Non-uniqueness results for the anisotropic Calder\’on problem at fixed energy.

Abstract: In its geometric formulation, the anisotropic Calder\’on problem consists in recovering up to some natural gauge equivalences the metric of a Riemannian manifold with boundary from the knowledge of the Dirichlet-to-Neumann map. I will survey some recent non-uniqueness results obtained in collaboration with Thierry Daud\’e (Cergy-Pontoise) and Francois Nicoleau (Nantes) for the anisotropic Calder\’on problem at fixed energy, in the case of disjoint or partial data. The underlying manifolds arising in these examples are diffeomorphic to toric cylinders with two connected boundary components. In the case of disjoint data the metric is a suitably chosen warped product metric which is everywhere smooth. For partial data, the metric, which is adapted from Miller’s example of an elliptic operator which fails to satisfy the unique continuation principle, is smooth in the interior of the manifold, but only H\”older continuous on one connected component of the boundary.

Mathematics

Audience: researchers in the topic

International Zoom Inverse Problems Seminar, UC Irvine