Elastic Neumann–Poincaré Operators on Three Dimensional Smooth Domains: Polynomial Compactness and Spectral Structure
Kazunori Ando, Hyeonbae Kang, Yoshihisa Miyanishi · International Mathematics Research Notices · 2017
We prove that the elastic Neumann–Poincaré (NP) operator defined on the smooth boundary of a bounded domain in three dimensions, which is known to be non-compact, is in fact polynomially compact. As a consequence, we prove that the spectrum of the elastic NP operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators, and the spectral mapping theorem.