Unified approach to critical-contrast homogenisation with explicit links to time-dispersive media

K. D. Cherednichenko ; Yu. Yu. Ershova ; A. V. Kiselev ; S. N. Naboko

Trudy Moskovskogo matematičeskogo obŝestva, Trudy Moskovskogo Matematicheskogo Obshchestva, Tome 80 (2019) no. 2, pp. 295-342.

Résumé

A novel approach to critical-contrast homogenisation is proposed. Norm-resolvent asymptotics are explicitly constructed. An essential feature of our approach is that it relates homogenisation limits to a class of time-dispersive media.

Bibliographie

[1] V. M. Adamyan, B. S. Pavlov, “Zero-radius potentials and M. G. Kreĭn's formula for generalized resolvents”, J. Soviet Math., 42:2 (1988), 1537–1550 | DOI | MR

[2] A. Bensoussan, J.-L. Lions, G. Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, 5, North-Holland Publishing Co., Amsterdam–New York, 1978 | MR | Zbl

[3] G. Berkolaiko, P. Kuchment, Introduction to quantum graphs, Mathematical Surveys and Monographs, 186, AMS, Providence, RI, 2013 | MR | Zbl

[4] M. Š. Birman, “On the theory of self-adjoint extensions of positive definite operators”, Mat. Sb. N.S., 38(80) (1956), 431–450 | MR | Zbl

[5] M. Sh. Birman, T. A. Suslina, “Second order periodic differential operators. Threshold properties and homogenization”, St. Petersburg Math. J., 15:5 (2004), 639–714 | DOI | MR | Zbl

[6] M. Sh. Birman, T. A. Suslina, “Homogenization with corrector term for periodic elliptic differential operators”, St. Petersburg Math. J., 17:6 (2006), 897–973 | DOI | MR | Zbl

[7] G. Bouchitté, D. Felbacq, “Homogenization near resonances and artificial magnetism from dielectrics”, C. R. Math. Acad. Sci. Paris, 339:5 (2004), 377–382 | DOI | MR | Zbl

[8] F. Capolino (ed.), Theory and phenomena of metamaterials, Taylor Francis, 2009

[9] K. Cherednichenko, Yu. Ershova, A. V. Kiselev, “Time-dispersive behavior as a feature of criticalcontrast media”, SIAM J. Appl. Math., 79:2 (2019), 690–715 | DOI | MR | Zbl

[10] K. D. Cherednichenko, Yu. Ershova, A. V. Kiselev, Effective behaviour of critical-contrast PDEs: micro-resonances, frequency conversion, and time dispersive properties. I, 2019, arXiv: 1808.03961

[11] K. D. Cherednichenko, A. V. Kiselev, “Norm-resolvent convergence of one-dimensional highcontrast periodic problems to a Kronig-Penney dipole-type model”, Comm. Math. Phys., 349:2 (2017), 441–480 | DOI | MR | Zbl

[12] K. D. Cherednichenko, A. V. Kiselev, L. O. Silva, “Functional model for extensions of symmetric operators and applications to scattering theory”, Netw. Heterog. Media, 13:2 (2018), 191–215 | DOI | MR | Zbl

[13] S. Datta, Electronic transport in mesoscopic systems, Cambridge University Press, 1995

[14] V. A. Derkach, M. M. Malamud, “Generalized resolvents and the boundary value problems for Hermitian operators with gaps”, J. Funct. Anal., 95 (1991), 1–95 | DOI | MR | Zbl

[15] Yu. Ershova, I. I. Karpenko, A. V. Kiselev, “Isospectrality for graph Laplacians under the change of coupling at graph vertices”, J. Spectr. Theory, 6:1 (2016), 43–66 | DOI | MR | Zbl

[16] P. Exner, O. Post, “Convergence of spectra of graph-like thin manifolds”, J. Geom. Phys., 54:1 (2005), 77–115 | DOI | MR | Zbl

[17] A. Figotin, J. H. Schenker, “Spectral theory of time dispersive and dissipative systems”, J. Stat. Phys., 118:1–2 (2005), 199–263 | DOI | MR | Zbl

[18] A. Figotin, J. H. Schenker, “Hamiltonian structure for dispersive and dissipative dynamical systems”, J. Stat. Phys., 128:4 (2007), 969–1056 | DOI | MR | Zbl

[19] L. Friedlander, “On the density of states of periodic media in the large coupling limit”, Comm. Partial Differential Equations, 27:1–2 (2002), 355–380 | DOI | MR | Zbl

[20] V. I. Gorbachuk, M. L. Gorbachuk, Boundary value problems for operator differential equations, Mathematics and its Applications (Soviet Series), 48, Kluwer Academic Publishers Group, Dordrecht, 1991 | MR | Zbl

[21] I. S. Gradshteyn, I. M. Ryzhik, Table of integrals, series, and products, 7th ed., Elsevier/Academic Press, Amsterdam, 2007 | MR | Zbl

[22] R. Hempel, K. Lienau, “Spectral properties of the periodic media in the large coupling limit”, Comm. Partial Differential Equations, 25:7–8 (2000), 1445–1470 | MR | Zbl

[23] L. Hörmander, The analysis of linear partial differential operators, v. III, Classics in Mathematics, Pseudo-differential operators, Springer, Berlin, 2007 | DOI | MR | Zbl

