Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

ISSN 1547-7371(online) ISSN 1061-0022(print)

Request Permissions   Purchase Content 
 
 

 

Homogenization of parabolic and elliptic periodic operators in $ L_2(\mathbb{R}^d)$ with the first and second correctors taken into account


Authors: E. S. Vasilevskaya and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 24 (2012), nomer 2.
Journal: St. Petersburg Math. J. 24 (2013), 185-261
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
Published electronically: January 22, 2013
MathSciNet review: 3013323
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: In the space $ L_2(\mathbb{R}^d;{\mathbb{C}}^n)$, a wide class of matrix elliptic second order differential operators (DO's) $ \mathcal {A}_\varepsilon $ is studied; the $ \mathcal {A}_\varepsilon $ are assumed to admit a factorization of the form $ \mathcal {A}_\varepsilon = \mathcal {X}_\varepsilon ^* \mathcal {X}_\varepsilon $, where $ \mathcal {X}_\varepsilon $ is a homogeneous first order DO. The coefficients of these operators are periodic and depend on $ \mathbf {x}/\varepsilon $, $ \varepsilon >0$. The behavior of the operator exponential $ e^{-\mathcal {A}_\varepsilon \tau }$, $ \tau >0$, and of the resolvent $ ({\mathcal {A}}_\varepsilon +I)^{-1}$ for small $ \varepsilon $ is investigated. An approximation for the exponential $ e^{-\mathcal {A}_\varepsilon \tau }$ in the operator norm in $ L_2(\mathbb{R}^d; \mathbb{C}^n)$ with an error term of order $ \tau ^{-3/2}\varepsilon ^3$ is obtained. For the resolvent $ ({\mathcal {A}}_\varepsilon +I)^{-1}$, approximation in the norm of operators acting from $ H^1(\mathbb{R}^d; \mathbb{C}^n)$ to $ L_2( \mathbb{R}^d; \mathbb{C}^n)$ is found with an error term of order $ \varepsilon ^3$. In these approximations, the first and second order correctors are taken into account.


References [Enhancements On Off] (What's this?)

  • [BaPa] N. S. Bakhvalov and G. O. Panasenko, Homogenization: averaging processes in periodic media. Mathematical problems in 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)
  • [BeLP] A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. Math. Appl., vol. 5, North-Holland Publ. Co., Amsterdam-New York, 1978. MR 503330 (82h:35001)
  • [BSu1] M. Sh. Birman and T. A. Suslina, 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)
  • [BSu2] -, 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)
  • [BSu3] -, 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)
  • [BSu4] -, Homogenization with corrctor 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., St. Petersburg Math. J. 18 (2007), no. 6, 857-955. MR 2307356 (2008d:35008)
  • [CoVa] C. Conca and M. Vanninathan, Homogenization of periodic structures via Bloch decomposition, SIAM J. Appl. Math. 57 (1997), no. 6, 1639-1659. MR 1484944 (98j:35017)
  • [V] E. S. Vasilevskaya, A periodic parabolic Cauchy problem: Homogenization with corrector, Algebra i Analiz 21 (2009), no. 1, 3-60; English transl., St. Petersburg Math. J. 21 (2010), no. 1, 1-41. MR 2553050 (2010k:35035)
  • [VSu] E. S. Vasilevskaya and T. A. Suslina, Threshold approximations of a factorized selfadjoint operator family taking into account the first and second correctors, Algebra i Analiz 23 (2011), no. 2, 102-146; English transl., St. Petersburg Math. J. 23 (2012), no. 2, 275-308. MR 2841674 (2012g:47052)
  • [Zh1] V. V. Zhikov, Spectral approach to asymptotic diffusion problems, Differentsial'nye Uravneniya 25 (1989), no. 1, 44-50; English transl., Differential Equations 25 (1989), no. 1, 33-39. MR 986395 (90a:35107)
  • [Zh2] -, On some estimates from homogenization theory, Dokl. Akad. Nauk 406 (2006), no. 5, 597-601. (Russian) MR 2347318 (2008d:35018)
  • [ZhKO] V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators, Nauka, Moscow, 1993; English transl., Springer-Verlag, Berlin, 1994. MR 1318242 (96h:35003a); MR 1329546 (96h:35003b)
  • [ZhPas1] 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 (2007c:35014)
  • [ZhPas2] -, Estimates of homogenization for a parabolic equation with periodic coefficients, Russ. J. Math. Phys. 13 (2006), no. 2, 224-237. MR 2262826 (2007k:35025)
  • [Se] 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 (1981), no. 2, 204-222; English transl., Math. USSR-Sb. 43 (1982), no. 2, 181-198. MR 622145 (83d:35038)
  • [Su1] T. A. Suslina, On the homogenization of periodic parabolic systems, Funktsional. Anal. i Prilozhen. 38 (2004), no. 4, 86-90; English transl., Funct. Anal. Appl. 38 (2004), no. 4, 309-312. MR 2117512 (2005j:35008)
  • [Su2] -, Homogenization of periodic parabolic Cauchy problem, Nonlinear Equations and Spectral Theory, Amer. Math. Soc. Transl. Ser. 2, vol. 220, Amer. Math. Soc., Providence, RI, 2007, pp. 201-233. MR 2343612 (2008k:35030)
  • [Su3] -, Homogenization of the parabolic Cauchy problem in the Sobolev class $ H^1(\mathbb{R}^d)$, Funktsional. Anal. i Prilozhen. 44 (2010), no. 4, 91-96; English transl., Funct. Anal. Appl. 44 (2010), no. 4, 318-322. MR 2768568 (2012k:35040)
  • [Su4] T. A. Suslina, Homogenization of a periodic parabolic Cauchy problem in the Sobolev space $ H^1(\mathbb{R}^d)$, Math. Model. Nat. Phenom. 5 (2010), no. 4, 390-447. MR 2662463 (2011g:35031)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35B27

Retrieve articles in all journals with MSC (2010): 35B27


Additional Information

E. S. Vasilevskaya
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: vasilevskaya-e@yandex.ru

T. A. Suslina
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: suslina@list.ru

DOI: https://doi.org/10.1090/S1061-0022-2013-01236-2
Keywords: Periodic differential operators, homogenization, effective operator, corrector
Received by editor(s): November 1, 2011
Published electronically: January 22, 2013
Additional Notes: The authors were supported by RFBR (grant no. 11-01-00458-a) and the Program of Support of Leading Scientific Schools (grant NSh-357.2012.1)
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society