The borderlines of the invisibility and visibility for Calderon's inverse problem

TL;DR

This paper investigates the limits of detectability and invisibility in Calderon's inverse conductivity problem, identifying the precise boundary between conductivities that can be uniquely reconstructed and those that can cloak or create illusions.

Contribution

It establishes the borderline conditions for conductivity invisibility and visibility, including the existence of electric holograms that challenge uniqueness in inverse problems.

Findings

Identified the boundary between visible and cloaking conductivities.

Discovered electric holograms that create illusions and challenge uniqueness.

Provided new counterexamples related to invisibility cloaking.

Abstract

We consider the determination of a conductivity function in a two-dimensional domain from the Cauchy data of the solutions of the conductivity equation on the boundary. We prove uniqueness results for this inverse problem, posed by Calderon, for conductivities that are degenerate, that is, they may not be bounded from above or below. In particular, for scalar conductivities we solve the inverse problem in a class which is larger than $L^\infty$. Also, we give new counterexamples for the uniqueness of the inverse conductivity problem. We say that a conductivity is visible if the inverse problem is solvable so that the inside of the domain can be uniquely determined, up to a change of coordinates, using the boundary measurements. The present counterexamples for the inverse problem have been related to the invisibility cloaking. This means that there are conductivities for which a part of the domain is shielded from detection via boundary measurements. Such conductivities are called invisibility cloaks. In the present paper we identify the borderline of the visible conductivities and the borderline of invisibility cloaking conductivities. Surprisingly, these borderlines are not the same. We show that between the visible and the cloaking conductivities there are the electric holograms, conductivities which create an illusion of a non-existing body. The electric holograms give counterexamples for the uniqueness of the inverse problem which are less degenerate than the previously known ones.