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Boundary rigidity with partial data


Authors: Plamen Stefanov, Gunther Uhlmann and Andras Vasy
Journal: J. Amer. Math. Soc. 29 (2016), 299-332
MSC (2010): Primary 53C24, 35R30
DOI: https://doi.org/10.1090/jams/846
Published electronically: November 17, 2015
MathSciNet review: 3454376
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Abstract: We study the boundary rigidity problem with partial data consisting of determining locally the Riemannian metric of a Riemannian manifold with boundary from the distance function measured at pairs of points near a fixed point on the boundary. We show that one can recover uniquely and in a stable way a conformal factor near a strictly convex point where we have the information. In particular, this implies that we can determine locally the isotropic sound speed of a medium by measuring the travel times of waves joining points close to a convex point on the boundary.

The local results lead to a global lens rigidity uniqueness and stability result assuming that the manifold is foliated by strictly convex hypersurfaces.


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Additional Information

Plamen Stefanov
Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
Email: Plamen-Stefanov@purdue.edu

Gunther Uhlmann
Affiliation: Department of Mathematics, University of Washington, Seattle, Washington 98195 and Department of Mathematics, University of Helsinki, Finland FI-00014
Email: gunther@math.washington.edu

Andras Vasy
Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
Email: andras@stanford.edu

DOI: https://doi.org/10.1090/jams/846
Received by editor(s): November 13, 2013
Published electronically: November 17, 2015
Additional Notes: The first author was partly supported by NSF Grant DMS-1301646
The second author was partly supported by NSF Grants CMG-1025259 and DMS-1265958, The Fondation Mathématiques de Paris, and a Simons fellowship
The third author was partly supported by NSF Grants CMG-1025259 and DMS-1068742.
Article copyright: © Copyright 2015 American Mathematical Society