ON NOVEL GEOMETRIC STRUCTURES OF LAPLACIAN EIGENFUNCTIONS IN ℝ³ AND APPLICATIONS TO INVERSE PROBLEMS

This is a continuation and an extension of our recent work [J. Math. Pures Appl., 143 (2020), pp. 116-161] on the geometric structures of Laplacian eigenfunctions and their applications to inverse scattering problems. In that work, we studied the analytic behavior of the Laplacian eigenfunctions at a point where two nodal or generalized singular lines intersect. The results reveal an important and intriguing property that the vanishing order of the eigenfunction at the intersecting point is closely related to the rationality of the intersecting angle. In this paper, we continue this development in three dimensions and study the analytic behaviors of the Laplacian eigenfunctions at places where nodal or generalized singular planes intersect. Compared with the two-dimensional case, the geometric situation is much more complicated, and so is the corresponding analysis: the intersection of two planes generates an edge corner, whereas the intersection of more than three planes generates a vertex corner. We provide a systematic and comprehensive characterization of the relations between the analytic behaviors of an eigenfunction at a corner point and the geometric quantities of that corner for all these geometric cases. Moreover, we apply the spectral results to establish some novel unique identifiability results for the geometric inverse problems of recovering the shape as well as the (possible) surface impedance coefficient by the associated scattering far-field measurements.