Hostname: page-component-76fb5796d-dfsvx Total loading time: 0 Render date: 2024-04-27T15:45:31.040Z Has data issue: false hasContentIssue false
  • English
  • Français

Homogenization of a Periodic Parabolic Cauchy Problem in the Sobolev Space H1 (ℝd)

Published online by Cambridge University Press:  12 May 2010

T. Suslina
T. Suslina*
Affiliation:
Department of Physics, St. Petersburg State University, Ul’yanovskaya 3, Petrodvorets, St. Petersburg, 198504, Russia
*
* E-mail: suslina@list.ru
Get access

Abstract

In L2(ℝd; ℂn), we consider a wide class of matrix elliptic second order differential operators $\mathcal{A}$ε with rapidly oscillating coefficients (depending on x/ε). For a fixed τ > 0 and small ε > 0, we find approximation of the operator exponential exp(− $\mathcal{A}$ετ) in the (L2(ℝd; ℂn) → H1(ℝd; ℂn))-operator norm with an error term of order ε. In this approximation, the corrector is taken into account. The results are applied to homogenization of a periodic parabolic Cauchy problem.

Keywords

periodic differential operatorsparabolic Cauchy problemhomogenizationeffective operatorcorrector
Type
Research Article
Information
Mathematical Modelling of Natural Phenomena , Volume 5 , Issue 4: Spectral problems. Issue dedicated to the memory of M. Birman , 2010 , pp. 390 - 447
Copyright
© EDP Sciences, 2010

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

Footnotes

To the memory of my dear Teacher Mikhail Shlemovich Birman

References

N. S. Bakhvalov, 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. Publishers Group, Dordrecht, 1989.
A. Bensoussan, J. L. Lions, G. Papanicolaou. Asymptotic analysis for periodic structures. Stud. Math. Appl., vol. 5, North-Holland Publishing Company, Amsterdam–New York, 1978, 700 pp.
M. Birman, 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.
Birman, M. Sh., Suslina, T. A.. 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.Google Scholar
Birman, M. Sh., Suslina, T. A.. 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., St. Petersburg Math. J.,17 (2006), no. 5, 745–762.Google Scholar
Birman, M. Sh., Suslina., T. A. Homogenization with corrector term for periodic elliptic differential operators , Algebra i Analiz, 17 (2005), no. 6, 1–104.Google Scholar
Birman, M. Sh., Suslina, T. A.. Homogenization with corrector for periodic differential operators. Approximation of solutions in the Sobolev class H 1( d ), Algebra i Analiz, 18 (2006), no. 6, 1–130; English transl., St. Petersburg Math. J., 18 (2007), no. 6, 857–955. Google Scholar
T. Kato. Perturbation theory for linear operators, 2nd ed. Grundlehren Math. Wiss., vol. 132, Springer-Verlag, Berlin-New York, 1976.
E. Sanchez-Palencia. Nonhomogeneous media and vibration theory. Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin–New York, 1980.
Suslina, T. A.. 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. Google Scholar
Suslina, T. A.. Homogenization of periodic parabolic Cauchy problem . Amer. Math. Soc. Transl., ser. 2, 220 (2007), 201–233. Google Scholar
Vasilevskaya, E. S.. A periodic parabolic Cauchy problem: homogenization with corrector . Algebra i Analiz, 20 (2009), no. 1, 3–60; English transl., St. Petersburg Math. J., 20 (2010), no. 1. Google Scholar
Zhikov, V. V., On some estimates of homogenization theory . Dokl. Ros. Akad. Nauk, 406 (2006), no. 5, 597–601; English transl., Dokl. Math., 73 (2006), 96–99. Google Scholar
V. V. Zhikov, S. M. Kozlov, O. A. Oleinik. Homogenization of differential operators. "Nauka", Moscow, 1993; English transl., Springer-Verlag, Berlin, 1994.
Zhikov, V. V., , S. E.. On operator estimates for some problems in homogenization theory . Russ. J. Math. Phys., 12 (2005), no. 4, 515–524. Google Scholar
Zhikov, V. V., Pastukhova, S. E.. Estimates of homogenization for a parabolic equation with periodic coefficients . Russ. J. Math. Phys., 13 (2006), no. 2, 251–265. CrossRefGoogle Scholar