A survey of non-uniqueness results for the anisotropic Calderón problem with disjoint data
Résumé
After giving a general introduction to the main known results on the anisotropic Calderón problem on n-dimensional compact Riemannian manifolds with boundary, we give a motivated review of some recent non-uniqueness results obtained in [5, 6] for the anisotropic Calderón problem at fixed frequency, in dimension n ≥ 3, when the Dirichlet and Neumann data are measured on disjoint subsets of the boundary. These non-uniqueness results are of the following nature: given a smooth compact connected Riemannian manifold with boundary (M, g) of dimension n ≥ 3, we first show that there exist in the conformal class of g an infinite number of Riemannian metrics˜gmetrics˜ metrics˜g such that their corresponding Dirichlet-to-Neumann maps at a fixed frequency coincide when the Dirichlet data ΓD and Neumann data ΓN are measured on disjoint sets and satisfy ΓD ∪ ΓN = ∂M. The corresponding conformal factors satisfy a nonlinear elliptic PDE of Yamabe type on (M, g) and arise from a natural but subtle gauge invariance of the Calderón when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n ≥ 3 to the anisotropic Calderón problem at fixed frequency with data on disjoint sets, which do not arise from this gauge invariance. They are given by cylindrical Riemannian manifolds with boundary having two ends, equipped with a suitably chosen warped product metric. This survey concludes with some remarks on the case of manifolds with corners.
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