.. .. Résultats, 7.1 Calcul numérique des valeurs propres de l'opérateur de Poincaré-Neumann dans un domaine borné, p.115

, Calcul numérique des valeurs propres de l'opérateur de Poincaré-Neumann dans un domaine non borné avec un opérateur de Dirichlet-to-Neumann

F. Abdmouleh, S. Charfi, A. Jeribi-;-habib-ammari, G. Ciraolo, H. Kang et al., Spectral theory of a Neumann-Poincaré-type operator and analysis of cloaking due to anomalous localized resonance, Journal of Mathematical Analysis and Applications, vol.386, issue.1, pp.667-692, 2012.

. Chérif, . Amrouche, . Vivette, J. Girault, and . Giroire, Dirichlet and Neumann exterior problems for the n-dimensional Laplace operator an approach in weighted Sobolev spaces, Journal de Mathématiques Pures et Appliquées, vol.76, issue.1, pp.55-81, 1997.

H. Ammari and H. Kang, Polarization and moment tensors: with applications to inverse problems and effective medium theory, Applied Mathematical Sciences, vol.162, 2007.

H. Habib-ammari, K. Kang, and . Touibi, Boundary layer techniques for deriving the effective properties of composite materials, Asymptotic Analysis, vol.41, issue.2, pp.119-140, 2005.

H. Ammari, P. Millien, M. Ruiz, and H. Zhang, Mathematical analysis of plasmonic nanoparticles: the scalar case. Archive for Rational Mechanics and, Analysis, vol.224, issue.2, pp.597-658, 2017.

M. Arw-+-16]-habib-ammari, W. Ruiz, S. Wu, H. Yu, and . Zhang, Mathematical and numerical framework for metasurfaces using thin layers of periodically distributed plasmonic nanoparticles, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, vol.472, 2016.

H. Brezis, P. G. Ciarlet, and J. L. Lions, Analyse fonctionnelle: théorie et applications, vol.91, 1999.

E. Bonnetier, C. Dapogny, and F. Triki, Homogenization of the eigenvalues of the Neumann-Poincaré operator, 2017.

L. Blanchet and G. Faye, Hadamard regularization, Journal of Mathematical Physics, vol.41, issue.11, pp.7675-7714, 2000.

G. Bouchitté and D. Felbacq, Homogenization near resonances and artificial magnetism from dielectrics, Comptes Rendus Mathématique, vol.339, issue.5, pp.377-382, 2004.

H. Brooks and M. Möller, Spectra of multiplication operators in Sobolev spaces, Results in Mathematics, vol.55, issue.3-4, pp.281-293, 2009.

I. Babu?ka and J. Osborn, Eigenvalue problems. Handbook of numerical analysis, vol.2, pp.641-787, 1991.

D. Boffi, Finite element approximation of eigenvalue problems, Acta Numerica, vol.19, pp.1-120, 2010.

B. Tahar-zamène, Inverted finite elements: a new method for solving elliptic problems in unbounded domains, 2005.

H. Brezis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, 2010.

E. Bonnetier and F. Triki, On the spectrum of the Poincaré variational problem for two close-to-touching inclusions in 2D. Archive for Rational Mechanics and Analysis, vol.209, pp.541-567, 2013.

E. Bonnetier, F. Triki, and C. Tsou, Eigenvalues of the Neumann-Poincaré operator for two inclusions, 2016.

Y. Chan, A. C. Fannjiang, H. Glaucio, B. Paulino, and . Feng, Finite part integrals and hypersingular kernels, Adv. Dyn. Syst, vol.14, issue.S2, pp.264-269, 2007.

H. Snorre and . Christiansen, Résolution des équations intégrales pour la diffraction d'ondes acoustiques et électromagnétiques-Stabilisation d'algorithmes itératifs et aspects de l'analyse numérique, 2002.

T. Charles, G. Campbell, and . Kim, Spr microscopy and its applications to high-throughput analyses of biomolecular binding events and their kinetics, Biomaterials, vol.28, issue.15, pp.2380-2392, 2007.

R. R. Coifman, A. Mcintosh, and Y. Meyer, L'intégrale de Cauchy définit un opérateur borné sur L 2 pour les courbes lipschitziennes, vol.116, pp.361-387, 1982.

M. Costabel, Boundary integral operators on Lipschitz domains: Elementary results, SIAM Journal on Mathematical Analysis, vol.19, issue.3, pp.613-626, 1988.

