A variational perspective on cloaking by anomalous localized resonance

TL;DR

This paper introduces a new variational approach to analyze cloaking by anomalous localized resonance, extending beyond radial geometries and highlighting the importance of source location in resonance phenomena.

Contribution

A novel dual variational framework is developed to study non-radial geometries in cloaking by anomalous localized resonance, overcoming limitations of previous separation of variables methods.

Findings

The variational approach applies to non-radial geometries.

Source location critically influences resonance occurrence.

The method generalizes analysis beyond radial symmetry.

Abstract

A body of literature has developed concerning "cloaking by anomalous localized resonance". The mathematical heart of the matter involves the behavior of a divergence-form elliptic equation in the plane, $\nabla\cdot (a(x)\nabla u(x)) = f(x)$. The complex-valued coefficient has a matrix-shell-core geometry, with real part equal to 1 in the matrix and the core, and -1 in the shell; one is interested in understanding the resonant behavior of the solution as the imaginary part of $a(x)$ decreases to zero (so that ellipticity is lost). Most analytical work in this area has relied on separation of variables, and has therefore been restricted to radial geometries. We introduce a new approach based on a pair of dual variational principles, and apply it to some non-radial examples. In our examples, as in the radial setting, the spatial location of the source $f$ plays a crucial role in determining whether or not resonance occurs.