# The Bilinear Strategy for Calder\'on's Problem

## TL;DR

This paper proves uniqueness in Calderón's inverse conductivity problem for higher dimensions with less regular conductivities, extending Tao's bilinear theorem to improve previous results.

## Contribution

It introduces a novel extension of Tao's bilinear theorem to establish uniqueness for less regular conductivities in higher dimensions, advancing prior work.

## Key findings

- Uniqueness holds for conductivities in specific Sobolev spaces in dimensions 5 and 6.
- The method extends Tao's bilinear theorem to the context of Calderón's problem.
- Improves upon recent results by Haberman, Ham, Kwon, and Lee.

## 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.