On the Hidden Mechanism Behind Non-uniqueness for the Anisotropic Calderon Problem with Data on Disjoint Sets

We show that there is generically non-uniqueness for the anisotropic Calderon problem at fixed frequency when the Dirichlet and Neumann data are measured on disjoint sets of the boundary of a given domain. More precisely, we first show that given a smooth compact connected Riemannian manifold with boundary (M,g) of dimension n3, there exist in the conformal class of g an infinite number of Riemannian metrics g such that their corresponding DN maps at a fixed frequency coincide when the Dirichlet data D and Neumann data N are measured on disjoint sets and satisfy DNM. The conformal factors that lead to these non-uniqueness results for the anisotropic Calderon problem satisfy a nonlinear elliptic PDE of Yamabe type on the original manifold (M,g) and are associated with a natural but subtle gauge invariance of the anisotropic Calderon problem with data on disjoint sets. We then construct a large class of counterexamples to uniqueness in dimension to the anisotropic Calderon problem at fixed frequency with data on disjoint sets and modulo this gauge invariance. This class consists in cylindrical Riemannian manifolds with boundary having two ends (meaning that the boundary has two connected components), equipped with a suitably chosen warped product metric.