Limiting Carleman weights and anisotropic inverse problems
The anisotropic Calderon problem and related inverse problems are considered in the article. This inverse problem is connected with so called Electrical Impedance Tomography (EIT), which consists in determining the conductivity of a medium by making voltage and current measurments at the boundary. Applications range from geophysical prospection to medical imaging. The approach is based on limiting Carleman weights [see \textit{C. E. Kenig, J. Sjöstrand} and \textit{G. Uhlmann}, Ann. Math. (2) 165, No. 2, 567--591 (2007; Zbl 1127.35079)]. The authors characterize those Riemannian manifolds which admit limiting Carleman weights, and give a complex geometrical optics construction for a class of such manifolds. This is used to prove uniqueness results for anisotropic inverse problems, via attenuated geodesic ray transform. The inverse problems considered are the recovery of an electric potential and a magnetic field from boundary measurements on an admissible Riemannian manifold, and determination of an admissible metric within a conformal class from the Dirichlet-to-Neumann map (DN map). Earlier results in dimension \(n \geq 3\) were restricted to real-analytic metrics.
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