论文信息 - Stability estimates for the X-ray transform of tensor fields and boundary rigidity

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 g.x; y/ known for all boundary points x andy? It was conjectured by Michel that this was true for simple metrics. In this paper, we study the linearized problem first which consists of determining a symmetric 2-tensor, up to a potential term, from its geodesic X-ray integral transformIg . We prove that the normal operator Ng D I g Ig is a pseudodifferential operator provided that g is simple, find its principal symbol, identify its kernel, and construct a microlocal parametrix. We prove 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 non-linear boundary rigidity problem near that g.

G. Uhlmann | P. Stefanov | Plamen Stefanov

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