[24] V. V. Jikov, S. M. Kozlov, O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994 | MR

[25] A. N. Kočubeĭ, “Extensions of symmetric operators and of symmetric binary relations”, Mat. Zametki, 17 (1975), 41–48 | MR

[26] A. N. Kočubeĭ, “Characteristic functions of symmetric operators and their extensions”, Izv. Akad. Nauk Arm. SSR. Ser. Mat., 15:3 (1980), 219–232 (Russian) | MR

[27] M. G. Kreĭn, “Theory of self-adjoint extensions of semibounded Hermitian operators and applications II”, Mat. Sb., 21:3 (1947), 365–404 | MR | Zbl

[28] P. Kuchment, H. Zeng, “Convergence of spectra of mesoscopic systems collapsing onto a graph”, J. Math. Anal. Appl., 258:2 (2001), 671–700 | DOI | MR | Zbl

[29] P. Kuchment, H. Zeng, “Asymptotics of spectra of Neumann Laplacians in thin domains”, Contemp. Math., 327, AMS, Providence, RI, 2003, 199–213 | DOI | MR | Zbl

[30] P. D. Lax, R. S. Phillips, Scattering theory, Pure and Applied Mathematics, 26, Academic Press, New York–London, 2010 | MR

[31] U. Leonhardt, T. Philbin, Geometry and light. The science of invisibility, Dover Publications, Inc., Mineola, NY, 2010 | MR | Zbl

[32] G. W. Milton, N.-A. P. Nicorovici, R. C. McPhedran, K. D. Cherednichenko, Z. Jacob, “Solutions in folded geometries and associated cloaking due to anomalous resonance”, New Journal of Physics, 10:11 (2008), 115021 | DOI

[33] M. Neumark, “Spectral functions of a symmetric operator”, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 4 (1940), 277–318 (Russian) | MR | Zbl

[34] M. Neumark, “Positive definite operator functions on a commutative group”, Bull. Acad. Sci. URSS. Sér. Math. [Izvestia Akad. Nauk SSSR], 7 (1943), 237–244 (Russian) | MR | Zbl

[35] V. Ryzhov, T. Philbin, “Functional model of a class of non-selfadjoint extensions of symmetric operators”, Oper. Theory Adv. Appl., 174, Birkhauser, Basel, 2007, 117–158 | DOI | MR | Zbl

[36] V. Ryzhov, “Weyl-Titchmarsh function of an abstract boundary value problem, operator colligations, and linear systems with boundary control”, Complex Anal. Oper. Theory, 3:1 (2009), 289–322 | DOI | MR | Zbl

[37] I. Schur, “Neue Begrundung der Theorie der Gruppencharaktere”, Sitzungsberichte der Königlich-Preussischen Akademie der Wissenschaften, Physikalisch-Mathematische Klasse, 1905, 406–436

[38] B. Simon, Functional integration and quantum physics, 2nd ed., AMS Chelsea Publishing, Providence, RI, 2005 | MR | Zbl

[39] A. A. Shkalikov, “Spectral portraits of the Orr-Sommerfeld operator at large Reynolds numbers”, J. Math. Sci. (N.Y.), 124:6 (2004), 5417–5441 | DOI | MR

[40] A. V. Štraus, “Generalized resolvents of symmetric operators”, Izv. Akad. Nauk SSSR, Ser. Mat., 18 (1954), 51–86 (Russian) | MR

[41] A. Tip, “Linear absorptive dielectrics”, Phys. Rev. A, 57 (1998), 4818–4841 | DOI

[42] A. Tip, “Some mathematical properties of Maxwell's equations for macroscopic dielectrics”, J. Math. Phys., 47:1 (2006), 012902, 22 pp. | DOI | MR | Zbl

[43] W. T. Tutte, Graph theory, Encyclopedia of Mathematics and its Applications, 21, Addison Wesley Publishing Company, Advanced Book Program, Reading, MA, 1984 | MR | Zbl

[44] V. G. Veselago, “The electrodynamics of substances with simultaneously negative values of $\varepsilon$ and $\mu$”, Soviet Physics Uspekhi, 10:4 (1968), 509–514 | DOI

[45] M. I. Višik, “On general boundary problems for elliptic differential equations”, Trudy Moskov. Mat. Obšč., 1, 1952, 187–246 (Russian) | MR | Zbl

[46] D. R. Yafaev, Mathematical scattering theory. General theory, Translations of Mathematical Monographs, 105, AMS, Providence, RI, 1992 | DOI | MR | Zbl

[47] V. V. Zhikov, “Spectral approach to asymptotic diffusion problems”, Differentsial'nye uravneniya, 25:1 (1989), 44–50 (Russian) | MR | Zbl

[48] V. V. Zhikov, “On an extension of the method of two-scale convergence and its applications”, Sbornik: Mathematics, 191:7–8 (2000), 973–1014 | DOI | MR | Zbl

[49] V. V. Zhikov, “Averaging of problems in the theory of elasticity on singular structures”, Izv. Math., 66:2 (2002), 299–365 | DOI | MR | Zbl

[50] V. V. Zhikov, “On gaps in the spectrum of some divergence elliptic operators with periodic coefficients”, St. Petersburg Math. J., 16:5 (2005), 773–719 | DOI | MR