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Geometric series expansion of the Neumann–Poincaré operator: Application to composite materials

Part of: Two-dimensional theory Eigenvalue problems Partial differential equations Geometric function theory

Published online by Cambridge University Press:  11 May 2021

ELENA CHERKAEV ,
MINWOO KIM  and
MIKYOUNG LIM Open the ORCID record for MIKYOUNG LIM [Opens in a new window]
ELENA CHERKAEV
Affiliation:
Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA email: elena@math.utah.edu
MINWOO KIM
Affiliation:
School of Electrical Engineering, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea email: epsilon4b@kaist.ac.kr
MIKYOUNG LIM
Affiliation:
Department of Mathematical Sciences, Korea Advanced Institute of Science and Technology, Daejeon 34141, Republic of Korea email: mklim@kaist.ac.kr
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Abstract

The Neumann–Poincaré (NP) operator, a singular integral operator on the boundary of a domain, naturally appears when one solves a conductivity transmission problem via the boundary integral formulation. Recently, a series expression of the NP operator was developed in two dimensions based on geometric function theory [34]. In this paper, we investigate geometric properties of composite materials using this series expansion. In particular, we obtain explicit formulas for the polarisation tensor and the effective conductivity for an inclusion or a periodic array of inclusions of arbitrary shape with extremal conductivity, in terms of the associated exterior conformal mapping. Also, we observe by numerical computations that the spectrum of the NP operator has a monotonic behaviour with respect to the shape deformation of the inclusion. Additionally, we derive inequality relations of the coefficients of the Riemann mapping of an arbitrary Lipschitz domain using the properties of the polarisation tensor corresponding to the domain.

Keywords

Homogenisationintegral equations methods in two dimensionseigenvalues of operatorsconformal mappings

MSC classification

Primary: 35B27: Homogenization; equations in media with periodic structure 31A10: Integral representations, integral operators, integral equations methods 45C05: Eigenvalue problems
Secondary: 30C35: General theory of conformal mappings
Type
Papers
Information
European Journal of Applied Mathematics , Volume 33 , Issue 3 , June 2022 , pp. 560 - 585
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Copyright
© The Author(s), 2021. Published by Cambridge University Press

