The Bilinear Strategy for Calder\'on's Problem

Abstract

Electrical Impedance Imaging would suffer a serious obstruction if for two different conductivities the potential and current measured at the boundary were the same. The Calder\'on's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In $\mathbb{R}^d$, for $d=5,6$, we show that uniqueness holds when the conductivity is in $W^{1+\frac{d-5}{2p}+, p}(\Omega)$, for $d\le p <\infty$. This improves on recent results of Haberman, and of Ham, Kwon and Lee. The main novelty of the proof is an extension of Tao's bilinear Theorem.