The global inverse fractional conductivity problem

We prove 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 nontrivial in the exterior of the domain and isotropic, 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 a novel exterior determination method from the partial exterior DN maps, which is a nonlocal analog of the classical boundary determination method by Kohn and Vogelius. Exterior determination is achieved independently of the UCP and despite the nonlocality of the equation.