# On non-uniqueness for the anisotropic Calder{\'o}n problem with partial data

## TL;DR

This paper demonstrates non-uniqueness in the anisotropic Calderón problem with partial data for certain Riemannian metrics, showing that different metrics can produce identical boundary measurements under specific conditions.

## Contribution

It provides explicit counterexamples in higher dimensions with metrics that are only Hölder continuous at the boundary, challenging uniqueness assumptions in inverse boundary value problems.

## Key findings

- Existence of non-unique metrics with identical partial Dirichlet-to-Neumann maps.
- Construction of counterexamples on cylindrical Riemannian manifolds with boundary.
- Metrics are smooth inside but Hölder continuous at the boundary, violating unique continuation.

## Abstract

We show that there is non-uniqueness for the Calder{\'o}n problem with partial data for Riemannian metrics with H{\"o}lder continuous coefficients in dimension greater or equal than three. We provide simple counterexamples in the case of cylindrical Riemannian manifolds with boundary having two ends. The coefficients of these metrics are smooth in the interior of the manifold and are only H{\"o}lder continuous of order $$\rho$\<1$ at the end where the measurements are made. More precisely, we construct a toroidal ring $(M, g)$ which is not a warped product manifold, and we show that there exist in the conformal class of g an infinite number of Riemannian $\tilde{g} = c^4 g such that their corresponding partial Dirichlet-to-Neumann maps at one end coincide. The corresponding smooth conformal factors are harmonic with respect to the metric g and do not satisfy the unique continuation principle.