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A Variational Perspective on Cloaking by Anomalous Localized Resonance

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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, \({{\rm div} (a(x) {\rm grad}\, u(x)) = f(x)}\). 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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References

  1. Ammari H., Ciraolo G., Kang H., Lee H., Milton G.W.: Spectral theory of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208(2), 667–692 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  2. Bouchitté G., Schweizer B.: Cloaking of small objects by anomalous localized resonance. Q. J. Mech. Appl. Math. 63(4), 437–463 (2010)

    Article  MATH  Google Scholar 

  3. Bouchitté G., Schweizer B.: Homogenization of Maxwell’s equations in a split ring geometry. Multiscale Model. Simul. 8(3), 717–750 (2010)

    Article  MATH  MathSciNet  Google Scholar 

  4. Bruno, O., Lintner, S.: Superlens-cloaking of small dielectric bodies in the quasistatic regime. J. Appl. Phys. 102(12), Art. No. 124502 (2007)

    Google Scholar 

  5. Cherkaev A.V., Gibiansky L.V.: Variational principles for complex conductivity, viscoelasticity, and similar problems in media with complex moduli. J. Math. Phys. 35(1), 127–145 (1994)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  6. Evans, L.C.: Partial Differential Equations, 2nd edn. Graduate Studies in Mathematics, Vol 19. Providence: AMS (2010)

  7. Grieser, D.: The plasmonic eigenvalue problem (2012). arXiv:1208.3120v1[math-ph]

  8. Milton G.W., Nicorovici N.-A.P.: On the cloaking effects associated with anomalous localized resonance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 462, 3027–3059 (2006)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  9. Milton G.W., Nicorovici N.-A.P., McPhedran R.C.: Opaque perfect lenses. Physica B 394, 171–175 (2007)

    Article  ADS  Google Scholar 

  10. Milton G.W., Nicorovici N.-A.P., McPhedran R.C., Podolskiy V.A.: A proof of superlensing in the quasistatic regime, and limitations of superlenses in this regime due to anomalous localized resonance. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 461(2064), 3999–4034 (2005)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  11. Milton G.W., Seppecher P., Bouchitté G.: Minimization variational principles for acoustics, elastodynamics and electromagnetism in lossy inhomogeneous bodies at fixed frequency. Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci. 465(2102), 367–396 (2009)

    Article  ADS  MATH  MathSciNet  Google Scholar 

  12. Nguyen, H.-M.: A study of negative index materials using transformation optics with applications to super lenses, cloaking, and illusion optics: the scalar case (2012). arXiv:1204.1518V1[math-ph]

  13. Nicorovici N.A., McPhedran R.C., Milton G.W.: Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49, 8479–8482 (1994)

    Article  ADS  Google Scholar 

  14. Smith D.R., Pendry J.B., Wiltshire M.C.K.: Metamaterials and negative refractive index. Science 305, 788–792 (2004)

    Article  ADS  Google Scholar 

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Authors and Affiliations

  1. Courant Institute of Mathematical Sciences, New York University, 251 Mercer Street, New York, NY, 10012, USA

    R. V. Kohn

  2. Mathematics Department, Duke University, Box 90320, Durham, NC, 27708, USA

    J. Lu

  3. Fakultät für Mathematik, Technische Universität Dortmund, Vogelpothsweg 87, 44227, Dortmund, Germany

    B. Schweizer

  4. Department of Applied Physics and Applied Mathematics, Columbia University, New York, NY, 10027, USA

    M. I. Weinstein

Authors
  1. R. V. Kohn
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  2. J. Lu
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  3. B. Schweizer
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  4. M. I. Weinstein
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Correspondence to R. V. Kohn.

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Communicated by L. Caffarelli

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Kohn, R.V., Lu, J., Schweizer, B. et al. A Variational Perspective on Cloaking by Anomalous Localized Resonance. Commun. Math. Phys. 328, 1–27 (2014). https://doi.org/10.1007/s00220-014-1943-y

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  • DOI: https://doi.org/10.1007/s00220-014-1943-y

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