Homogenization with corrector term for periodic elliptic differential operators

Birman, M.; Suslina, T.

doi:10.1090/S1061-0022-06-00935-6

Homogenization with corrector term for periodic elliptic differential operators
HTML articles powered by AMS MathViewer

by M. Sh. Birman and T. A. Suslina
Translated by: T. A. Suslina

St. Petersburg Math. J. 17 (2006), 897-973

DOI: https://doi.org/10.1090/S1061-0022-06-00935-6

Published electronically: September 21, 2006

PDF | Request permission

Abstract:

We continue to study the class of matrix periodic elliptic differential operators ${{\mathcal {A}}_\varepsilon }$ in ${\mathbb {R}^d}$ with coefficients oscillating rapidly (i.e., depending on ${{\mathbf {x}}/\varepsilon }$). This class was introduced in the authors’ earlier work of 2001 and 2003. The problem of homogenization in the small period limit is considered. Approximation for the resolvent ${({\mathcal {A}}_\varepsilon + I)^{-1}}$ is obtained in the operator norm in ${L_2(\mathbb {R}^d)}$ with error term of order ${\varepsilon ^2}$. The so-called corrector is taken into account. We develop the approach of our paper of 2003, where approximation with no corrector term but with remainder term of order ${\varepsilon }$ was found. The paper is based on the operator-theoretic material obtained in our paper in the previous issue of this journal. Though the present paper is a continuation of the earlier work, it can be read independently.

