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

A variational perspective on cloaking by anomalous localized resonance