Global counterexamples to uniqueness for a Calder\'on problem with $C^k$ conductivities

TL;DR

This paper constructs infinitely many pairs of non-isometric $C^k$ conductivities close to a given smooth conductivity in a bounded domain, which produce identical Dirichlet-to-Neumann maps at a fixed non-zero frequency, challenging uniqueness in inverse problems.

Contribution

It demonstrates the existence of global counterexamples to uniqueness for the Calderón problem with $C^k$ conductivities at a fixed frequency.

Findings

Existence of infinitely many non-isometric conductivities with identical DN maps.

Counterexamples are constructed close to a given smooth conductivity.

Results hold for all $k \\geq 1$ and fixed non-zero frequency.

Abstract

Let $\Omega \subset R^n$, $n \geq 3$, be a fixed smooth bounded domain, and let $\gamma$ be a smooth conductivity in $\overline{\Omega}$. Consider a non-zero frequency $\lambda_0$ which does not belong to the Dirichlet spectrum of $L_\gamma = -{\rm div} (\gamma \nabla \cdot)$. Then, for all $k \geq 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $(\gamma_1, \gamma_2)$ on $\overline{\Omega}$, which are close to $\gamma$ such that the associated DN maps at frequency $\lambda_0$ satisfy \begin{equation*} \Lambda_{\gamma_1,\lambda_0} = \Lambda_{\gamma_2,\lambda_0}. \end{equation*}