Elastic Neumann-Poincare operators on three dimensional smooth domains: Polynomial compactness and spectral structure

We prove that the elastic Neumann--Poincare 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 Neumann-Poincare operator consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lame parameters. These results are proved using the surface Riesz transform, calculus of pseudo-differential operators and the spectral mapping theorem.