The Calderón problem in transversally anisotropic geometries
Ferreira, D. D. S., Kurylev, Y., Lassas, M., & Salo, M. (2016). The Calderón problem in transversally anisotropic geometries. Journal of the European Mathematical Society, 18(11), 2579-2626. https://doi.org/10.4171/JEMS/649
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Journal of the European Mathematical SocietyDate
2016Copyright
© 2016 EMS Publishing House. This is a final draft version of an article whose final and definitive form has been published by EMS Publishing House. Published in this repository with the kind permission of the publisher.
We consider the anisotropic Calder´on problem of recovering
a conductivity matrix or a Riemannian metric from electrical boundary
measurements in three and higher dimensions. In the earlier work [13],
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 Calder´on problem and Gel’fand’s
inverse problem for the wave equation and the boundary control method.
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European Mathematical Society Publishing House; European Mathematical SocietyISSN Search the Publication Forum
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