Boundary rigidity with partial data

Stefanov, Plamen; Uhlmann, Gunther; Vasy, Andras

doi:10.1090/jams/846

Boundary rigidity with partial data
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by Plamen Stefanov, Gunther Uhlmann and Andras Vasy

J. Amer. Math. Soc. 29 (2016), 299-332

DOI: https://doi.org/10.1090/jams/846

Published electronically: November 17, 2015

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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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