Calderón’s inverse problem: imaging and invisibility

Summary

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. In the first sections of the paper we consider this inverse problem, posed by Calderón, for conductivities that are in L^∞ and are bounded from below by a positive constant. After this we consider uniqueness results and counterexamples for conductivities that are degenerate, that is, not necessarily bounded from above or below. Elliptic equations with such coefficient functions are essential for physical models used in transformation optics and metamaterial constructions. The present counterexamples for the inverse problem have been related to 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. At the end of the paper we consider the borderline of the smoothness required for the visible conductivities and the borderline of smoothness needed for invisibility cloaking conductivities.

In electrical impedance tomography one aims to determine the internal structure of a body from electrical measurements on its surface. To consider the precise mathematical formulation of the electrical impedance tomography problem, suppose that Ω ⊂ ℝⁿ is a bounded domain with connected complement and let us start with the case whenσ: Ω → (0;∞) be a measurable function that is bounded away from zero and infinity.

The authors were supported by the Academy of Finland, the Finnish Centres of Excellence in Analysis and Dynamics and Inverse Problems, and the Mathematical Sciences Research Institute.