Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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



Homogenization of an elliptic system under condensing perforation of the domain

Authors: S. A. Nazarov and A. S. Slutskii
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 17 (2005), nomer 6.
Journal: St. Petersburg Math. J. 17 (2006), 989-1014
MSC (2000): Primary 35J99
Published electronically: September 20, 2006
MathSciNet review: 2202047
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Homogenization of a system of second-order differential equations is performed in the case of a nonuniformly perforated rectangle where the sizes of the holes and the distances between pairs of them decrease as the distance from one of the bases of the rectangle increases. The Neumann conditions are assumed on the boundaries of the holes. The formal asymptotics of the solution is constructed, which involves the usual Ansatz of homogenization theory and also some Ansätze typical of solutions of boundary-value problems in thin domains, in particular, exponential boundary layers. Justification of the asymptotics is done with the help of the Korn inequality, which is proved for the perforated domain $ \Omega(h)$. Depending on the properties of the right-hand side, the norm of the difference between the true and the approximate solutions in the Sobolev space $ H^1(\Omega(h))$ is estimated by the quantity $ ch^\varkappa$ with $ \varkappa\in(0,1/2]$.

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

  • 1. O. A. Ladyzhenskaya, \cyr Kraevye zadachi matematicheskoĭ fiziki., Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
  • 2. Jindřich Nečas, Les méthodes directes en théorie des équations elliptiques, Masson et Cie, Éditeurs, Paris; Academia, Éditeurs, Prague, 1967 (French). MR 0227584
  • 3. S. A. Nazarov, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes, Uspekhi Mat. Nauk 54 (1999), no. 5(329), 77–142 (Russian); English transl., Russian Math. Surveys 54 (1999), no. 5, 947–1014. MR 1741662,
  • 4. S. A. Nazarov, Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, J. Math. Sci. (New York) 92 (1998), no. 6, 4338–4353. Some questions of mathematical physics and function theory. MR 1668418,
  • 5. V. A. Marchenko and E. Ya. Khruslov, \cyr Kraevye zadachi v oblastyakh s melkozernistoĭ granitseĭ., Izdat. “Naukova Dumka”, Kiev, 1974 (Russian). MR 0601059
  • 6. N. Bakhvalov and G. Panasenko, Homogenisation: averaging processes in periodic media, Mathematics and its Applications (Soviet Series), vol. 36, Kluwer Academic Publishers Group, Dordrecht, 1989. Mathematical problems in the mechanics of composite materials; Translated from the Russian by D. Leĭtes. MR 1112788
  • 7. E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 0578345 (82j:35010)
  • 8. O. A. Oleĭnik, G. A. Iosif′yan, and A. S. Shamaev, \cyr Matematicheskie zadachi teorii sil′no neodnorodnykh uprugikh sred, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR 1115306
  • 9. V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, \cyr Usrednenie differentsial′nykh operatorov, “Nauka”, Moscow, 1993 (Russian, with English and Russian summaries). MR 1318242
    V. V. Jikov, S. M. Kozlov, and O. A. Oleĭnik, Homogenization of differential operators and integral functionals, Springer-Verlag, Berlin, 1994. Translated from the Russian by G. A. Yosifian [G. A. Iosif′yan]. MR 1329546
  • 10. Doina Cioranescu and Jeannine Saint Jean Paulin, Homogenization of reticulated structures, Applied Mathematical Sciences, vol. 136, Springer-Verlag, New York, 1999. MR 1676922
  • 11. Serguei M. Kozlov, Harmonization and homogenization on fractals, Comm. Math. Phys. 153 (1993), no. 2, 339–357. MR 1218305
  • 12. V. V. Zhikov, Connectedness and averaging. Examples of fractal conductivity, Mat. Sb. 187 (1996), no. 8, 3–40 (Russian, with Russian summary); English transl., Sb. Math. 187 (1996), no. 8, 1109–1147. MR 1418340,
  • 13. S. A. Nazarov, A general scheme for averaging selfadjoint elliptic systems in multidimensional domains, including thin domains, Algebra i Analiz 7 (1995), no. 5, 1–92 (Russian, with Russian summary); English transl., St. Petersburg Math. J. 7 (1996), no. 5, 681–748. MR 1365812
  • 14. -, Asymptotic theory of thin plates and rods. Dimension reduction and integral estimates. Vol. 1, ``Nauchn. Kniga,'' Novosibirsk, 2002. (Russian)
  • 15. S. A. Nazarov and A. S. Slutskiĭ, Branching periodicity: homogenization of the Dirichlet problem for an elliptic system, Dokl. Akad. Nauk 397 (2004), no. 6, 743–747 (Russian). MR 2120175
  • 16. S. A. Nazarov, Elliptic boundary value problems with periodic coefficients in a cylinder, Izv. Akad. Nauk SSSR Ser. Mat. 45 (1981), no. 1, 101-112; English transl. in Math. USSR-Izv. 18 (1982), no. 1. MR 0607578 (82e:35035)
  • 17. P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4, 3-52; English transl., Russian Math. Surveys 37 (1982), no. 4, 60-94. MR 0667973 (84b:35018)
  • 18. S. A. Nazarov, Nonselfadjoint elliptic problems with the polynomial property in domains possessing cylindrical outlets to infinity, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 249 (1997), no. Kraev. Zadachi Mat. Fiz. i Smezh. Vopr. Teor. Funkts. 29, 212–230, 316–317 (Russian, with English and Russian summaries); English transl., J. Math. Sci. (New York) 101 (2000), no. 5, 3512–3522. MR 1698519,
  • 19. Sergey A. Nazarov and Boris A. Plamenevsky, Elliptic problems in domains with piecewise smooth boundaries, De Gruyter Expositions in Mathematics, vol. 13, Walter de Gruyter & Co., Berlin, 1994. MR 1283387

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2000): 35J99

Retrieve articles in all journals with MSC (2000): 35J99

Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia

A. S. Slutskii
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia

Keywords: Fractal type perforated domain, homogenization, Korn's inequality, corrector
Received by editor(s): September 13, 2004
Published electronically: September 20, 2006
Additional Notes: Supported by RFBR (project no. 03-01-00838)
Article copyright: © Copyright 2006 American Mathematical Society

American Mathematical Society