Low regularity theory for the inverse fractional conductivity problem

TL;DR

This paper advances the understanding of the inverse fractional conductivity problem by establishing partial data uniqueness under lower regularity assumptions, constructing counterexamples, and providing a new proof approach, extending previous results.

Contribution

It extends partial data uniqueness results to lower regularity $H^{s,n/s}$ conductivities, constructs counterexamples in specific cases, and offers a new proof method not relying on Runge approximation.

Findings

Partial data uniqueness characterized for $H^{s,n/s}$ conductivities.

Counterexamples constructed for domains with measurements in disjoint sets.

New proof of uniqueness not based on Runge approximation.

Abstract

We characterize partial data uniqueness for the inverse fractional conductivity problem with $H^{s,n/s}$ regularity assumptions in all dimensions. This extends the earlier results for $H^{2s,\frac{n}{2s}}\cap H^s$ conductivities by Covi and the authors. We construct counterexamples to uniqueness on domains bounded in one direction whenever measurements are performed in disjoint open sets having positive distance to the domain. In particular, we provide counterexamples in the special cases $s \in (n/4,1)$, $n=2,3$, missing in the literature due to the earlier regularity conditions. We also give a new proof of the uniqueness result which is not based on the Runge approximation property. Our work can be seen as a fractional counterpart of Haberman's uniqueness theorem for the classical Calder\'on problem with $W^{1,n}$ conductivities when $n=3,4$. One motivation of this work is Brown's conjecture that uniqueness for the classical Calder\'on problem holds for $W^{1,n}$ conductivities also in dimensions $n \geq 5$.