ABSTRACT: Negative-index metamaterials are engineered structures
whose refractive index has a negative value over some frequency range.
Their existence was postulated by Veselago in 1964 and confirmed
experimentally by Shelby, Smith, and Schultz in 2001. Negative-index
metamaterial research has been a very active topic of investigation not
only because of potentially interesting applications but also because of
challenges in understanding their surprising properties. From a
mathematical point of view, the subtlety and the challenges in the
study of negative-index metamaterials are from the sign-changing
coefficients in the modelling equations, hence the ellipticity and the
compactness are lost in general. Moreover, localize resonance, i.e., a
phenomenon for which the field explodes in some regions and remains
bounded in some others as the loss (the damping/viscosity coefficient)
goes to 0, might occur. In this talk, I discuss superlensing and
cloaking applications of negative-index metamaterials, and the stability
of associated fields in the frequency domain. Some mathematical
ideas/techniques/tools used to analyse these phenomena are mentioned.
These involves deriving/analysing Cauchy’s problems,
applying/establising three-sphere inequalities (with partial data), and
introducing the removing localized singularity technique.