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Contents1 Introduction and Preliminaries 11.1 Overview 11.2 Layer Potentials in Electro-Magnetic System 213 Layer Potentials in Elastic System 41.4 Bessel and Neumann Functions 62 Mathematical Theory of Plasmon/Polariton Resonances in Quasi-Static Regime 92.1 Maxwell's Problem 92.1.1 Introduction to Plasmonic Resonances 92.1.2 Drude's Model for the Electric Permittivity and Magnetic Permeability 122.1.3 Boundary Integral Operators and Resolvent Estimates 152.1.4 Layer Potential Formulation 242.1.5 Derivation of the Asymptotic Formula 252.1.6 Numerical Ilustrations 462.1.7 Concluding Remarks 482.2 Elastic Problem 482.2.1 Layer Potential Techniques 512.2.2 Asymptotics for the Integral Operators 532.23 Far Field Expansion 572.2.5 Resolvent Analysis 672.2.6 Polariton Resonance for Elastic Nanoparticles 693 Anomalous Localized Resonances and Their Cloaking Effect 773.1 Elastostatic Problem 773.1.1 Ma