Inverse Problems for Linear and Non-linear Elliptic Equations

Abstract

An inverse problem is a mathematical framework that is used to obtain information about a physical object or system from observed measurements. A typical inverse problem is to recover the coefficients of a partial differential equation from measurements on the boundary of the domain. The study of conditions under which such a recovery is possible is of considerable interest and has seen a lot of work in the past few years. This thesis research makes two primary contributions to uniqueness aspects of elliptic inverse problems. First, we prove that the knowledge of Dirichlet-to-Neumann map for a rough first order perturbation of the poly-harmonic operator in a bounded domain uniquely determines the perturbation. This is relevant as the result generalizes previous work on unique recovery of perturbations of the poly-harmonic operator. Second, we show that for a quasi-linear elliptic equation, a perfect cloak can be obtained via a singular change of variables scheme and an approximate cloak can be achieved via a regular change of variables scheme. These approximate cloaks turn out to be anisotropic. We also show that it is possible to get isotropic regular approximate cloaks using a homogenization framework. This work generalizes to quasi-linear settings previous work on cloaking in the context of Electrical Impedance Tomography for the conductivity equation.