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Inverse Boundary Problems for Biharmonic Operators and Nonlinear PDEs on Riemannian Manifolds


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.

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