The PI proposes to study the mathematical theory behind non-invasive methods of detection via probing by electromagnetic waves. The goal of such methods is to determine the interior composition of an object by making only surface measurements and without drilling holes or taking samples. The theory has potential applications in the early detection of malignant breast tumors and can lead to more economical oil exploration techniques. The idea is to apply an electromagnetic wave on the surface of the object and analyze its interaction with the interior material via surface measurements. One of the objectives of this project is to show that when the surface measurements are taken on only a part of the surface one can still recover the interior composition of the material.
The PI proposes to study non-invasive methods of detection from the point of view of inverse boundary value problems for partial differential equations (PDEs), where one solves a boundary value problem and measures the normal derivative of the solution at the boundary. The question is whether one can recover the coefficients of the PDE from these measurements. The PI proposes to address this problem for the conductivity equation and the Pauli-Dirac system. The PI proposes to use variational convergence methods and Carleman estimates for systems of PDEs.