Archive for Rational Mechanics and Analysis

, Volume 234, Issue 2, pp 777–855 | Cite as

Homogenization of the Eigenvalues of the Neumann–Poincaré Operator

  • Éric Bonnetier
  • Charles DapognyEmail author
  • Faouzi Triki


In this article, we investigate the spectrum of the Neumann–Poincaré operator \({{\mathcal {K}}}_\varepsilon ^*\) (or equivalently, that of the associated Poincaré variational operator \(T_\varepsilon \)) associated to a periodic distribution of small inclusions with size \(\varepsilon \), and its asymptotic behavior as the parameter \(\varepsilon \) vanishes. Combining techniques pertaining to the fields of homogenization and potential theory, we prove that the limit spectrum is composed of the ‘trivial’ eigenvalues 0 and 1, and of a subset which stays bounded away from 0 and 1 uniformly with respect to \(\varepsilon \). This non trivial part is the reunion of the Bloch spectrum, accounting for the collective resonances between collections of inclusions, and of the boundary layer spectrum, associated to eigenfunctions which spend a not too small part of their energies near the boundary of the macroscopic device. These results shed new light on the issue of the homogenization of the voltage potential \(u_\varepsilon \) caused by a given source in a medium composed of a periodic distribution of small inclusions with an arbitrary (possible negative) conductivity a, surrounded by a dielectric medium, with unit conductivity. In particular, we prove that the limit behavior of \(u_\varepsilon \) is strongly related to the (possibly ill-defined) homogenized diffusion matrix predicted by the homogenization theory in the standard elliptic case. Additionally, we prove that the homogenization of \(u_\varepsilon \) is always possible when a is either positive, or negative with a ‘small’ or ‘large’ modulus.



The authors were partially supported by the AGIR-HOMONIM grant from Université Grenoble-Alpes, and by the Labex PERSYVAL-Lab (ANR-11-LABX-0025-01).

