Uniqueness in the Calder\'on problem via infinitesimally bounded potentials

TL;DR

This paper proves uniqueness in the Calderón inverse problem for scalar conductivities in Sobolev spaces with minimal regularity, extending previous results to higher dimensions and broader function classes.

Contribution

It generalizes prior uniqueness results to conductivities in W^{1,p} spaces for p ≥ n in dimensions n ≥ 3, using a novel combination of Fourier and analytic methods.

Findings

Proves uniqueness for scalar conductivities in W^{1,p} with p ≥ n in dimensions n ≥ 3.

Extends previous results from n=3,4 to higher dimensions.

Introduces a new method combining Fourier series with infinitesimal boundedness criteria.

Abstract

The Calder\'on problem is an inverse problem with applications to electrical impedance tomography and geophysical prospection. We prove uniqueness in the Calder\'on problem in spatial dimension $n \geq 3$ for scalar conductivities in the Sobolev space $W^{1,p}$ with $p \geq n$. This generalizes a result of Haberman who considered the case $p \geq n$ and $n=3$ or $4$. Our method of proof combines a Fourier series approach with an analytic criterion for infinitesimal boundedness of potentials appearing in a Schr\"odinger equation with respect to the Laplacian.