On spectral properties of Neuman–Poincaré operator and plasmonic resonances in 3D elastostatics

Abstract

We consider plasmon resonances and cloaking for the elastostatic system in ${\mathbb{R}}^3$ via the spectral theory of the Neumann–Poincaré operator. We first derive the full spectral properties of the Neumann–Poincaré operator for the 3D elastostatic system in the spherical geometry. The spectral result is of significant interest for its own sake, and serves as a highly nontrivial extension of the corresponding 2D study in [4]. The derivation of the spectral result in 3D involves much more complicated and subtle calculations and arguments than that for the 2D case. Then we consider a 3D plasmonic structure in elastostatics which takes a general core-shell-matrix form with the metamaterial located in the shell. Using the obtained spectral result, we provide an accurate characterisation of the anomalous localised resonance and cloaking associated to such a plasmonic structure.