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

2018/06/06 by Kang, Hyeonbae, Kawagoe, Daisuke
#35P05 #42B20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.1806.02026

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^∞-smooth. We extend this result to less smooth boundaries, namely, C^{1, α}-smooth boundaries for some α> 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.

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