In this dissertation we study inverse boundary problems for elliptic PDE in the low regularity setting. The new results established are concerned with the uniqueness and stability issues. Specifically, in Chapter 2 we analyze inverse boundary problems for first order perturbations of the Laplacian, which arise as model operators in the acoustic tomography of a moving fluid. We show that the knowledge of the Dirichlet-to-Neumann map on the boundary of a bounded domain in \mathbb{R}^n, n \geq 3, determines the first order perturbation of low regularity up to a natural gauge transformation, which sometimes is trivial. As an application, we recover the fluid parameters of low regularity from boundary measurements, allowing some of them to be discontinuous.

In Chapter 3, we study inverse boundary problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. We establish stability estimates in the high frequency regime, with an explicit dependence on the frequency parameter, under mild regularity assumptions on the potentials.

This thesis compiles my work on three projects.

In my first project, we proved a global uniqueness result for an inverse boundary problem for a first order perturbation of the biharmonic operator on a conformally transversally anisotropic (CTA) Riemannian manifold of dimension $n \ge 3$. Specifically, we established that a continuous first order perturbation can be determined uniquely from the knowledge of the Cauchy data set of solutions of the perturbed biharmonic operator on the boundary of the manifold provided that the geodesic $X$-ray transform on the transversal manifold is injective.

In my second project, we showed that a continuous potential can be constructively determined from the Cauchy data set of solutions to the perturbed biharmonic equation on a CTA Riemannian manifold of dimension $\ge 3$ with boundary, assuming that the geodesic $X$-ray transform on the transversal manifold is constructively invertible. This is a constructive counterpart of our uniqueness result \cite{Yan_2020}. In particular, our result is applicable and new in the case of smooth bounded domains in the $3$--dimensional Euclidean space as well as in the case of $3$--dimensional CTA manifolds with simple transversal manifold.

In my third project joint with Katya Krupchyk and Gunther Uhlmann, we solved an inverse boundary problem for the nonlinear magnetic Schr\"odinger operator on a compact complex manifold, equipped with a K\"ahler metric and admitting sufficiently many global holomorphic functions.