Uniqueness results for fractional Calderón problems

Abstract

This dissertation studies the inverse problem for a specific partial differential equation, the so called fractional Calderón problem or inverse problem for the fractional Schrödinger equation. The dissertation focuses mainly on uniqueness results for inverse problems involving the Dirichlet to Neumann map, the object encoding exterior measurements in the model. The included articles show how this information suffices to determine the parameters involved in the problems considered. The first article considers a fractional version of the inverse problem for the conductivity equation, showing that the unknown conductivity can be recovered from the DN map even in the case of a single measurement. The technique employed is the fractional Liouville reduction, which allows one to state the problem in terms of the fractional Schrödinger equation. The second article extends the known result for the fractional Schrödinger equation to the magnetic case, showing how a nonlocal perturbation and a potential can be both recovered up to a natural gauge. This resembles the results known for the local case. The third article explores the fractional Schrödinger equation in a high order regime, proving the injectivity of the relative DN map in both the perturbed and unperturbed cases. This requires a high order Poincaré inequality, which has been studied in the same paper. The fifth article follows the third one, extending the study to general local high-order perturbations: the coefficients of any local lower order operator are shown to be recoverable from the DN map. The fourth article studies the perturbed fractional Calderón problem by means of the Caffarelli-Silvestre extension, transforming it into a local problem with mixed Robin boundary conditions, eventually showing that the bulk and boundary potentials can be recovered simultaneously. This requires some technical Carleman estimates and the construction of a new class of CGO solutions. The introduction of the dissertation contains a survey of the literature related to both the classical and fractional Calderón problems, as well as a collection of the definitions of the function spaces appearing in the articles. The appendix is an informal introduction to key concepts in inverse problems and EIT, thought for the use of the general public.