The global inverse fractional conductivity problem

Abstract: We prove \emph{global} uniqueness for an inverse problem for the fractional conductivity equation on domains that are bounded in one direction. The conductivities are assumed to be isotropic and nontrivial in the exterior of the domain, while the data is given in the form of partial Dirichlet-to-Neumann (DN) maps measured in nondisjoint open subsets of the exterior. This can be seen as the fractional counterpart of the classical inverse conductivity problem. The proof is based on a unique continuation property (UCP) for the DN maps and an exterior determination method from the partial exterior DN maps. This is analogous to the classical boundary determination method by Kohn and Vogelius. The most important technical novelty is the construction of sequences of special solutions to the fractional conductivity equation whose Dirichlet energies in the limit can be concentrated at any given point in the exterior. This is achieved independently of the UCP and despite the nonlocality of the equation. Due to the recent counterexamples by the last two authors, our results almost completely characterize uniqueness for the inverse fractional conductivity problem with partial data for isotropic global conductivities.