Global counterexamples to uniqueness for a Calderón problem with $C^k$ conductivities

Abstract

Let $\hat{I}\text{c◯}\subset R^n$, $n\ge 3$, be a fixed smooth bounded domain, and let ${\hat{I}}^3$ be a smooth conductivity in $\overline{\hat{I}}\text{c◯}$. Consider a non-zero frequency $\hat{I}{»}_0$ which does not belong to the Dirichlet spectrum of $L_{\hat{I}}^3=-\mathrm{div}({\hat{I}}^3\nabla \cdot )$. Then, for all $k\ge 1$, there exists an infinite number of pairs of non-isometric $C^k$ conductivities $({\hat{I}}_1^3,{\hat{I}}_2^3)$ on $\overline{\hat{I}}\text{c◯}$, which are close to ${\hat{I}}^3$ such that the associated DN maps at frequency $\hat{I}{»}_0$ satisfy \begin{equation*} Λ_{γ_1,λ_0} = Λ_{γ_2,λ_0}. \end{equation*}

Minor typos corrected