Recent progress on the mathematical study of anomalous localized resonance in elasticity

Hongjie Li; Hongjie Li

doi:10.3934/era.2020069

Electronic Research Archive

2020, Volume 28, Issue 3: 1257-1272. doi: 10.3934/era.2020069

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Recent progress on the mathematical study of anomalous localized resonance in elasticity

Hongjie Li ^,

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong SAR, China

Received: 01 May 2020
Primary: 35R30, 35B30, 35Q60, 47G40

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.
- Anomalous localized resonance,
- negative material,
- Neumann-Poincaré operator,
- spectral,
- variational
Citation: Hongjie Li. Recent progress on the mathematical study of anomalous localized resonance in elasticity[J]. Electronic Research Archive, 2020, 28(3): 1257-1272. doi: 10.3934/era.2020069

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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.

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