Inverse problems for fractional operators involving a magnetic potential

Abstract

In this thesis, we study forward and inverse problems for fractional operators involving a magnetic potential. We show that many properties of fractional operators are preserved under the perturbation by a magnetic potential. Besides, we carefully use Runge approximation properties to obtain strong results when we study inverse problems. More precisely, we determine both the magnetic potential and the electric potential from exterior partial measurements of the Dirichlet-to-Neumann map in the linear fractional inverse problem by using the Runge approximation property and an integral identity; we also determine both the magnetic potential and the non-linearity in the semi-linear fractional inverse problem by using the Runge approximation property and a first order linearization.