## 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.

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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