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On the Helmholtz equations with sign changing coefficients

Banff International Research Station for Mathematical Innovation and Discovery

This talk is devoted to various properties and applica- tions of the Helmholtz equations with sign changing coefficients. These equations are used to model negative index materials which are artifi- cial structures whose refractive index are negative over some frequency range. These materials were first investigated theoretically by Veselago in 1964 and their existence was confirmed experimentally by Shelby et al. in 2001. The study of these equations faces two difficulties. First the ellipticity and the compactness are lost in general due to the changing sign coefficients. Second, the localized resonance, i.e., the fields blow up in some regions and remain bounded in some others as the loss (the viscosity) goes to 0, might occur. In this talk, I will present several remarkable properties of these equations and their applications such as cloaking and superlensing using complementary media and cloaking an object via anomalous localized resonance. Various conditions under which the equations are stable are also presented. Some useful tech- niques dealing with these equations such as reflecting and removing localized singularity techniques are mentioned.

Mathematics; Partial differential equations; Calculus of variations and optimal control; optimization

video/mp4

Author affiliation: Ecole Polytechnique Federale de Lausanne, EPFL

2016-03-11

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/57142

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/