Uniqueness in Calder\'on's problem for conductivities with unbounded gradient

TL;DR

This paper establishes the uniqueness of solutions in Calderón's inverse conductivity problem for conductivities with unbounded gradients in certain Sobolev spaces, extending previous results to less regular conductivities.

Contribution

It proves uniqueness for conductivities in $W^{s,p}( abla)$ spaces that are not necessarily Lipschitz, including $W^{1,n}$ for $n=3,4$, improving prior regularity assumptions.

Findings

Uniqueness holds for conductivities in $W^{s,p}$ with unbounded gradients.

Extends Calderón's problem results to less regular conductivities.

Includes new results for conductivities in $W^{1,n}$ for $n=3,4$.

Abstract

We prove uniqueness in the inverse conductivity problem for uniformly elliptic conductivities in $W^{s,p}(\Omega)$, where $\Omega \subset \mathbb R^n$ is Lipschitz, $3\leq n \leq 6$, and $s$ and $p$ are such that $ W^{s,p}(\Omega)\not \subset W^{1,\infty}(\Omega)$. In particular, we obtain uniqueness for conductivities in $W^{1,n}(\Omega)$ ($n=3,4$). This improves on the result of the author and Tataru, who assumed that the conductivity is Lipschitz.