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15 June 2004 Stability estimates for the X-ray transform of tensor fields and boundary rigidity
Plamen Stefanov, Gunther Uhlmann
Duke Math. J. 123(3): 445-467 (15 June 2004). DOI: 10.1215/S0012-7094-04-12332-2

Abstract

We study the boundary rigidity problem for domains in Rn: Is a Riemannian metric uniquely determined, up to an action of diffeomorphism fixing the boundary, by the distance function ρg(x,y) known for all boundary points x and y? It was conjectured by Michel [M] that this was true for simple metrics. In this paper, we first study the linearized problem that consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transform Ig. We prove that the normal operator Ng=I*gIg is a pseudodifferential operator (ΨDO) provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove a hypoelliptic type of stability estimate related to the linear problem. Next, we apply this estimate to show that unique solvability of the linear problem for a given simple metric g, up to potential terms, implies local uniqueness for the nonlinear boundary rigidity problem near that g.

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Plamen Stefanov. Gunther Uhlmann. "Stability estimates for the X-ray transform of tensor fields and boundary rigidity." Duke Math. J. 123 (3) 445 - 467, 15 June 2004. https://doi.org/10.1215/S0012-7094-04-12332-2

Information

Published: 15 June 2004
First available in Project Euclid: 11 June 2004

zbMATH: 1058.44003
MathSciNet: MR2068966
Digital Object Identifier: 10.1215/S0012-7094-04-12332-2

Subjects:
Primary: 53C65
Secondary: 35R30, 53C21, 53C24

Rights: Copyright © 2004 Duke University Press

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Vol.123 • No. 3 • 15 June 2004
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