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Recent progress on the mathematical study of anomalous localized resonance in elasticity. (English) Zbl 1450.35253

Author’s abstract: We consider the anomalous localized resonance induced by negative elastic metamaterials and its application in invisibility cloaking. We survey the recent mathematical developments in the literature and discuss two mathematical strategies that have been developed for tackling this peculiar resonance phenomenon. The first one is the spectral method, which explores the anomalous localized resonance through investigating the spectral system of the associated Neumann-Poincaré operator. The other one is the variational method, which considers the anomalous localized resonance via calculating the nontrivial kernels of a non-elliptic partial differential operator. The advantages and the relationship between the two methods are discussed. Finally, we propose some open problems for the future study.

MSC:

35Q74 PDEs in connection with mechanics of deformable solids
35R30 Inverse problems for PDEs
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
47G40 Potential operators
35B34 Resonance in context of PDEs
35A15 Variational methods applied to PDEs
35P15 Estimates of eigenvalues in context of PDEs
74B10 Linear elasticity with initial stresses
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