Spectrum of Neumann-Poincaré operator on annuli and cloaking by anomalous localized resonance for linear elasticity

We investigate anomalous localized resonance on the circular coated structure and cloaking due to it in the context of elastostatic systems. The structure consists of the circular core with constant Lamé parameters and the circular shell with negative Lamé parameters proportional to those of the core. We show that there is a nonzero number k0 determined by Lamé parameters such that two nonempty eigenvalue sequences of the Neumann{Poincaré operator associated with the structure converge to k0 and -k0, respectively, and derive precise asymptotics of the convergence. We then show by qualitative estimates based on asymptotics of eigenvalues that cloaking by anomalous localized resonance takes place if and only if the dipole-type source lies inside critical radii determined by radii of the core and the shell. The critical radii corresponding to k0 and -k0 are different.