M. Costabel, On the spectrum of volume integral operators in acoustic scattering, Integral Methods in Science and Engineering, pp.119-127, 2015.
URL : https://hal.archives-ouvertes.fr/hal-01098834

M. Costabel and E. Stephan, A direct boundary integral equation method for transmission problems, Journal of mathematical analysis and applications, vol.106, issue.2, pp.367-413, 1985.

M. Daa-+-99]-r.-dautray, J. C. Artola, J. L. Amson, P. Lions, M. Benilan et al., Mathematical Analysis and Numerical Methods for Science and Technology: Volume 4 Integral Equations and Numerical Methods. Mathematical Analysis and Numerical Methods for Science and Technology, Heidelberg, 1999.

A. Dhia, L. Chesnel, and P. Ciarlet, Tcoercivity for scalar interface problems between dielectrics and metamaterials, ESAIM: Mathematical Modelling and Numerical Analysis, vol.46, issue.06, pp.1363-1387, 2012.
URL : https://hal.archives-ouvertes.fr/hal-00564312

A. Bonnet-ben-dhia, C. Carvalho, and P. Ciarlet, Mesh requirements for the finite element approximation of problems with sign-changing coefficients, Numerische Mathematik, vol.138, issue.4, pp.801-838, 2018.
URL : https://hal.archives-ouvertes.fr/hal-01335153

A. Dhia, P. Ciarlet, and C. M. Zwölf, Time harmonic wave diffraction problems in materials with signshifting coefficients, Journal of Computational and Applied Mathematics, vol.234, issue.6, pp.1912-1919, 2010.
URL : https://hal.archives-ouvertes.fr/hal-00975073

D. Edmunds and D. Evans, Spectral theory and differential operators, 2018.

A. Ern and J. Guermond, Theory and practice of finite elements, vol.159, 2013.

C. Lawrence, R. F. Evans, and . Gariepy, Measure theory and fine properties of functions, 2015.

C. Thomas-w-ebbesen, . Genet, . Sergey, and . Bozhevolnyi, Surfaceplasmon circuitry, Physics Today, vol.61, issue.5, p.44, 2008.

P. G. Etchegoin, E. C. Le-ru, and M. Meyer, An analytic model for the optical properties of gold, The Journal of Chemical Physics, vol.125, issue.16, p.164705, 2006.

D. Felbacq and G. Bouchitté, Homogenization of a set of parallel fibres, Waves in Random Media, vol.7, pp.245-256, 1997.
URL : https://hal.archives-ouvertes.fr/hal-01283233

B. Gerald and . Folland, Introduction to partial differential equations, 1995.

D. Givoli, Numerical Methods for Problems in Infinite Domains, Studies in Applied Mechanics. Elsevier Science, 2013.

D. Givoli, I. Patlashenko, and J. Keller, Discrete Dirichlet-toNeumann maps for unbounded domains, Computer methods in applied mechanics and engineering, vol.164, issue.1-2, pp.173-185, 1998.

F. Hecht, New development in FreeFem++, J. Numer. Math, vol.20, issue.3-4, pp.251-265, 2012.
URL : https://hal.archives-ouvertes.fr/hal-01476313

P. Henrici, Discrete Fourier analysis, Cauchy integrals, construction of conformal maps, univalent functions, vol.3, 1993.

J. Homola, Surface Plasmon Resonance Based Sensors, Springer Series on Chemical Sensors and Biosensors, 2006.

A. Henrot and M. Pierre, Variation et optimisation de formes: une analyse géométrique, vol.48, 2006.

J. David and J. , , 1999.

P. B. Johnson and R. W. Christy, Optical constants of the noble metals, Phys. Rev. B, vol.6, pp.4370-4379, 1972.

H. Kang, The layer potential technique for the inverse conductivity problem, Inverse Problems, vol.12, issue.3, pp.267-278, 1996.

H. Kang, Layer potential approaches to interface problems. Inverse problems and imaging, 2015.

T. Kim-chuan and S. Mukherjee, Hypersingular and finite part integrals in the boundary element method, International Journal of Solids and Structures, vol.31, issue.17, pp.2299-2312, 1994.

K. Oliver-dimon, Foundations of Potential Theory. Grundlehren der mathematischen Wissenschaften, 2012.

R. Kress, V. Maz'ya, and V. Kozlov, Linear integral equations, Applied Mathematical Sciences, vol.17, 1989.

D. Khavinson, M. Putinar, and H. Shapiro, Poincaré's variational problem in potential theory. Archive for rational mechanics and analysis, vol.185, pp.143-184, 2007.

A. Kufner, Weighted Sobolev Spaces, 1985.

C. Luo, S. G. Johnson, J. D. Joannopoulos, and J. B. Pendry, Subwavelength imaging in photonic crystals, Phys. Rev. B, vol.68, p.45115, 2003.

