Hostname: page-component-78c5997874-fbnjt Total loading time: 0 Render date: 2024-11-11T08:38:58.813Z Has data issue: false hasContentIssue false
  • English
  • Français

Composites with invisible inclusions: Eigenvalues of ℝ-linear problem

Published online by Cambridge University Press:  31 March 2016

V. V. MITYUSHEV
V. V. MITYUSHEV*
Affiliation:
Department of Computer Science and Computational Methods, Pedagogical University, Krakow, Poland e-mail: mityu@up.krakow.pl
Get access Rights & Permissions [Opens in a new window]

Abstract

A new eigenvalue ℝ-linear problem arisen in the theory of metamaterials and neutral inclusions is reduced to integral equations. The problem is constructively investigated for circular non-overlapping inclusions. An asymptotic formula for eigenvalues is deduced when the radii of inclusions tend to zero. The nodal domains conjecture related to univalent eigenfunctions is posed. Demonstration of the conjecture allows to justify that a set of inclusions can be made neutral by surrounding it with an appropriate coating.

Keywords

invisible inclusionsℝ-linear problemmetamaterialsunivalent eigenfunction
Type
Papers
Information
European Journal of Applied Mathematics , Volume 27 , Issue 5 , October 2016 , pp. 796 - 806
Copyright
Copyright © Cambridge University Press 2016 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