Supplementary material


  1. 1.
    Adams, R.A., Fournier, J.F.: Sobolev Spaces, 2nd edn. Academic Press, London 2003zbMATHGoogle Scholar
  2. 2.
    Aguirre, F., Conca, C.: Eigenfrequencies of a tube bundle immersed in a fluid. Appl. Math. Optim. 18, 1–38, 1988MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23(6), 1482–1518, 1992MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Allaire, G.: Shape Optimization by the Homogenization Method. Springer, New York 2001zbMATHGoogle Scholar
  5. 5.
    Allaire, G., Briane, M., Vanninathan, M.: A comparison between two-scale asymptotic expansions and Bloch wave expansions for the homogenization of periodic structures. SEMA J. 73(3), 237–259, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Allaire, G., Conca, C.: Bloch wave homogenization for a spectral problem in fluid–solid structures. Arch. Ration. Mech. Anal. 135, 197–257, 1996MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Allaire, G., Conca, C.: Bloch wave homogenization and spectral asymptotic analysis. J. Math. Pures Appl. 77, 153–208, 1998MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    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, 667–692, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Ammari, H., Ciraolo, G., Kang, H., Lee, H., Milton, G.W.: Spectral analysis of a Neumann–Poincaré-type operator and analysis of cloaking due to anomalous localized resonance II. Contemp. Math. 615, 1–14, 2014CrossRefzbMATHGoogle Scholar
  10. 10.
    Ammari, H., Deng, Y., Millien, P.: Surface plasmon resonance of nanoparticles and applications in imaging. Arch. Ration. Mech. Anal. 220, 109–153, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Ammari, H., Kang, H.: Polarization and Moment Tensors; With Applications to Inverse Problems and Effective Medium Theory. Springer Applied Mathematical Sciences, vol. 162, 2007Google Scholar
  12. 12.
    Ammari, H., Kang, H., Lee, H.: Layer Potential Techniques in Spectral Analysis, Mathematical Surveys and Monographs, vol. 153. American Mathematical Society, Providence RI 2009CrossRefGoogle Scholar
  13. 13.
    Ammari, H., Kang, H., Lim, M.: Gradient estimates for solutions to the conductivity problem. Math. Ann. 332, 277–286, 2005MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Ammari, H., Millien, P., Ruiz, M., Zhang, H.: Mathematical analysis of plasmonic nanoparticles: the scalar case, 2016. arXiv:1506.00866
  15. 15.
    Ammari, H., Ruiz, M., Yu, S., Zhang, H.: Mathematical analysis of plasmonic resonances for nanoparticles: the full Maxwell equations, 2016. arXiv:1511.06817
  16. 16.
    Ammari, H., Kang, H., Touibi, K.: Boundary layer techniques for deriving the effective properties of composite materials. Asymptot. Anal. 41(2), 119–140, 2005MathSciNetzbMATHGoogle Scholar
  17. 17.
    Ammari, H., Seo, J.K.: An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30(4), 679–705, 2003MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Ando, K., Kang, H.: Analysis of plasmon resonance on smooth domains using spectral properties of the Neumann–Poincaré operator. J. Math. Anal. Appl. 435, 162–178, 2016MathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Arbogast Jr., T., Douglas, J., Hornung, U.: Derivation of the double porosity model of single phase flow via homogenization theory. SIAM J. Math. Anal. 21(4), 823–836, 1990MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Bao, E.S., Li, Y., Yin, B.: Gradient estimates for the perfect conductivity problem. Arch. Ration. Mech. Anal. 193, 195–226, 2009MathSciNetCrossRefzbMATHGoogle Scholar
  21. 21.
    Bensoussan, A., Lions, J.-L., Papanicolau, G.: Asymptotic Analysis of Periodic Structures. North Holland, Amsterdam 1978Google Scholar
  22. 22.
    Bonnet-Ben Dhia, A.-S., Ciarlet Jr., P., Zwölf, C.-M.: Time harmonic wave diffraction problems in materials with sign-shifting coefficients. J. Comput. Appl. Math. 234, 1912–1919, 2007. Corrigendum 2616 (2010)Google Scholar
  23. 23.
    Bonnetier, E., Nguyen, H.-M.: Superlensing using hyperbolic metamaterials: the scalar case. J de l’École polytechnique - Mathématiques 4(2017), 973–1003, 2017MathSciNetCrossRefzbMATHGoogle Scholar
