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A variational perspective on cloaking by anomalous localized resonance. (English) Zbl 1366.78002

This work studies the cloaking by anomalous localized resonance. It focuses on the model of a two-dimensional elliptic equation in divergence-form, with a complex-valued coefficient. The coefficient is defined on a matrix-shell-core geometry, with real part equal to \(1\) in the matrix and the core, and \(-1\) in the shell. The main task is to understand the resonant behavior of the solution to the model when the imaginary part of the coefficient approaches zero. In this limit, the solution oscillates; it is known under radial geometries that the resonance depends strongly on the location of the source, and the oscillations are localized spatially. This work confirms that the former strong dependence is also true for some non-radial geometries. The main analytical tool in most existing works on the topic is by separation of variables and applies only to radial geometries. This work develops a novel analytical tool and studies the cloaking by anomalous localized resonance for some non-radial geometries from a variational perspective. The main results are derived almost exclusively for a circular outer shell boundary, which is essential to the concerned variational principle as perfect plasmon waves are used on the outer shell boundary in the construction of comparison functions.

MSC:

78A10 Physical optics
78M30 Variational methods applied to problems in optics and electromagnetic theory
35Q60 PDEs in connection with optics and electromagnetic theory
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References:

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