References

N. S. Bahvalov, Averaged characteristics of bodies with periodic structure, Dokl. Akad. Nauk SSSR 218 (1974), 1046–1048 (Russian). MR 0395435
N. S. Bahvalov, Averaging of partial differential equations with rapidly oscillating coefficients, Dokl. Akad. Nauk SSSR 221 (1975), no. 3, 516–519 (Russian). MR 0380075
N. S. Bakhvalov and G. P. Panasenko, Osrednenie protsessov v periodicheskikh sredakh, “Nauka”, Moscow, 1984 (Russian). Matematicheskie zadachi mekhaniki kompozitsionnykh materialov. [Mathematical problems of the mechanics of composite materials]. MR 797571
Alain Bensoussan, Jacques-Louis Lions, and George Papanicolaou, Asymptotic analysis for periodic structures, Studies in Mathematics and its Applications, vol. 5, North-Holland Publishing Co., Amsterdam-New York, 1978. MR 503330
M. Sh. Birman, On the averaging procedure for periodic operators in a neighborhood of an edge of an internal gap, Algebra i Analiz 15 (2003), no. 4, 61–71 (Russian); English transl., St. Petersburg Math. J. 15 (2004), no. 4, 507–513. MR 2068979, DOI 10.1090/S1061-0022-04-00819-2
Michael Birman and Tatyana Suslina, Threshold effects near the lower edge of the spectrum for periodic differential operators of mathematical physics, Systems, approximation, singular integral operators, and related topics (Bordeaux, 2000) Oper. Theory Adv. Appl., vol. 129, Birkhäuser, Basel, 2001, pp. 71–107. MR 1882692
M. Sh. Birman and T. A. Suslina, Periodic second-order differential operators. Threshold properties and averaging, Algebra i Analiz 15 (2003), no. 5, 1–108 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 15 (2004), no. 5, 639–714. MR 2068790, DOI 10.1090/S1061-0022-04-00827-1
M. Sh. Birman and T. A. Suslina, Homogenization of a multidimensional periodic elliptic operator in a neighborhood of an edge of an inner gap, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 318 (2004), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 36 [35], 60–74, 309 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 136 (2006), no. 2, 3682–3690. MR 2120232, DOI 10.1007/s10958-006-0192-9
M. Sh. Birman (joint work with T. A. Suslina), Homogenization of periodic differential operators in $\mathbb {R}^d$ as a spectral threshold effect, Mathematisches Forschungsinstitut Oberwolfach, Report No. 53/2004, Spectral Analysis of Partial Differential Equations, November 28th – December 4th, 2004, pp. 2847–2849.
M. Sh. Birman and T. A. Suslina, Threshold approximations with corrector term for the resolvent of a factorized selfadjoint operator family, Algebra i Analiz 17 (2005), no. 5, 69–90; English transl. in St. Petersburg Math. J. 17 (2006), no. 5.
Carlos Conca, Rafael Orive, and Muthusamy Vanninathan, Bloch approximation in homogenization and applications, SIAM J. Math. Anal. 33 (2002), no. 5, 1166–1198. MR 1897707, DOI 10.1137/S0036141001382200
Georges Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal. 40 (2004), no. 3-4, 269–286. MR 2107633
Georges Griso, Interior error estimate for periodic homogenization, C. R. Math. Acad. Sci. Paris 340 (2005), no. 3, 251–254 (English, with English and French summaries). MR 2123038, DOI 10.1016/j.crma.2004.10.027
V. V. Zhikov, Spectral approach to asymptotic diffusion problems, Differentsial′nye Uravneniya 25 (1989), no. 1, 44–50, 180 (Russian); English transl., Differential Equations 25 (1989), no. 1, 33–39. MR 986395
V. V. Zhikov, On operator estimates in homogenization theory, Dokl. Akad. Nauk 403 (2005), no. 3, 305–308 (Russian). MR 2164541
V. V. Zhikov, On the spectral method in homogenization theory, Tr. Mat. Inst. Steklova 250 (2005), no. Differ. Uravn. i Din. Sist., 95–104 (Russian, with Russian summary); English transl., Proc. Steklov Inst. Math. 3(250) (2005), 85–94. MR 2200910
—, On some estimates of the homogenization theory, Dokl. Akad. Nauk (to appear).
V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242
V. V. Zhikov and S. E. Pastukhova, On operator estimates for some problems in homogenization theory, Russ. J. Math. Phys. 12 (2005), no. 4, 515–524. MR 2201316
Tosio Kato, Perturbation theory for linear operators, 2nd ed., Grundlehren der Mathematischen Wissenschaften, Band 132, Springer-Verlag, Berlin-New York, 1976. MR 0407617
O. A. Ladyzhenskaya and N. N. Ural′tseva, Lineĭnye i kvazilineĭnye uravneniya èllipticheskogo tipa, Izdat. “Nauka”, Moscow, 1973 (Russian). Second edition, revised. MR 0509265
J. Marcinkiewicz, Sur les multiplicateurs des series de Fourier, Studia Math. 8 (1939), 78–91.
S. G. Mihlin, Mnogomernye singulyarnye integraly i integral′nye uravneniya, Gosudarstv. Izdat. Fiz.-Mat. Lit., Moscow, 1962 (Russian). MR 0155165
S. E. Pastukhova, On some estimates for homogenization problems of elasticity theory, Dokl. Akad. Nauk (to appear).
E. V. Sevost′yanova, Asymptotic expansion of the solution of a second-order elliptic equation with periodic rapidly oscillating coefficients, Mat. Sb. (N.S.) 115(157) (1981), no. 2, 204–222, 319 (Russian). MR 622145
T. A. Suslina, On the averaging of a periodic Maxwell system, Funktsional. Anal. i Prilozhen. 38 (2004), no. 3, 90–94 (Russian); English transl., Funct. Anal. Appl. 38 (2004), no. 3, 234–237. MR 2095137, DOI 10.1023/B:FAIA.0000042808.32919.b7
T. A. Suslina, Averaging of a stationary periodic Maxwell system, Algebra i Analiz 16 (2004), no. 5, 162–244 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 5, 863–922. MR 2106671, DOI 10.1090/S1061-0022-05-00883-6
—, Homogenization of periodic parabolic systems, POMI Preprint no. 15/2004. (Russian)
T. A. Suslina, On the averaging of periodic parabolic systems, Funktsional. Anal. i Prilozhen. 38 (2004), no. 4, 86–90 (Russian); English transl., Funct. Anal. Appl. 38 (2004), no. 4, 309–312. MR 2117512, DOI 10.1007/s10688-005-0010-z
R. G. Shterenberg, An example of a periodic magnetic Schrödinger operator with a degenerate lower edge of the spectrum, Algebra i Analiz 16 (2004), no. 2, 177–185 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 16 (2005), no. 2, 417–422. MR 2068347, DOI 10.1090/S1061-0022-05-00858-7
—, On the structure of the lower edge of the spectrum of the periodic magnetic Schrödinger operator with small magnetic potential, Algebra i Analiz 17 (2005), no. 5, 232–243; English transl. in St. Petersburg Math. J. 17 (2006), no. 5.