  24. 24.
    Bonnetier, E., Triki, F.: Pointwise bounds on the gradient and the spectrum of the Neumann–Poincaré operator: the case of 2 discs. Contemp. Math. 577, 81–92, 2012CrossRefzbMATHGoogle Scholar
  25. 25.
    Bonnetier, E., Triki, F.: On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2d. Arch. Ration. Mech. Anal. 209, 541–567, 2013MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Bonnetier, E., Triki, F., Tsou, C.H.: Eigenvalues of the Neumann-Poincaré operator for two inclusions with contact of order m: a numerical study. J. Comput. Math. 36, 17–28, 2018MathSciNetCrossRefzbMATHGoogle Scholar
  27. 27.
    Bouchitté, G., Schweizer, B.: Cloaking of small objects by anomalous localized resonance. Q. J. Mech. Appl. Math. 63, 437–463, 2010MathSciNetCrossRefzbMATHGoogle Scholar
  28. 28.
    Braides, A., Briane, M., Casado-Diaz, J.: Homogenization of non-uniformly bounded periodic diffusion energies in dimension two. Nonlinearity 22, 1459–1480, 2009ADSMathSciNetCrossRefzbMATHGoogle Scholar
  29. 29.
    Briane, M., Casado-Diaz, J.: Uniform convergence of sequences of solutions of two-dimensional linear elliptic equations with unbounded coefficients. J. Differ. Equ. 245, 2038–2054, 2008ADSMathSciNetCrossRefzbMATHGoogle Scholar
  30. 30.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer, Berlin 2000Google Scholar
  31. 31.
    Bunoiu, R., Ramdani, K.: Homogenization of materials with sign changing coefficients, 2015 (submitted)Google Scholar
  32. 32.
    Castro, C., Zuazua, E.: Une remarque sur l’analyse asymptotique spectrale en homogénéisation. C. R. Acad. Sci. Paris Ser. I 335, 99–104, 2002CrossRefGoogle Scholar
  33. 33.
    Cioranescu, D., Damlamian, A., Griso, G.: Periodic unfolding and homogenization. C. R. Acad. Sci. Paris Ser. I 322, 1043–1047, 1996zbMATHGoogle Scholar
  34. 34.
    Cioranescu, D., Damlamian, A., Griso, G.: The periodic unfolding method in homogenization. SIAM J. Math. Anal. 40(4), 1585–1620, 2008MathSciNetCrossRefzbMATHGoogle Scholar
  35. 35.
    Cioranescu, D., Damlamian, A., Li, T.: Periodic homogenization for inner boundary conditions with equi-valued surfaces: the unfolding approach. In: Partial Differential Equations: Theory, Control and Approximation. pp 183–209. Springer, Berlin, Heidelberg 2014Google Scholar
  36. 36.
    Conca, C., Planchard, J., Vanninathan, M.: Fluids and Periodic Structures, RMA, 38. Wiley & Masson, London 1995zbMATHGoogle Scholar
  37. 37.
    Conca, C., Vanninathan, M.: Homogenization of periodic structures via Bloch decomposition. SIAM J. Appl. Math. 57, 1639–1659, 1997MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Costabel, M., Stephan, E.: A direct boundary integral equation method for transmission problems. J. Math. Anal. Appl. 106, 367–413, 1985MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    El-Sayed, I.H., Huang, X., El-Sayed, M.A.: Surface plasmon resonance scattering and absorption of anti-EGFR antibody conjugated gold nanoparticles in cancer diagnostics: applications in oral cancer. Nano Lett. 5(5), 829–834, 2005ADSCrossRefGoogle Scholar
  40. 40.
    Folland, G.B.: Introduction to Partial Differential Equations, 2nd edn. Princeton University Press, Princeton 1995zbMATHGoogle Scholar
  41. 41.
    Gérard, P.: Mesures semi-classiques et ondes de Bloch. Séminaire Équations aux Dérivées Partielles 1990–1991, volume 16, Ecole Polytechnique, Palaiseau, 1991Google Scholar
  42. 42.
    Grieser, D.: The plasmonic eigenvalue problem. Rev. Math. Phys. 26, 1450005, 2014MathSciNetCrossRefzbMATHGoogle Scholar
  43. 43.
    Griso, G.: Analyse asymptotique de structures réticulées. Thèse de l’Université Pierre et Marie Curie (Paris VI), 1996Google Scholar
  44. 44.
    Jikov, V.V., Kozlov, S.M., Oleinik, O.A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin 1994CrossRefGoogle Scholar
  45. 45.
    John, F.: The Dirichlet problem for a hyperbolic equation. Am. J. Math. 63(1), 141–154, 1941MathSciNetCrossRefzbMATHGoogle Scholar
  46. 46.
