Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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

 
 

 

Operator error estimates for homogenization of the elliptic Dirichlet problem in a bounded domain


Authors: M. A. Pakhnin and T. A. Suslina
Translated by: T. A. Suslina
Original publication: Algebra i Analiz, tom 24 (2012), nomer 6.
Journal: St. Petersburg Math. J. 24 (2013), 949-976
MSC (2010): Primary 35B27
DOI: https://doi.org/10.1090/S1061-0022-2013-01274-X
Published electronically: September 23, 2013
MathSciNet review: 3097556
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let $ \mathcal {O} \subset \mathbb{R}^d$ be a bounded domain of class $ C^{1,1}$. In the Hilbert space $ L_2(\mathcal {O};\mathbb{C}^n)$, a matrix elliptic second order differential operator $ \mathcal {A}_{D,\varepsilon }$ is considered with the Dirichlet boundary condition. Here $ \varepsilon >0$ is a small parameter. The coefficients of the operator are periodic and depend on $ \mathbf {x}/\varepsilon $. Approximation is found for the operator $ \mathcal {A}_{D,\varepsilon }^{-1}$ in the norm of operators acting from $ L_2(\mathcal {O};\mathbb{C}^n)$ to the Sobolev space $ H^1(\mathcal {O};\mathbb{C}^n)$ with an error term of $ O(\sqrt {\varepsilon })$. This approximation is given by the sum of the operator $ (\mathcal {A}^0_D)^{-1}$ and the first order corrector, where $ \mathcal {A}^0_D$ is the effective operator with constant coefficients and with the Dirichlet boundary condition.


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

  • [BaPa] N. S. Bakhvalov and G. P. Panasenko, Homogenization of processes in periodic media, Nauka, Moscow, 1984; English transl., Math. Appl. (Soviet Ser.), vol. 36, Kluwer Acad. Publ. Group, Dordrecht, 1989. MR 797571 (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 050330 (82h:35001)
  • [BSu1] M. Birman and T. 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 (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 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)
  • [Gr1] G. Griso, Error estimate and unfolding for periodic homogenization, Asymptot. Anal. 40 (2004), 269-286. MR 2107633 (2006a:35015)
  • [Gr2] -, Interior error estimate for periodic homogenization, Anal. Appl. (Singap.) 4 (2006), no. 1, 61-79. MR 2199793 (2007d:35014)
  • [Zh1] V. V. Zhikov, On operator estimates in homogenization theory, Dokl. Akad. Nauk 403 (2005), no. 3, 305-308; English transl., Dokl. Math. 72 (2005), 535-538. MR 2164541
  • [Zh2] -, On some estimates from homogenization theory, Dokl. Akad. Nauk 406 (2006), no. 5, 597-601; English transl., Dokl. Math. 73 (2006), 96-99. MR 2347318 (2008d:35018)
  • [ZhKO] V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of differential operators, Nauka, Moscow, 1993; English transl., Springer-Verlag, Berlin, 1994. MR 1318242 (96h:35003a); MR 1329546 (96h:35003b)
  • [ZhPas] 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)
  • [McL] W. McLean, Strongly elliptic systems and boundary integral equations, Cambridge Univ. Press, Cambridge, 2000. MR 1742312 (2001a:35051)
  • [Pas] S. E. Pastukhova, On some estimates from the homogenization of problems in plasticity theory, Dokl. Akad. Nauk 406 (2006), no. 5, 604-608; English transl., Dokl. Math. 73 (2006), 102-106. MR 2347320 (2008d:35016)
  • [PSu] M. A. Pakhnin and T. A. Suslina, Homogenization of the elliptic Dirichlet problem: error estimates in $ (L_2 \to H^1)$-norm, Funktsional. Anal. i Prilozhen. 46 (2012), no. 2, 92-96; English transl., Funct. Anal. Appl. 46 (2012), no. 2, 155-159. MR 2978064
  • [Su1] T. A. Suslina, Operator error estimates in $ L_2$ for homogenization of the elliptic Dirichlet problem, Funktsional. Anal. i Prilozhen. 46 (2012), no. 3, 92-96; English transl., Funct. Anal. Appl. 46 (2012), no. 3, 234-238. MR 2978064
  • [Su2] -, Homogenization of the Dirichlet problem for elliptic systems: $ L_2$-operator error estimates, Mathematika available at CJO2013. doi:10.1112/S0025579312001131.

Similar Articles

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

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


Additional Information

M. A. Pakhnin
Affiliation: Department of Physics, St. Petersburg State University, Ul′yanovskaya 3, Petrodvorets, St. Petersburg 198504, Russia
Email: mpakhnin@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/S1061-0022-2013-01274-X
Keywords: Periodic differential operators, homogenization, effective operator, operator error estimates
Received by editor(s): July 2, 2012
Published electronically: September 23, 2013
Additional Notes: Supported by RFBR (grant no. 11-01-00458-a) and the Program of support for leading scientific schools (grant NSh-357.2012.1).
Article copyright: © Copyright 2013 American Mathematical Society

American Mathematical Society