A survey of non-uniqueness results for the anisotropic Calder{\'o}n problem with disjoint data

TL;DR

This survey reviews recent non-uniqueness results for the anisotropic Calderón problem on Riemannian manifolds, highlighting counterexamples and gauge invariances when Dirichlet and Neumann data are measured on disjoint boundary sets.

Contribution

It provides a comprehensive overview of non-uniqueness phenomena and counterexamples in the anisotropic Calderón problem with disjoint boundary data, including new insights into gauge invariance and specific manifold constructions.

Findings

Existence of multiple conformal metrics with identical Dirichlet-to-Neumann maps on disjoint boundary sets.

Counterexamples using warped product metrics on cylindrical manifolds with boundary.

Identification of gauge invariance related to nonlinear elliptic PDEs of Yamabe type.

Abstract

After giving a general introduction to the main known results on the anisotropic Calder{\'o}n 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 Calder{\'o}n problem at fixed frequency, in dimension n $\ge$ 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 $\ge$ 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 $\Gamma$D and Neumann data $\Gamma$N are measured on disjoint sets and satisfy $\Gamma$D $\cup$ $\Gamma$N = $\partial$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 Calder{\'o}n when the data are given on disjoint sets. We then present counterexamples to uniqueness in dimension n $\ge$ 3 to the anisotropic Calder{\'o}n 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.