Eigenvalue Problems for Wave Equations with Applications in Invisibility and Inverse Scattering

This project is concerned with some eigenvalue problems arising from wave equations and their applications in areas such as invisibility and inverse scattering problems. We shall consider the conventional eigenvalue problems such as the Dirichlet and Neumann eigenvalue problem, the interior transmission eigenvalue problem for medium and medium containing obstacles as well as the exterior transmission eigenvalue problems. We shall also propose new model of transmission eigenvalue problems for geometric structure such as local rough surfaces and periodic structures, which are essentially different from the interior or exterior of a single bounded domains. Starting from the Helmholtz equation for acoustic wave , we shall move on to the Maxwell’s equations for electromagnetic wave and the Navier equation for elastic wave.

Through theoretical analysis and numerical computation, we shall first establish the properties for some of the eigenvalue problems, such as the existence and discreteness of the eigenvalues. More importantly, we shall investigate the geometric structure of the eigenfunctions for certain domains, which is a new contribution to the field. The connections among the eigenvalue problems, their associated scattering/diffraction problems and the Herglotz wave approximation theory will then be utilized to give rise to new notions of invisibility, which differ from the conventional ones in a few major aspects such as the requirement of the cloaking layer and the material properties.

For the Dirichlet and Neumann eigenvalue problems, those connections will also lead us to a novel computational method for the inverse obstacle scattering problems. In view of the PI’s recent findings on the geometric structure for interior transmission eigenfunctions, we shall see the method can be modified for solving the inverse medium scattering problems. This method are fundamentally different from conventional ones with salient features such as the ability to resolve corners, edges and narrow regions, which are difficult for conventional methods.