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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
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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]$.


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  • 1. O. A. Ladyzhenskaya, Kraevye zadachi matematicheskoi fiziki, Izdat. “Nauka”, Moscow, 1973 (Russian). MR 0599579
    O. A. Ladyzhenskaya, The boundary value problems of mathematical physics, Applied Mathematical Sciences, vol. 49, Springer-Verlag, New York, 1985. Translated from the Russian by Jack Lohwater [Arthur J. Lohwater]. MR 793735
  • 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, 10.1070/rm1999v054n05ABEH000204
  • 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, 10.1007/BF02433440
  • 5. V. A. Marčenko and E. Ya. Khruslov, Kraevye zadachi v oblastyakh s melkozernistoi granitsei, Izdat. “Naukova Dumka”, Kiev, 1974 (Russian). MR 0601059
  • 6. 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
    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. Enrique Sánchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Physics, vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 578345
  • 8. O. A. Oleĭnik, G. A. Iosif′yan, and A. S. Shamaev, Matematicheskie zadachi teorii silno neodnorodnykh uprugikh sred, Moskov. Gos. Univ., Moscow, 1990 (Russian). MR 1115306
  • 9. V. V. Zhikov, S. M. Kozlov, and O. A. Oleĭnik, Usrednenie differentsialnykh 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, 10.1070/SM1996v187n08ABEH000150
  • 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, 239 (Russian). MR 607578
  • 17. P. A. Kuchment, Floquet theory for partial differential equations, Uspekhi Mat. Nauk 37 (1982), no. 4(226), 3–52, 240 (Russian). MR 667973
  • 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, 10.1007/BF02680148
  • 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

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Additional Information

S. A. Nazarov
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia
Email: serna@snark.ipme.ru

A. S. Slutskii
Affiliation: Institute of Mechanical Engineering Problems, Bol′shoĭ pr. V. O. 61, 199178 St. Petersburg, Russia
Email: slutskii@snark.ipme.ru

DOI: https://doi.org/10.1090/S1061-0022-06-00937-X
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