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Homogenization with corrector for a stationary periodic Maxwell system


Author: T. A. Suslina
Translated by: the author
Original publication: Algebra i Analiz, tom 19 (2007), nomer 3.
Journal: St. Petersburg Math. J. 19 (2008), 455-494
MSC (2000): Primary 35P20, 35Q60
DOI: https://doi.org/10.1090/S1061-0022-08-01006-6
Published electronically: March 21, 2008
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Abstract | References | Similar Articles | Additional Information

Abstract: The homogenization problem in the small period limit for a stationary periodic Maxwell system in $ {\mathbb{R}^3}$ is studied. It is assumed that the dielectric permittivity and the magnetic permeability are rapidly oscillating (depending on $ \mathbf{x}/\varepsilon$), positive definite, and bounded matrix-valued functions. For all four physical fields (the strength of the electric field, the strength of the magnetic field, the electric displacement vector, and the magnetic displacement vector), uniform approximations in the $ {L_2(\mathbb{R}^3)}$-norm are obtained with the (order-sharp) error term of order $ \varepsilon$. Besides solutions of the homogenized Maxwell system, the approximations contain rapidly oscillating terms of zero order that weakly tend to zero. These terms can be interpreted as correctors of zero order.


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  • [BaPa] N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging processes in periodic media. Mathematical problems in the mechanics of composite materials, ``Nauka'', Moscow, 1984; English transl., Math. Appl. (Soviet Ser.), vol. 36, Kluwer Acad. Publ. Group, Dordrecht, 1989. MR 0797571 (86m:73049); MR 1112788 (92d:73002)
  • [BeLP] A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 0503330 (82h:35001)
  • [BSu1] M. Sh. Birman and T. A. Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, Approximations, Singular Integral Operators and Related Topics (Bordeaux, 2000), Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71-107. MR 1882692 (2003f:35220)
  • [BSu2] -, Second order periodic differential operators. Threshold properties and homogenization, Algebra i Analiz 15 (2003), no. 5, 1-108; English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639-714. MR 2068790 (2005k:47097)
  • [BSu3] -, Threshold approximations with corrector for the resolvent of a factorized selfadjoint operator family, Algebra i Analiz 17 (2005), no. 5, 69-90; English transl., St. Petersburg Math. J. 17 (2006), no. 5, 745-762. MR 2241423(2008d:47047)
  • [BSu4] -, Homogenization with corrector term for periodic elliptic differential operators, Algebra i Analiz 17 (2005), no. 6, 1-104; English transl., St. Petersburg Math. J. 17 (2006), no. 6, 897-973. MR 2202045 (2006k:35011)
  • [BSu5] -, Homogenization with corrector term for periodic differential operators. Approximation of solutions in the Sobolev class $ H^1(\mathbb{R}^d)$, Algebra i Analiz 18 (2006), no. 6, 1-130; English transl. in St. Petersburg Math. J. 18 (2007), no. 6, 857-955. MR 2307356 (2008d:35008)
  • [BSu6] -, Homogenization of a stationary periodic Maxwell system in the case of constant magnetic permeability, Funktsional. Anal. i Prilozhen. 41 (2007), no. 2, 3-23; English transl. in Funct. Anal. Appl. 41 (2007), no. 2. MR 2345036
  • [H] Q. Huang, Estimates on the generalized Morrey spaces $ L^{2,\lambda}_\varphi$ and $ \hbox{\rm BMO}_\psi$ for linear elliptic systems, Indiana Univ. Math. J. 45 (1996), no. 2, 397-439. MR 1414336 (97i:35033)
  • [ZhKO] V. V. Zhikov, S. M. Kozlov, and O. A. Oleınik, Homogenization of differential operators, ``Nauka'', Moscow, 1993; English transl., Springer-Verlag, Berlin, 1994. MR 1318242 (96h:35003a); MR 1329546 (96h: 35003b)
  • [LaU] O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, ``Nauka'', Moscow, 1964; English transl., Acad. Press, New York-London, 1968. MR 0211073 (35:1955); MR 0244627 (39:5941)
  • [Sa] E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 0578345 (82j:35010)
  • [Su1] T. A. Suslina, On the homogenization of a periodic Maxwell system, Funktsional. Anal. i Prilozhen. 38 (2004), no. 3, 90-94; English transl., Funct. Anal. Appl. 38 (2004), no. 3, 234-237. MR 2095137 (2005g:35017)
  • [Su2] -, Homogenization of a stationary periodic Maxwell system, Algebra i Analiz 16 (2004), no. 5, 162-244; English transl., St. Petersburg Math. J. 16 (2005), no. 5, 863-922. MR 2106671 (2005h:35019)

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Additional Information

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, 198504 St. Petersburg, Russia
Email: suslina@list.ru

DOI: https://doi.org/10.1090/S1061-0022-08-01006-6
Keywords: Periodic Maxwell operator, homogenization, effective medium, corrector
Received by editor(s): February 8, 2007
Published electronically: March 21, 2008
Additional Notes: Supported by RFBR (grant no. 05-01-01076-a) and the President grant “Scientific Schools” (grant no. 5403.2006.1).
Article copyright: © Copyright 2008 American Mathematical Society

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