Negative index materials and their applications: Recent mathematics progress
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Negative index materials are artificial structures whose refractive index has negative value over some frequency range. These materials were first investigated theoretically by Veselago in 1946 and were confirmed experimentally by Shelby, Smith, and Schultz in 2001. Mathematically, the study of negative index materials faces two difficulties. Firstly, the equations describing the phenomenon have sign changing coefficients, hence the ellipticity and the compactness are lost in general. Secondly, the localized resonance, i.e., the field explodes in some regions and remains bounded in some others as the loss goes to 0, might appear. In this survey, the author discusses recent mathematics progress in understanding properties of negative index materials and their applications. The topics are reflecting complementary media, superlensing and cloaking by using complementary media, cloaking a source via anomalous localized resonance, the limiting absorption principle and the well-posedness of the Helmholtz equation with sign changing coefficients.
KeywordsNegative index materials Localized resonance Cloaking Superlensing
2000 MR Subject Classification35B34 35B35 35B40 35J05 78A25
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The author thanks Professor Haïm Brezis deeply for his encouragement, his guidance, and his support from his Ph.D studies. His joint work with Bethuel and Hélein on the Ginzburg-Landau equation in  inspired the author to introduce the removing localized singularity technique to handle the localized resonance which appears naturally in the study of negative index materials. The author also thanks Professor Graeme Milton for useful discussions on the subject.
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