Conductivity Imaging from One Interior Measurement in the Presence of Perfectly Conducting and Insulating Inclusions

Abstract

We consider the problem of recovering an isotropic conductivity outside some perfectly conducting inclusions or insulating inclusions from the interior measurement of the magnitude of one current density field $|J|$. We show that the conductivity outside the inclusions and the shape and position of the inclusions are uniquely determined (except in an exceptional case) by the magnitude of the current generated by imposing a given boundary voltage. Our results show that even when the minimizer of the least gradient problem $\min \int_{\Omega} a |\nabla u|$ with $u|_{\partial \Omega}=f$ exhibits flat regions (i.e., regions with $\nabla u=0$) it can be identified as the voltage potential of a conductivity problem with perfectly conducting inclusions.