Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets

Thierry Daudé (1) , Niky Kamran (2) , Francois Nicoleau (3)

1 AGM - UMR 8088 - Analyse, Géométrie et Modélisation
2 Department of Mathematics and Statistics [Montréal]
3 LMJL - Laboratoire de Mathématiques Jean Leray

In this paper, we give some simple counterexamples to uniqueness for the Calderon problem on Riemannian manifolds with boundary when the Dirichlet and Neumann data are measured on disjoint sets of the boundary. We provide counterexamples in the case of two and three dimensional Riemannian manifolds with boundary having the topology of circular cylinders in dimension two and toric cylinders in dimension three. The construction could be easily extended to higher dimensional Riemannian manifolds.

Mots clés: Helmholtz equation on a Riemannian manifold; Sturm-Liouville problems; Weyl-Titchmarsh function; Anisotropic Calderon problem

Domaines: Physique mathématique [math-ph]; Théorie spectrale [math.SP]

Thierry Daudé, Niky Kamran, Francois Nicoleau. Non-uniqueness results for the anisotropic Calderon problem with data measured on disjoint sets. Annales de l'Institut Fourier, 2019, Volume 69 (n°1), pp.119-170. ⟨hal-01219221⟩