Integral geometry and unique continuation principles

Abstract

In this thesis we study inverse problems in integral geometry and non-local partial differential equations. We will study these rather different areas of mathematical inverse problems by using the theory of non-local fractional operators. This thesis mainly focuses on proving different kind of unique continuation results of fractional operators which are then used to prove uniqueness results for fractional Calderón problems and partial data problems in scalar and vector field tomography. The introductory part of the thesis contains a general introduction and review of inverse problems arising in medical and seismic imaging. The included articles are divided into three classes which are then presented in their own sections and studied in different levels of detail. In the articles [A, B, C, G] we consider partial data problems in the X-ray tomography of scalar and vector fields. In the first article [A] we prove unique continuation for certain Riesz potentials and apply it to partial data problems of scalar fields. In the second article [B] we prove unique continuation results for higher order fractional Laplacians which are then used in proving uniqueness for partial data problems of d-plane transforms. In the third article [C] we study partial data problems of vector fields and we prove unique continuation of the normal operator of vector fields which implies uniqueness for the partial data problems. In the seventh article [G] we generalize the unique continuation result of fractional Laplacians proved in [B] and use it to prove uniqueness for partial data problems of scalar and vector fields, extending the partial data results of the articles [A, B, C] to more general cases. In the articles [B, D] we consider higher order fractional Calderón problems. In the second article [B] we use the unique continuation of higher order fractional Laplacians to prove uniqueness for the Calderón problem of the higher order fractional (magnetic) Schrödinger equation. In the fourth article [D] we generalize the uniqueness result proved in [B] to include general lower order local perturbations of the fractional Laplacian. In the articles [E, F] we consider the travel time tomography problem and its different linearized versions. In the fifth article [E] we study mixing ray transforms which are generalizations of the geodesic ray transform. We prove solenoidal injectivity results for them in various different cases. In the sixth article [F] we study the boundary rigidity problem on certain non-reversible Finsler manifolds which are also called Randers manifolds. We prove that if the Randers metric consists of a boundary rigid Riemannian metric and a closed 1-form, then the boundary distances determine the Randers metric uniquely up to a natural gauge.