recorded by the source corpus.

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 limit measure) of an unknown coefficient. In particular, one obtains uniqueness results whenever the geodesic ray transform on the transversal manifold is injective. The second result shows that the boundary measurements in an infinite cylinder uniquely determine the transversal metric. The first result is proved by using complex geometrical optics solutions involving Gaussian beam quasimodes, and the second result follows from a connection between the Calderon problem and Gel'fand's inverse problem for the wave equation and the boundary control method.

Subfields: Computational Theory and Mathematics, Mathematical Physics, Mechanics of Materials

View Paper: https://www.semanticscholar.org/paper/be3e4cafaf86f2bb2462832b9d4c28989b3164e9

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In Computational Theory and Mathematics

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In Mechanics of Materials

Citations in the corpus

An inverse problem for a semi-linear elliptic equation in Riemannian geometries (2019, unknown): “The presentation here follows [9, 23] with some modifications.”
The Calderón problem with partial data on manifolds and applications (2012, background): “Salo thanks David dos Santos Ferreira, Yaroslav Kurylev, and Matti Lassas for several helpful discussions and for allowing the authors to use arguments from [Ferreira 2013], which was in preparation simultaneously with this manuscript and involved Gaussian beam quasimodes and a reduction from the at”
Fractional anisotropic Calderón problem on closed Riemannian manifolds (2021, background): “In the C case, the uniqueness for the anisotropic Calderón problem has only been established in [10], [11] for metrics in a fixed conformal class in the case of a conformally transversally anisotropic manifold 1”