Surface Riesz transforms and spectral property of elastic Neumann--Poinca\'e operators on less smooth domains in three dimensions

Abstract

It is known that the Neumann--Poincar\'e operator for the Lam\'e 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 Lam\'e 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.