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

TL;DR

This paper presents counterexamples demonstrating non-uniqueness in the anisotropic Calderon problem when Dirichlet and Neumann data are measured on disjoint boundary sets, challenging previous uniqueness assumptions.

Contribution

It provides explicit counterexamples in 2D and 3D Riemannian manifolds with boundary, illustrating non-uniqueness in the anisotropic Calderon problem.

Findings

Counterexamples in 2D and 3D manifolds

Non-uniqueness when data are on disjoint sets

Construction extendable to higher dimensions

Abstract

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.