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References

Abramowitz, M. & Stegun, I. A. (1964) Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. National Bureau of Standards Applied Mathematics Series, Vol. 55. For sale by the Superintendent of Documents, US Government Printing Office, Washington, DC.Google Scholar
Ammari, H., Capdeboscq, Y., Kang, H., Kim, E. & Lim, M. (2006) Attainability by simply connected domains of optimal bounds for the polarization tensor. Eur. J. Appl. Math. 17(2), 201–219.CrossRefGoogle Scholar
Ammari, H., Ciraolo, G., Kang, H., Lee, H. & Milton, G. W. (2013) 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.CrossRefGoogle Scholar
Ammari, H., Garnier, J., Jing, W., Kang, H., Lim, M., Sølna, K. & Wang, H. (2013) Mathematical and Statistical Methods for Multistatic Imaging, Vol. 2098, Springer, Cham.CrossRefGoogle Scholar
Ammari, H. & Kang, H. (2004) Reconstruction of Small Inhomogeneities from Boundary Measurements, Vol. 1846, Springer, Berlin.CrossRefGoogle Scholar
Ammari, H. & Kang, H. (2007) Polarization and Moment Tensors. Applied Mathematical Sciences, Vol. 162, Springer, New York. With applications to inverse problems and effective medium theory.Google Scholar
Ammari, H., Kang, H. & Lee, H. (2007) A boundary integral method for computing elastic moment tensors for ellipses and ellipsoids. J. Comput. Math. 25(1), 2–12.Google Scholar
Ammari, H., Kang, H. & Lim, M. (2005) Polarization tensors and their applications. J. Phys. Conf. Ser. 12, 13–22.CrossRefGoogle Scholar
Ammari, H., Kang, H. & Lim, M. (2006) Effective parameters of elastic composites. Indiana Univ. Math. J. 55(3), 903–922.CrossRefGoogle Scholar
Ammari, H., Kang, H. & Touibi, K. (2005) Boundary layer techniques for deriving the effective properties of composite materials. Asymptot. Anal. 41(2), 119–140.Google Scholar
Ammari, H., Putinar, M., Ruiz, M., Yu, S. & Zhang, H. (2019) Shape reconstruction of nanoparticles from their associated plasmonic resonances. J. Math. Pures Appl. 122, 23–48.CrossRefGoogle Scholar
Ando, K., Kang, H. & Liu, H. (2016) Plasmon resonance with finite frequencies: a validation of the quasi-static approximation for diametrically small inclusions. SIAM J. Appl. Math. 76(2), 731–749.CrossRefGoogle Scholar
Ando, K., Kang, H. & Miyanishi, Y. (2018) Exponential decay estimates of the eigenvalues for the Neumann-Poincaré operator on analytic boundaries in two dimensions. J. Integr. Equations Appl. 30(4), 473–489.CrossRefGoogle Scholar
Ando, K., Kang, H., Miyanishi, Y. & Putinar, M. (2020) Spectral analysis of Neumann-Poincaré operator. arXiv preprint arXiv:2003.14387.Google Scholar
Bieberbach, L. (1916) Über die Koeffizienten derjenigen Potenzreihen, welche eine schlichte Abbildung des Einheitskreises vermitteln, Vol. 138. Preussische Akademie der Wissenschaften Berlin: Sitzungsberichte der Preußischen Akademie der Wissenschaften zu Berlin.Google Scholar
Blumenfeld, J. & Mayer, W. (1914) Über Poincarésche Fundamentalfunktionen. Sitz. Wien. Akad. Wiss. Math. Nat. Klasse Abt. IIa 122, 2011–2047.Google Scholar
Böttcher, A., Chithra, A. V. & Namboodiri, M. N. N. (2001) Approximation of approximation numbers by truncation. Integr. Equations Oper. Theory 39(4), 387–395.CrossRefGoogle Scholar
Capdeboscq, Y. & Vogelius, M. S. (2004) A review of some recent work on impedance imaging for inhomogeneities of low volume fraction. In: Partial Differential Equations and Inverse Problems. Contemporary Mathematics, Vol. 362, American Mathematical Society, Providence, RI, pp. 69–87.CrossRefGoogle Scholar
Carathéodory, C. (1913) Über die gegenseitige Beziehung der Ränder bei der konformen Abbildung des Inneren einer Jordanschen Kurve auf einen Kreis. Math. Ann. 73(2), 305–320.Google Scholar
Choi, D. S., Helsing, J. & Lim, M. (2018) Corner effects on the perturbation of an electric potential. SIAM J. Appl. Math. 78(3), 1577–1601.CrossRefGoogle Scholar
Choi, D., Kim, J. & Lim, M. (to appear) Analytical shape recovery of a conductivity inclusion based on Faber polynomials. Mathematische Annalen. doi: 10.1007/s00208-020-02041-1.CrossRefGoogle Scholar
Choi, D., Kim, J. & Lim, M. (2018) Geometric multipole expansion and its application to neutral inclusions of general shape. arXiv preprint arXiv:1808.02446.Google Scholar
Duren, P. L. (1983) Univalent Functions. Grundlehren der mathematischen Wissenschaften, Vol. 259, Springer-Verlag, New York.Google Scholar
Escauriaza, L., Fabes, E. B. & Verchota, G. (1992) On a regularity theorem for weak solutions to transmission problems with internal Lipschitz boundaries. Proc. Amer. Math. Soc. 115(4), 1069–1076.CrossRefGoogle Scholar
Faber, G. (1903) Über polynomische Entwickelungen. Mathematische Annalen 57(3), 389–408.CrossRefGoogle Scholar
Garboczi, E. J. & Douglas, J. F. (1996) Intrinsic conductivity of objects having arbitrary shape and conductivity. Phys. Rev. E 53, 6169–6180.CrossRefGoogle ScholarPubMed
Grieser, D. (2014) The plasmonic eigenvalue problem. Rev. Math. Phys. 26(3), 1450005.CrossRefGoogle Scholar
Helsing, J., Kang, H. & Lim, M. (2017) Classification of spectra of the Neumann-Poincaré operator on planar domains with corners by resonance. Ann. Inst. H. Poincaré Anal. Non Linéaire 34(4), 991–1011.CrossRefGoogle Scholar
