The anisotropic Calderón problem on 3-dimensional conformally Stäckel manifolds
Résumé
Conformally Stäckel manifolds can be characterized as the class of n-dimensional pseudo-Riemannian manifolds (M, G) on which the Hamilton-Jacobi equation G(∇u, ∇u) = 0 for null geodesics and the Laplace equation −∆ G ψ = 0 are solvable by R-separation of variables. In the particular case in which the metric has Riemannian signature, they provide explicit examples of metrics admitting a set of n−1 commuting conformal symmetry operators for the Laplace-Beltrami operator ∆ G. In this paper, we solve the anisotropic Calderón problem on compact 3-dimensional Riemannian manifolds with boundary which are conformally Stäckel, that is we show that the metric of such manifolds is uniquely determined by the Dirichlet-to-Neumann map measured on the boundary of the manifold, up to dieomorphims that preserve the boundary.
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