## Duke Mathematical Journal

### Stability estimates for the X-ray transform of tensor fields and boundary rigidity

#### 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.

#### Article information

Source
Duke Math. J. Volume 123, Number 3 (2004), 445-467.

Dates
First available in Project Euclid: 11 June 2004

http://projecteuclid.org/euclid.dmj/1086957713

Digital Object Identifier
doi:10.1215/S0012-7094-04-12332-2

Mathematical Reviews number (MathSciNet)
MR2068966

Zentralblatt MATH identifier
1058.44003

#### Citation

Stefanov, Plamen; Uhlmann, Gunther. Stability estimates for the X-ray transform of tensor fields and boundary rigidity. Duke Math. J. 123 (2004), no. 3, 445--467. doi:10.1215/S0012-7094-04-12332-2. http://projecteuclid.org/euclid.dmj/1086957713.

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