Classification of spectra of the Neumann–Poincaré operator on planar domains with corners by resonance

Helsing, Johan; Kang, Hyeonbae; Lim, Mikyoung

Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 4, pp. 991-1011.

Nous étudions les propriétés spectrales de l'opérateur Neumann–Poincaré sur les domaines planaires avec coins. Un accent particulier est mis sur l'existence d'un spectre continu et d'un point isolé du spectre. Nous montrons que le taux de résonance du spectre continu est différent de celui des valeurs propres. Nous dérivons ensuite une méthode pour distinguer spectre continu et valeurs propres. Nous effectuons des expériences numériques afin de voir si le spectre continu et les valeurs propres apparaissent pour des domaines avec coins. Pour les calculs, nous utilisons une modification de la méthode de Nyström. Elle permet la discrétisation convergente de l'opérateur Neumann–Poincaré d'ordre élevé sur des domaines avec coins. Les résultats des expériences montrent que tous les trois spectres possibles, spectre absolument continu, spectre singulier et point isolé du spectre, peuvent apparaître en fonction des domaines. Nous montrons aussi rigoureusement deux propriétés spectrales qui sont suggérées par des expériences numériques : la symétrie du spectre (y compris spectre continu), et de l'existence des valeurs propres sur des rectangles ayant des rapports d'aspect élevés.

We study spectral properties of the Neumann–Poincaré operator on planar domains with corners with particular emphasis on existence of continuous spectrum and pure point spectrum. We show that the rate of resonance at continuous spectrum is different from that at eigenvalues, and then derive a method to distinguish continuous spectrum from eigenvalues. We perform computational experiments using the method to see whether continuous spectrum and pure point spectrum appear on domains with corners. For the computations we use a modification of the Nyström method which makes it possible to construct high-order convergent discretizations of the Neumann–Poincaré operator on domains with corners. The results of experiments show that all three possible spectra, absolutely continuous spectrum, singularly continuous spectrum, and pure point spectrum, may appear depending on domains. We also prove rigorously two properties of spectrum which are suggested by numerical experiments: symmetry of spectrum (including continuous spectrum), and existence of eigenvalues on rectangles of high aspect ratio.

Classification : 35P05, 45B05
Mots clés : Neumann–Poincaré operator, Lipschitz domain, Spectrum, RCIP method, Resonance

[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., Volume 208 (2013), pp. 667–692 | DOI | MR | Zbl

[2] Ammari, H.; Kang, H. Polarization and Moment Tensors with Applications to Inverse Problems and Effective Medium Theory, Applied Mathematical Sciences, vol. 162, Springer-Verlag, New York, 2007 | MR | Zbl

[3] Atkinson, K.E. The Numerical Solution of Integral Equations of the Second Kind, Cambridge University Press, Cambridge, 1997 | DOI | MR | Zbl

[4] Carleman, T. Über das Neumann–Poincarésche Problem für ein Gebiet mit Ecken, Almquist and Wiksells, Uppsala, 1916 | JFM

[5] Folland, G.B. Introduction to Partial Differential Equations, Princeton Univ. Press, Princeton, 1995 | MR | Zbl

[6] Helsing, J. The effective conductivity of arrays of squares: large random unit cells and extreme contrast ratios, J. Comput. Phys., Volume 230 (2011), pp. 7533–7547 | DOI | Zbl

[7] Helsing, J. Solving integral equations on piecewise smooth boundaries using the RCIP method: a tutorial, Abstr. Appl. Anal., Volume 2013 (2013) | DOI | MR | Zbl

[8] Helsing, J.; Karlsson, A. Determination of normalized electric eigenfields in microwave cavities with sharp edges, J. Comput. Phys., Volume 304 (2016), pp. 465–486 | DOI | MR | Zbl

[9] Helsing, J.; McPhedran, R.C.; Milton, G.W. Spectral super-resolution in metamaterial composites, New J. Phys., Volume 13 (2011) | DOI

[10] Helsing, J.; Ojala, R. Corner singularities for elliptic problems: integral equations, graded meshes, quadrature, and compressed inverse preconditioning, J. Comput. Phys., Volume 227 (2008), pp. 8820–8840 | DOI | MR | Zbl

[11] Helsing, J.; Perfekt, K. On the polarizability and capacitance of the cube, Appl. Comput. Harmon. Anal., Volume 34 (2013), pp. 445–468 | DOI | MR | Zbl

[12] Kang, H. Layer potential approaches to interface problems, Inverse Problems and Imaging, Panoramas et Syntheses, vol. 44, Soc. Math. France, 2015 | MR | Zbl

[13] Kang, H.; Kim, K.; Lee, H.; Shin, J.; Yu, S. Spectral properties of the Neumann–Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients, J. Lond. Math. Soc., Volume 93 (2016), pp. 519–546 | DOI | MR | Zbl

[14] Kang, H.; Lim, M.; Yu, S. Spectral resolution of the Neumann–Poincaré operator on intersecting disks and analysis of plasmon resonance | arXiv | DOI | Zbl

[15] Kellogg, O.D. Foundations of Potential Theory, Dover, New York, 1953 | MR | Zbl

[16] Khavinson, D.; Putinar, M.; Shapiro, H.S. Poincaré's variational problem in potential theory, Arch. Ration. Mech. Anal., Volume 185 (2007), pp. 143–184 | DOI | MR | Zbl

[17] Lei, D.Y.; Aubry, A.; Luo, Y.; Maier, S.A.; Pendry, J.B. Plasmonic interaction between overlapping nanowires, ACS Nano, Volume 5 (2011) no. 1, pp. 597–607

[18] Lim, M. Symmetry of a boundary integral operator and a characterization of a ball, Ill. J. Math., Volume 45 (2001), pp. 537–543 | MR | Zbl

[19] Mayergoyz, I.D.; Fredkin, D.R.; Zhang, Z. Electrostatic (plasmon) resonances in nanoparticles, Phys. Rev. B, Volume 72 (2005) | DOI

[20] Milton, G.W.; Nicorovici, N.-A.P. On the cloaking effects associated with anomalous localized resonance, Proc. R. Soc. A, Volume 462 (2006), pp. 3027–3059 | DOI | MR | Zbl

[21] Miyanishi, Y.; Suzuki, T. Eigenvalues and eigenfunctions of double layer potentials, Trans. Am. Math. Soc. (2016) (in press) | arXiv | MR

[22] Perfekt, K.; Putinar, M. Spectral bounds for the Neumann–Poincaré operator on planar domains with corners, J. Anal. Math., Volume 124 (2014), pp. 39–57 | DOI | MR | Zbl

[23] Perfekt, K.; Putinar, M. The essential spectrum of the Neumann–Poincaré operator on a domain with corners | arXiv | DOI | Zbl

[24] Reed, M.; Simon, B. Methods of Modern Mathematical Physics. I. Functional Analysis, Academic Press, New York, 1980 | Zbl

[25] Schiffer, M. The Fredholm eigenvalues of plane domains, Pac. J. Math., Volume 7 (1957), pp. 1187–1225 | DOI | MR | Zbl

[26] Yosida, K. Functional Analysis, Springer, Berlin, 1974 | MR