Duke Mathematical Journal
- Duke Math. J.
- Volume 123, Number 3 (2004), 445-467.
Stability estimates for the X-ray transform of tensor fields and boundary rigidity
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.
Duke Math. J. Volume 123, Number 3 (2004), 445-467.
First available in Project Euclid: 11 June 2004
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 53C65: Integral geometry [See also 52A22, 60D05]; differential forms, currents, etc. [See mainly 58Axx]
Secondary: 53C24: Rigidity results 53C21: Methods of Riemannian geometry, including PDE methods; curvature restrictions [See also 58J60] 35R30: Inverse problems
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.