Mathematical Analysis on Scattering from Corner Singularities, Inverse Shape Problems and Geometric Structures of Transmission Eigenfunctions 

Description

We consider three different but closely connected topics associated with the Maxwell system and the magnetic Schrodinger equation. They are the scattering from corner singularities, inverse shape determination of an inhomogeneity by a single far-field pattern, and the geometric structures of transmission eigenfunctions. This project is a natural continuation and extension of the PI's work on related studies for acoustic scattering governed by the Helmholtz system. We propose to conduct a systematic and comprehensive study on those fundamental issues to direct and inverse scattering theory. For the electromagnetic scattering, we consider a generic inhomogeneous medium which possesses a corner singularity on its support. In such a case, we establish the failure of the real-analytic extension across the corner, of certain electromagnetic fields satisfying the generalized transmission eigenvalue problem. This result has two important consequences. First, we can show that the corner medium scatters any incident electromagnetic waves nontrivially. Second, we can derive the unique recovery of the convex polyhedral support of a medium without knowing its contents by a single far-field pattern. Furthermore, if the medium parameters are assumed to be piecewise-constant supported in either polyhedral-cell or polyhedral-nest geometries, then the parameters can be uniquely recovered as well. Those recovery results are completely new and in sharp difference to the existing results in the literature where one basically makes use of infinitely many measurements. They also shed light and provide inspiration on a longstanding geometrical shape problem in inverse scattering theory. To push our study even further, we intend to quantify the corner scattering and the shape recovery results by establishing sharp stability estimates. The obtained quantitative results have important implications to invisibility cloaking. Moreover, they pave the way to study the geometric structures of transmission eigenfunctions. We consider both interior and exterior transmission eigenvalue problems and show that the eigenfunctions either vanish or localize near a corner point depending on the corner angle. The results are not only of significant theoretical interest, but can also be used to produce super-resolution reconstructions for inverse problems. Finally, we extend some of the proposed studies to the case with quasi-corner points. The proposed studies on exterior transmission eigenfunctions and quasi-corners are new to literature and were not ever considered even for the acoustic scattering. For the magnetic Schrodinger equation, we focus on the unique recovery of polyhedral-supported electric and magnetic potentials from a single far-field pattern, which is mainly of mathematical interest.