Mathematical Studies of Surface-localized Transmission Eigenstates and Applications

Description

In this project, we aim to fully uncover and comprehensively study a peculiar spectral phenomenon of a global nature for the transmission eigenvalue problems, i.e. the surface localization of transmission eigenfunctions. Furthermore, we shall consider the corresponding applications of the surface-localized transmission eigenstates for problems of practical significance in inverse scattering theory and wave imaging associated with acoustics, electromagnetism and elasticity.  The transmission eigenvalue problems are a type of non-elliptic, non-selfadjoint and nonlinear (in a certain sense) spectral problems that arise in the theory of wave scattering. They connect to the direct and inverse scattering problems in many aspects in delicate ways. The spectral properties of the transmission eigenvalues have received considerable attentions in the literature over the years due to their theoretical and practical importance. Recently, several studies reveal that the transmission eigenfunctions possess rich geometric structures. Nonetheless, most of the existing and ongoing studies are concerned with the local geometric properties of the transmission eigenfunctions, showing that they generically vanish around the corner or highly-curved parts of the underlying boundary. Meanwhile, a completely new and highly intriguing global pattern of the transmission eigenfunctions was discovered by chance in our recent study. It is found that the transmission eigenfunctions show a certain localization/concentration phenomenon (in terms of the $L^2$-energy) around the boundary of the underlying domain. That is, though vanishing around the corner or highly-curved parts of the boundary according to the local geometric property, ``many" transmission eigenfunctions tend to localize/concentrate around the boundary of the underlying domain. The corresponding finding, though still rather preliminary, has opened up an exciting direction of research for many potential developments. In this project, we aim to fully develop the corresponding study along this direction to completely uncover such a peculiar spectral phenomenon and gain a thorough mathematical understanding. We shall provide rigorous justifications and theoretical understandings of the surface-localized transmission eigenstates associated with the Helmholtz, Maxwell and Lame systems that respectively arise in acoustics, electromagnetism and elasticity. Then we shall study their qualitative and quantitative relationships to the underlying domains as well as the medium parameters. Furthermore, we shall consider the corresponding applications in super-resolution imaging, sensing and artificial mirage. The results achieved in this project shall not only enrich the geometric understanding of the spectral theory for transmission eigenfunctions in a global view, but also provide a new perspective on the wave localization, which is a central topic in mathematical physics.