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Journal of the American Mathematical Society

Published by the American Mathematical Society, the Journal of the American Mathematical Society (JAMS) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6834 (online) ISSN 0894-0347 (print)

The 2020 MCQ for Journal of the American Mathematical Society is 4.79.

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Boundary rigidity with partial data
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by Plamen Stefanov, Gunther Uhlmann and Andras Vasy PDF
J. Amer. Math. Soc. 29 (2016), 299-332 Request permission

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.

References
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Additional Information
  • Plamen Stefanov
  • Affiliation: Department of Mathematics, Purdue University, West Lafayette, Indiana 47907
  • MR Author ID: 166695
  • 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
  • MR Author ID: 175790
  • Email: gunther@math.washington.edu
  • Andras Vasy
  • Affiliation: Department of Mathematics, Stanford University, Stanford, California 94305
  • MR Author ID: 616271
  • Email: andras@stanford.edu
  • 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.
  • © Copyright 2015 American Mathematical Society
  • Journal: J. Amer. Math. Soc. 29 (2016), 299-332
  • MSC (2010): Primary 53C24, 35R30
  • DOI: https://doi.org/10.1090/jams/846
  • MathSciNet review: 3454376