Surface Riesz transforms and spectral property of elastic Neumann--Poincaé operators on less smooth domains in three dimensions

Abstract

It is known that the Neumann--Poincaré operator for the Lamé system of linear elasticity is polynomially compact and, as a consequence, that its spectrum consists of three non-empty sequences of eigenvalues accumulating to certain numbers determined by Lamé parameters, if the boundary of the domain where the operator is defined is $C^{\infty }$-smooth. We extend this result to less smooth boundaries, namely, $C^{1,\alpha }$-smooth boundaries for some $\alpha >0$. The results are obtained by proving certain identities for surface Riesz transforms, which are singular integral operators of nonconvolution type, defined by the matrix tensor on a given surface.

14 pages