Spectral properties of the Neumann-Poincar\'e operator and cloaking by anomalous localized resonance for the elasto-static system

TL;DR

This paper analyzes the spectral properties of the Neumann-Poincaré operator in elastostatics, revealing eigenvalue behavior and demonstrating cloaking via anomalous localized resonance on elliptical domains.

Contribution

It provides a detailed spectral analysis of the NP operator for the Lamé system, including explicit eigenvalues and eigenfunctions, and links these to cloaking phenomena.

Findings

Eigenvalues accumulate at two constants determined by Lamé constants.

Cloaking by anomalous localized resonance occurs at eigenvalue accumulation points.

Explicit eigenfunctions are derived for disks and ellipses.

Abstract

We first investigate spectral properties of the Neumann-Poincar\'e (NP) operator for the Lam\'e system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, its spectrum consists of eigenvalues which accumulates to two numbers determined by Lam\'e constants. We then derive explicitly eigenvalues and eigenfunctions on disks and ellipses. We then investigate resonance occurring at eigenvalues and anomalous localized resonance at accumulation points of eigenvalues. We show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.