    Kang, H.: Layer potential approaches to interface problems. Inverse Problems and Imaging: Panoramas et synthèses, 44. Société Mathématique de France, 2013Google Scholar
  47. 47.
    Khavinson, D., Putinar, M., Shapiro, H.S.: On Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185, 143–184, 2007MathSciNetCrossRefzbMATHGoogle Scholar
  48. 48.
    Kohn, R.V., Milton, G.W.: On bounding the effective conductivity of anisotropic composites. Homogenization and Effective Moduli of Materials and Media, Vol. 1 IMA Volumes in Mathematics and Its Applications (Eds. Ericksen, J.L., Kinderlehrer, D., Kohn, R., Lions, J.-L.) Springer, Berlin, 97–125, 1986Google Scholar
  49. 49.
    Kuchment, P.: Floquet Theory for Partial Differential Equations. Birkhäuser, Basel 1993CrossRefzbMATHGoogle Scholar
  50. 50.
    Lipton, R., Viator, R.: Bloch waves in crystals and periodic high contrast media, 2016 (submitted)Google Scholar
  51. 51.
    Maier, S.A.: Plasmonics: Fundamentals and Applications. Springer, Berlin 2007CrossRefGoogle Scholar
  52. 52.
    Manley, P., Burger, S., Schmidt, F., Schmid, M.: Design principles for plasmonic nanoparticle devices. Progress in Nonlinear Nano-Optics Part of the Series Nano-Optics and Nanophotonics, 223–247, 2015Google Scholar
  53. 53.
    Mc Lean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge 2000Google Scholar
  54. 54.
    Moskow, S., Vogelius, M.S.: First-order corrections to the homogenised eigenvalues of a periodic composite medium. A convergence proof. Proc. R. Soc. Edinb. 127, 1263–1299, 1997MathSciNetCrossRefzbMATHGoogle Scholar
  55. 55.
    Nicorovici, N.A., McPhedran, R.C., Milton, G.M.: Optical and dielectric properties of partially resonant composites. Phys. Rev. B 49, 8479–8482, 1994ADSCrossRefGoogle Scholar
  56. 56.
    Nguyen, H.-M.: Cloaking using complementary media in the quasistatic regime. Ann. I. H. Poincaré (C) Non Linear Anal. 32, 471–484, 2015ADSCrossRefGoogle Scholar
  57. 57.
    Nguyen, H.-M.: Negative index materials and their applications: recent mathematics progress. Chin. Ann. Math. 2016. Google Scholar
  58. 58.
    Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal 20(3), 608–623, 1989MathSciNetCrossRefzbMATHGoogle Scholar
  59. 59.
    Planchard, J.: Global behaviour of large elastic tube bundles immersed in a fluid. Comput. Mech. 2, 105–118, 1987CrossRefzbMATHGoogle Scholar
  60. 60.
    Otomori, M., Yamada, T., Izui, K., Nishiwaki, S., Andkjær, J.: Topology optimization of hyperbolic metamaterials for an optical hyperlens. Struct. Multidiscip. Optim. 2016.
  61. 61.
    Patching, S.G.: Surface plasmon resonance spectroscopy for characterisation of membrane protein–ligand interactions and its potential for drug discovery. Biochim. Biophys. Acta (BBA) - Biomembr. 1838(1), 43–55, 2014CrossRefGoogle Scholar
  62. 62.
    Poddubny, A., Iorsh, I., Belov, P., Kivshar, Y.: Hyperbolic metamaterials. Nat. Photon. 7, 948–957, 2013ADSCrossRefGoogle Scholar
  63. 63.
    Reed, M., Simon, B.: Methods of Modern Mathematical Physics, IV, Analysis of operators. Academic Press, New York 1978zbMATHGoogle Scholar
  64. 64.
    Rudin, W.: Functional Analysis, 2nd edn. International Series in Pure and Applied MathematicsMcGraw-Hill, New York, NY 1991zbMATHGoogle Scholar
  65. 65.
    Sauter, S.A., Schwab, C.: Boundary Element Methods. Springer Series in Computational Mathematics, Springer, GmbH & Co. K, Berlin, Heidelberg 2010Google Scholar
  66. 66.
    Shekhar, P., Atkinson, J., Jacob, Z.: Hyperbolic metamaterials: fundamentals and applications. Nano Converg. 1, 1–14, 2014CrossRefGoogle Scholar
  67. 67.
    Triki, F., Vauthrin, M.: Mathematical modeling of the Photoacoustic effect generated by the heating of metallic nanoparticles. Q. Appl. Math. 76, 673–698, 2018MathSciNetCrossRefzbMATHGoogle Scholar
  68. 68.
    Wilcox, C.: Theory of Bloch waves. J. Anal. Math. 33, 146–167, 1978MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Institut FourierUniversité Grenoble AlpesGrenoble Cedex 9France
  2. 2.Laboratoire Jean KuntzmannUniversité Grenoble AlpesGrenoble Cedex 9France

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