G. Lumer and M. Rosenblum, Linear operator equations, Proceedings of the American Mathematical Society, vol.10, issue.1, pp.32-41, 1959.

S. A. Maier, Plasmonics -Fundamentals and Applications, 2007.

C. H. William and . Mclean, Strongly Elliptic Systems and Boundary Integral Equations, 2000.

M. Miniaci, A. Krushynska, F. Bosia, and N. M. Pugno, Large scale mechanical metamaterials as seismic shields, New Journal of Physics, vol.18, issue.8, p.83041, 2016.

Y. Maekawa and H. Miura, On Poisson operators and Dirichlet-Neumann maps in H s for divergence form elliptic operators with Lipschitz coefficients, vol.368, p.2013

J. Nédélec, Acoustic and electromagnetic equations: integral representations for harmonic problems, vol.144, 2001.

S. Nicaise and J. Venel, A posteriori error estimates for a finite element approximation of transmission problems with sign changing coefficients, Journal of Computational and Applied Mathematics, vol.235, issue.14, pp.4272-4282, 2011.
URL : https://hal.archives-ouvertes.fr/hal-00517989

J. Sophocles and . Orfanidis, Electromagnetic Waves and Antennas, 2016.

E. Ozbay, Plasmonics: merging photonics and electronics at nanoscale dimensions. science, vol.311, pp.189-193, 2006.

J. Plemelj, Problems in the sense of Riemann and Klein, 1964.

K. Perfekt and M. Putinar, Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. Journal d'Analyse Mathématique, vol.124, pp.39-57, 2014.

G. Alexander and . Ramm, A simple proof of the fredholm alternative and a characterization of the fredholm operators, The American Mathematical Monthly, vol.108, issue.9, pp.855-860, 2001.

M. Rosenblum, On the operator equation BX ? XA Q, Duke Math. J, vol.23, issue.2, p.1956

M. Reed and B. Simon, Methods of Modern Mathematical Physics. IV Analysis of Operators, 1978.

M. Vladimir, W. Shalaev, . Cai, K. Uday, H. Chettiar et al., Negative index of refraction in optical metamaterials, SCC + 05, vol.30, pp.3356-3358, 2005.

A. Sellier, Hadamard's finite part concept in dimension n ? 2, distributional definition, regularization forms and distributional derivatives, Proceedings: Mathematical and Physical Sciences, vol.445, pp.69-98, 1923.

M. Vladimir and . Shalaev, Optical negative-index metamaterials, Nature photonics, vol.1, issue.1, p.41, 2007.

. Smj-+-06]-david-schurig, . Mock, . Justice, A. Steven, J. B. Cummer et al., Metamaterial electromagnetic cloak at microwave frequencies, Science, vol.314, issue.5801, pp.977-980, 2006.

R. David, J. B. Smith, M. Pendry, and . Wiltshire, Metamaterials and negative refractive index, Science, vol.305, issue.5685, pp.788-792, 2004.

V. Siahpoush, T. Søndergaard, and J. Jung, Green's function approach to investigate the excitation of surface plasmon polaritons in a nanometer-thin metal film, Phys. Rev. B, vol.85, p.75305, 2012.

J. A. Stratton, Electromagnetic theory, 2007.

J. Sylvester and G. Uhlmann, The Dirichlet to Neumann map and applications. Inverse problems in partial differential equations, pp.101-139, 1990.

L. , R. Scott, and S. Zhang, Finite element interpolation of nonsmooth functions satisfying boundary conditions, Mathematics of Computation, vol.54, pp.483-493, 1990.

F. Triki and É. Bonnetier, Asymptotic of plasmonic resonances, 2016.

A. E. Taylor and D. C. Lay, Introduction to Functional Analysis, 1986.

F. Triki and M. Vauthrin, Mathematical modelization of the photoacoustic effect generated by the heating of metallic nanoparticles, Quarterly of Applied Mathematics, vol.01, p.2017

J. L. Volakis, C. Chen, and K. Fujimoto, Small antennas: miniaturization techniques & applications, vol.1, 2010.

A. Vial, A. Grimault, D. Macías, D. Barchiesi, and M. Lamy-de-la-chapelle, Improved analytical fit of gold dispersion: Application to the modeling of extinction spectra with a finite-difference time-domain method, VGM + 05, vol.71, p.85416, 2005.

C. M. Zwölf, Méthodes variationnelles pour la modélisation des problèmes de transmission d'onde électromagnétique entre diélectrique et méta-matériau, 2008.