[1] Alu, A. & Engheta, N. (2005) Achieving transparency with plasmonic and metamaterial coatings. Phys. Rev. E 72, 016623.Google Scholar
[2] Ammari, H., Kang, H., Lee, H. & Lim, M. (2013) Enhancement of near cloaking using generalized polarization tensors vanishing structures. Part I: The conductivity problem. Comm. Math. Phys. 317, 253–266.Google Scholar
[3] Ammari, H., Ciraolo, G., Kang, H., Lee, H. & Milton, G. W. (2013) Spectral theory of a Neumann-Poincaree-type operator and analysis of cloaking due to anomalous localized resonance. Arch. Ration. Mech. Anal. 208, 667–692 Google Scholar
[4] Asatryan, A. A., Botten, L. C., Fang, K., Fan, S. & McPhedran, R. C. (2014) Two-dimensional Green's tensor for gyrotropic clusters composed of circular cylinders. J. Opt. Soc. Am. A Opt. Image Sci. Vis. 31, 2294–303.CrossRefGoogle ScholarPubMed
[5] Banuelos, R., Kulczycki, T., Polterovich, I. & Siudeja, B. (2010) Eigenvalue inequalities for mixed Steklov problems. In “Operator Theory and Its Applications: In Memory of V. B. Lidskii (1924–2008)'' edited by M. Levitin, D. Vassiliev, AMS Providence, Rhode Island, pp. 19–34.Google Scholar
[6] Bogatyrev, A. (2000) Poincaré-Steklov integral equations and the Riemann monodromy problem. Funct. Anal. Appl. 34, 86–97.Google Scholar
[7] Bojarski, B. (1960) On generalized Hilbert boundary-value problem. Soobsch. Akad. Nauk Gruz. SSR 25, 385–390.Google Scholar
[8] Bojarski, B. & Mityushev, V. (2013) R-linear problem for multiply connected domains and alternating method of Schwarz. J. Math. Sci. 189, 68–77.Google Scholar
[9] Chavel, I. (1984) Eigenvalues in Riemannian Geometry, London, Academic Press, pp. 7–24.Google Scholar
[10] Colquitt, D. J., Jones, I. S., Movchan, N. V., Movchan, A. B., Brun, M. & McPhedran, R. C. (2013) Making waves round a structured cloak: Lattices, negative refraction and fringes. Proc. Roy. Soc. A469, 20130218.Google Scholar
[11] Drygas, P. (2007) A functional-differential equation in a class of analytic functions and its application. Aequationes Math. 73, 222–232.Google Scholar
[12] Gakhov, F. D. (1977) Boundary Problems, Moscow, Nauka, pp. 37–43.Google Scholar
[13] Jarczyk, P. & Mityushev, V. (2011) Neutral coated inclusions of finite conductivity. Proc. R. Soc. Lond. A468, 954–970.Google Scholar
[14] Kang, H. & Lee, H. (2014) Coated inclusions of finite conductivity neutral to multiple fields in two dimensional conductivity or anti-plane elasticity. Euro. J. Appl. Math. 25, 329–338.Google Scholar
[15] Kang, H., Lee, H. & Sakaguchi, S. (2014) An over-determined boundary value problem arising from neutrally coated inclusions in three dimensions. Annali della Scuola Normale Superiore di Pisa, Classe di Scienze, to appear. arXiv 1501.07465Google Scholar
[16] Kalamkarov, A. L., Andrianov, I. V. & Danishevskyy, V. V. (2009) Asymptotic homogenization of composite materials and structures, Appl. Mech. Rev. 62, 030802-1–20.Google Scholar
[17] Kerker, M. (1975) Invisible bodies. J. Opt. Soc. Am. 65, 376–379.CrossRefGoogle Scholar
[18] Kolmogorov, A. N. & Fomin, S. V. (1999) Elements of the Theory of Functions and Functional Analysis, New York, Dover Publications, pp. 246–253.Google Scholar
[19] Lebedev, V. I. & Agoshkov, V. I. (1983) Poincaré - Steklov Operators and their Applications in Analysis, Moscow, Akad. Nauk SSSR, Vychisl. TCentr Publ, pp. 6–78.Google Scholar
[20] Milton, G. W. & Serkov, S. K. (2001) Neutral coated inclusions in conductivity and anti-plane elasticity. Proc. R. Soc. Lond. A457, 1973–1997.Google Scholar
[21] Mityushev, V. V. (1984) Boundary Value Problems and Functional Equations with Shifts in Domains, Byelorussian State University, Minsk, PhD Thesis.Google Scholar
[22] Mityushev, V. V. (1992) Eigenvalues of the ℝ-linear problems. Izvestia vuzov. Math. 11, 35–38.Google Scholar
[23] Mityushev, V. V. & Rogosin, S. V. (2000) Constructive Methods for Linear and Nonlinear Boundary Value Problems for Analytic Functions. Theory and Applications, Chapman & Hall / CRC, Boca Raton, London, New York, Washington DC, pp. 133–201.Google Scholar
[24] Mityushev, V. V. (2011) Riemann-Hilbert problems for multiply connected domains and circular slit maps. Comp. Meth. Func. Theory 11, 557–90.Google Scholar
[25] Mityushev, V. & Rylko, N. (2012) Optimal distribution of the non-overlapping conducting disks. Multiscale Model. Simul. 12, 180–90.Google Scholar
[26] O'Neill, J., Selsil, Ö., McPhedran, R. C., Movchan, A. B. & Movchan, N. V. (2015) Active cloaking of inclusions for flexural waves in thin elastic plates. Q. J. Mech. Appl. Math. published online June 25, doi:10.1093/qjmam/hbv007.Google Scholar
[27] Poulton, C. G., Movchan, A. B., Movchan, N. V. & McPhedran, R. C. (2012) Analytic theory of defects in periodically structured elastic plates. Proc. R. Soc. A471, 1196–16.Google Scholar
[28] Rylko, N. (2000) Transport properties of a rectangular array of highly conducting cylinders. J. Eng. Math. 38, 1–12.Google Scholar
[29] Rylko, N. (2008) Structure of the scalar field around unidirectional circular cylinders. Proc. R. Soc. A464, 391–407.Google Scholar
[30] Rylko, N. (2013) Effective anti-plane properties of piezoelectric fibrous composites. Acta Mech. 224, 2719–34.Google Scholar
[31] Rylko, N. (2015) Fractal local fields in random composites. Comput. Math. Appl. 69, 247–54.CrossRefGoogle Scholar
[32] Schiffer, M. (1959) Fredholm eigen values of multiply-connected domains. J. d'Anal. Math. 9, 211–69.Google Scholar