TY - JOUR

T1 - A variational perspective on cloaking by anomalous localized resonance

AU - Kohn, R. V.

AU - Lu, J.

AU - Schweizer, B.

AU - Weinstein, M. I.

N1 - Publisher Copyright:
© Springer-Verlag Berlin Heidelberg 2014.
Copyright:
Copyright 2018 Elsevier B.V., All rights reserved.

PY - 2014/3/14

Y1 - 2014/3/14

N2 - 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, div (a(x) grad 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.

AB - 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, div (a(x) grad 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.

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U2 - 10.1007/s00220-014-1943-y

DO - 10.1007/s00220-014-1943-y

M3 - Article

AN - SCOPUS:84895855605

VL - 328

SP - 1

EP - 27

JO - Communications in Mathematical Physics

JF - Communications in Mathematical Physics

SN - 0010-3616

IS - 1

ER -