Hofmann, S., Mitrea, M. & Taylor, M. (2007) Geometric and transformational properties of Lipschitz domains, Semmes-Kenig-Toro domains, and other classes of finite perimeter domains. J. Geom. Anal. 17(4), 593–647.CrossRefGoogle Scholar
Jeffrey, D. J. (1973) Conduction through a random suspension of spheres. Proc. R. Soc. London Ser. A Math. Phys. Sci. 335(1602), 355–367.Google Scholar
Jikov, V. V., Kozlov, S. M. & Olenik, O. A. (1994) Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin.CrossRefGoogle Scholar
Jung, Y. & Lim, M. (2018) Spectral analysis of the Neumann-Poincaré operator on touching disks and analysis of plasmon resonance. arXiv preprint arXiv:1810.12486.Google Scholar
Jung, Y. & Lim, M. (2020) A decay estimate for the eigenvalues of the Neumann-Poincaré operator using the Grunsky coefficients. Proc. Amer. Math. Soc. 148(2), 591–600.Google Scholar
Jung, Y. & Lim, M. (2021) Series expansions of the layer potential operators using the Faber polynomials and their applications to the transmission problem. SIAM J. Math. Anal. 53(2), 1630–1669.CrossRefGoogle Scholar
Kang, H., Kim, K., Lee, H., Shin, J. & Yu, S. (2016) Spectral properties of the Neumann-Poincaré operator and uniformity of estimates for the conductivity equation with complex coefficients. J. Lond. Math. Soc. (2) 93(2), 519–545.CrossRefGoogle Scholar
Kang, H., Lee, H. & Lim, M. (2015) Construction of conformal mappings by generalized polarization tensors. Math. Methods Appl. Sci. 38(9), 1847–1854.CrossRefGoogle Scholar
Kang, H., Lim, M. & Yu, S. (2017) Spectral resolution of the Neumann-Poincaré operator on intersecting disks and analysis of plasmon resonance. Arch. Ration. Mech. Anal. 226(1), 83–115.CrossRefGoogle Scholar
Kang, H. & Milton, G. W. (2008) Solutions to the Pólya–Szegö conjecture and the Weak Eshelby Conjecture. Arch. Ration. Mech. Anal. 188(1), 93–116.CrossRefGoogle Scholar
Kang, H. & Putinar, M. (2018) Spectral permanence in a space with two norms. Rev. Mat. Iberoam. 34(2), 621–635.CrossRefGoogle Scholar
Kellogg, O. D. (1953) Foundations of Potential Theory, Dover, New York. Originally published in 1929 by J. Springer.CrossRefGoogle Scholar
Khavinson, D., Putinar, M. & Shapiro, H. S. (2007) Poincaré’s variational problem in potential theory. Arch. Ration. Mech. Anal. 185(1), 143–184.CrossRefGoogle Scholar
Kiryatski, È. G. (1990) Some functionals on a class of univalent functions. Litovsk. Mat. Sb. 30(2), 261–267.Google Scholar
Krein, M. G. (1998) Compact linear operators on functional spaces with two norms. Integr. Equations Oper. Theory 30(2), 140–162.CrossRefGoogle Scholar
Lewin, M. (1971) Bounds for functionals of a certain class of analytic functions. Bull. London Math. Soc. 3, 329–330.CrossRefGoogle Scholar
Li, W. & Shipman, S. P. (2019) Embedded eigenvalues for the Neumann-Poincaré operator. J. Integral Equations Appl. 31(4), 505–534.CrossRefGoogle Scholar
Lim, M. (2001) Symmetry of a boundary integral operator and a characterization of a ball. Illinois J. Math. 45(2), 537–543.CrossRefGoogle Scholar
Lipton, R. (1993) Inequalities for electric and elastic polarization tensors with applications to random composites. J. Mech. Phys. Solids 41(5), 809–833.CrossRefGoogle Scholar
Mayergoyz, I. D., Fredkin, D. R. & Zhang, Z. (2005) Electrostatic (plasmon) resonances in nanoparticles. Phys. Rev. B 72(15), 155412.CrossRefGoogle Scholar
Perfekt, K.-M. & Putinar, M. (2014) Spectral bounds for the Neumann-Poincaré operator on planar domains with corners. J. Anal. Math. 124, 39–57.CrossRefGoogle Scholar
Perfekt, K.-M. & Putinar, M. (2017) The essential spectrum of the Neumann-Poincaré operator on a domain with corners. Arch. Ration. Mech. Anal. 223(2), 1019–1033.CrossRefGoogle Scholar
Pólya, G. & Szegö, G. (1951) Isoperimetric Inequalities in Mathematical Physics. Annals of Mathematics Studies, Vol. 27, Princeton University Press, Princeton, NJ.Google Scholar
Pommerenke, C. (1975) Univalent Functions. Vandenhoeck & Ruprecht, Göttingen. With a chapter on quadratic differentials by Gerd Jensen, Studia Mathematica/Mathematische Lehrbücher, Band XXV.Google Scholar
Pommerenke, C. (1992) Boundary Behaviour of Conformal Maps. Grundlehren der mathematischen Wissenschaften, Vol. 299, Springer-Verlag, Berlin, Germany.CrossRefGoogle Scholar
Sangani, A. S. (1990) Conductivity of N-dimensional composites containing hyperspherical inclusion. SIAM J. Appl. Math. 50(1), 64–73.CrossRefGoogle Scholar
Schaeffer, A. C. & Spencer, D. C. (1950) Coefficient Regions for Schlicht Functions. American Mathematical Society Colloquium Publications, Vol. 35, American Mathematical Society, New York, NY. With a Chapter on the Region of the Derivative of a Schlicht Function by Arthur Grad.Google Scholar
Thomas, D. K., Tuneski, N. & Vasudevarao, A. (2018) Univalent Functions. De Gruyter Studies in Mathematics, Vol. 69, De Gruyter, Berlin. A primer.CrossRefGoogle Scholar
Verchota, G. (1984) Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611.CrossRefGoogle Scholar
Yu, S. & Lim, M. (2017) Shielding at a distance due to anomalous resonance. New J. Phys. 19(3), 033018.CrossRefGoogle Scholar