A bilinear strategy for Calderón's problem

Abstract

Electrical impedance imaging would suffer a serious obstruction if two different conductivities yielded the same measurements of potential and current at the boundary. The Calderón's problem is to decide whether the conductivity is indeed uniquely determined by the data at the boundary. In ${\mathbb{R}}^d$, for $d\ge 5$, we show that uniqueness holds when the conductivity is in $W^{1+(d-5)/(2p)+,p}(\Omega )$, 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.