Inverse boundary value problems for polyharmonic operators in two dimensions

Abstract

In this thesis, I study an inverse boundary value problem in two dimensions for a polyharmonic operator of the form \begin{equation*} \mathcal{L} = \partial^{m}\bar{\partial}^{m} + \sum_{j,k = 0}^{m-1}{A_{j,k}\partial^{j}\bar{\partial}^{k}}, \quad m\geq 2. \end{equation*} The inverse problem is whether we can recover uniquely the coefficients $A_{j,k}$ from the set of Cauchy data \begin{equation*} \mathcal{C}(\mathcal{L}) = \left\{ \left( u|_{\partial\Omega}, \partial_{\nu}u|_{\partial\Omega}, \partial_{\nu}^{2}u|_{\partial\Omega} \ldots, \partial_\nu^{(2m-1)}u|_{\partial\Omega} \right): u \in H^{2m}(\Omega), \mathcal{L}u =0\right\}, \end{equation*} where $\nu$ is an outer unit normal to $\partial\Omega$.

In joint work with Prof. Krishnan and Dr. Rahul Raju Pattar, I establish that the Cauchy data for a polyharmonic operator uniquely determines all anisotropic perturbations of order at most $m-1$ and several perturbations of orders $m$ to $2m-2$ with some restrictions. This restriction is captured in the following representation of the operator $\mathcal{L}$ as \[(\partial \bar{\partial})^{m} + A_{m-1,m-1}(\partial\bar{\partial})^{m-1} + \sum_{l=1}^{m-2}{\left(\sum_{j+k=m-l-1}{A_{j+l,k+l}\partial^j\bar\partial^k}\right)(\partial\bar\partial)^l } + \sum_{l=0}^{m-1}\sum_{j + k = l}{A_{j,k}\partial^{j}\bar{\partial}^{k}}. \]

We start this thesis with the Calderón inverse problem, where we followed Prof. Pedro Caro's class lecture notes. This is the first problem that represents many ideas. The proof given here is based on Carleman estimates which is different from the original proof of Sylvester and Uhlmannn. It is the primary technique in partial data inverse problems for constructing CGO solutions. After that, we study the inverse boundary value problem for Schr\"{o}dinger operator, which will help to discuss our original contribution.