The anisotropic Calderon problem for singular metrics of warped product type: the borderline between uniqueness and invisibility

In this paper, we investigate the anisotropic Calderon problem on cylindrical Riemannian manifolds with boundary having two ends and equipped with singular metrics of warped product type, that is whose coefficients only depend on the horizontal direction of the cylinder. By singular, we mean that these coefficients are positive almost everywhere and belong to some L-p, 1 <= p <= infinity spaces only. Using the recent developments on Weyl-Titchmarsh's theory for singular Sturm-Liouville operators, we prove that the local Dirichlet to Neumann maps at each end are well defined and determine the metric uniquely if 1. (doubly warped product case) the coefficients of the metric are L-infinity and bounded from below by a positive constant; 2. (warped product case) the coefficients of the metrics belong to a critical L-p space where p < infinity depends on the dimension of the transversal directions of the cylinder. Eventually, we show (in the warped product case and for zero frequency) that these uniqueness results are sharp by giving simple counterexamples for a class of singular metrics whose coefficients do not belong to the critical L-p space. All these counterexamples lead in fact to a region of space that is invisible to boundary measurements.