Geometric Properties of Three Classes of Spectral Problems with Applications to Inverse Problems and Material Sciences

Description

This project is concerned with the mathematical study of three classes of spectral problems. We aim to characterise the geometric structures of various associated eigenfunctions as well as to consider the corresponding applications to inverse problems and material sciences. It is a continuation and an extension of the PI's recent work initiated with his coauthors on this intriguing and important topic. The first class of spectral problems is concerned with the eigenfunctions associated to three partial differential operators (PDOs) including the Laplacian, Lame and Maxwellian operators. We shall study the geometric properties of the nodal or generalised singular sets of the corresponding eigenfunctions. Indeed, we shall establish the sharp quantitative relationship between the analytic behaviours of the eigenfunctions at certain nodal/generalised singular points and the curvature of the nodal/generalised singular curve/surface at those points. In particular, in the corner/edge point case, we shall accurately characterise the vanishing order of the eigenfunction and its relationship to the corresponding corner/edge angle. The results obtained can be used to establish novel unique identifiability results for several geometrical inverse scattering problems by at most a few far-field measurements, as well as to produce important applications for the Pompeiu's problem in integral geometry. The second class of spectral problems is concerned with the interior transmission eigenfunctions associated to acoustic and elastic wave scattering. We aim to characterise the geometric behaviours of the transmission eigenfunctions at a corner point or an admissible high-curvature point. For the acoustic problem, we are mainly interested in the anisotropic case, whereas for the elastic problem, we shall study the case that the Lame parameters possess jump singularities. The results obtained have fundamental implications to invisibility cloaking. They can also be used to establish novel unique identifiability results in determining the shapes/supports of a general class of inhomogeneous media, independent of their contents, by a single far-field measurement.The third class of spectral problems is concerned with the eigenfunctions of the Neumann-Poincare (NP) operators. The NP operators are a type of boundary layer potential operators. We aim to quantitatively characterise the localisation/concentration behaviours ("quantum ergodicity") of the NP eigenfunctions at high-curvature boundary points within both quasi-static regime and the frequency regime beyond the quasistatic approximation. The results obtained shall have significant implications to plasmonic resonances and plasmonic imaging.