Note on Calder\'on's inverse problem for measurable conductivities

TL;DR

This paper proposes a novel approach using Clifford algebras and higher-dimensional Beltrami equations to address Calderón's inverse conductivity problem for measurable conductivities, aiming to establish uniqueness in higher dimensions.

Contribution

Introduces a new strategy based on Clifford algebras and higher-dimensional Beltrami equations for Calderón's problem in multiple dimensions.

Findings

Suggests a potential pathway for proving uniqueness in three or more dimensions.

Develops a higher-dimensional analogue of the Beltrami equation.

Provides a new mathematical framework for inverse conductivity problems.

Abstract

The unique determination of a measurable conductivity from the Dirichlet-to-Neumann map of the equation $\mathrm{div} (\sigma \nabla u) = 0$ is the subject of this note. A new strategy, based on Clifford algebras and a higher dimensional analogue of the Beltrami equation, is here proposed. This represents a possible first step for a proof of uniqueness for the Calder\'on problem in three and higher dimensions in the $L^\infty$ case.