Remote Access St. Petersburg Mathematical Journal

St. Petersburg Mathematical Journal

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



Homogenization of the mixed boundary-value problem for a formally selfadjoint elliptic system in a periodically punched domain

Authors: G. Cardone, A. Corbo Esposito and S. A. Nazarov
Translated by: A. Plotkin
Original publication: Algebra i Analiz, tom 21 (2009), nomer 4.
Journal: St. Petersburg Math. J. 21 (2010), 601-634
MSC (2010): Primary 35J57
Published electronically: May 20, 2010
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: A generalized Gårding-Korn inequality is established in a domain $ \Omega(h)\subset{\mathbb{R}}^n$ with a small, of size $ O(h)$, periodic perforation, without any restrictions on the shape of the periodicity cell, except for the usual assumptions that the boundary is Lipschitzian, which ensures the Korn inequality in a general domain. Homogenization is performed for a formally selfadjoint elliptic system of second order differential equations with the Dirichlet or Neumann conditions on the outer or inner parts of the boundary, respectively; the data of the problem are assumed to satisfy assumptions of two types: additional smoothness is required from the dependence on either the ``slow'' variables $ x$, or the ``fast'' variables $ y=h^{-1}x$. It is checked that the exponent $ \delta\in(0,1/2]$ in the accuracy $ O(h^\delta)$ of homogenization depends on the smoothness properties of the problem data.

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

  • 1. O. A. Ladyzhenskaya, Boundary value problems of mathematical physics, Nauka, Moscow, 1973; English transl., Appl. Math. Sci., vol. 49, Springer-Verlag, New York, 1985. MR 0599579 (58:29032); MR 0793735 (87f:35001)
  • 2. J. Nečas, Les méthodes directes en théorie des équations elliptiques, Masson, Paris; Academia, Prague, 1967. MR 0227584 (37:3168)
  • 3. S. A. Nazarov, Self-adjoint elliptic boundary-value problems. The polynomial property and formally positive operators, Probl. Mat. Anal., No. 16, S.-Peterburg. Univ., St. Petersburg, 1997, pp. 167-192; English transl., J. Math. Sci. (New York) 92 (1998), no. 6, 4338-4353. MR 1668418 (2000a:35042)
  • 4. -, Polynomial property of selfadjoint elliptic boundary value problems, and the algebraic description of their attributes, Uspekhi Mat. Nauk 54 (1999), no. 5, 77-142; English transl., Russian Math. Surveys 54 (1999), no. 5, 947-1014. MR 1741662 (2001k:35073)
  • 5. S. G. Lekhnit'skiĭ, Theory of elasticity of an anisotropic body, 2nd ed., Nauka, Moscow, 1977. (Russian) MR 0502604 (58:19575)
  • 6. S. A. Nazarov, Asymptotic theory of thin plates and rods. Dimension reduction and integral estimates. Vol. 1, Nauchn. Kniga, Novosibirsk, 2002. (Russian)
  • 7. V. Z. Parton and B. A. Kudryavtsev, Electro-elasticity of piezoelectric and electroconductive bodies, Nauka, Moscow, 1988. (Russian)
  • 8. S. A. Nazarov, Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigenoscillations of a piezoelectric plate, Probl. Mat. Anal., No. 25, Tamara Rozhkovskaya, Novosibirsk, 2003, pp. 99-188; English transl., J. Math. Sci. (New York) 114 (2003), no. 5, 1657-1725. MR 1981301 (2004e:35028)
  • 9. E. A. Akimova, S. A. Nazarov, and G. A. Chechkin, The weighted Korn inequality: the ``tetris'' procedure that serves an arbitrary periodic plate, Dokl. Akad. Nauk 380 (2001), no. 4, 439-442. (Russian) MR 1875497
  • 10. -, Asymptotics of the solution of the problem of the deformation of an arbitrary locally periodic thin plate, Tr. Mosk. Mat. Obshch. 65 (2004), 3-34; English transl., Trans. Moscow Math. Soc. 2004, 1-29. MR 2193435 (2006j:74056)
  • 11. S. A. Nazarov, Korn's inequalities for elastic joints of massive bodies, thin plates, and rods, Uspekhi Mat. Nauk 63 (2008), no. 1, 37-110; English transl., Russian Math. Surveys 63 (2008), no. 1, 35-107. MR 2406182 (2009c:35457)
  • 12. -, Justification of the asymptotic theory of thin rods. Integral and pointwise estimates, Probl. Mat. Anal., No. 17, S.-Peterburg. Univ., St. Petersburg, 1997, pp. 101-152; English transl., J. Math. Sci (New York) 97 (1999), no. 4, 4245-4279. MR 1788230 (2002d:74045)
  • 13. A. Bensoussan, J.-L. Lions, and G. Papanicolaou, Asymptotic analysis for periodic structures, Stud. in Math. Appl., vol. 5, North-Holland, Amsterdam-New York, 1978. MR 0503330 (82h:35001)
  • 14. N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging processes in periodic media, Mathematical Problems of the 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); MR 1112788 (92d:73002)
  • 15. E. Sanchez-Palencia, Nonhomogeneous media and vibration theory, Lecture Notes in Phys., vol. 127, Springer-Verlag, Berlin-New York, 1980. MR 0578345 (82j:35010)
  • 16. O. A. Oleĭnik, G. A. Iosif'yan, and A. S. Shamaev, Mathematical problems in the theory of strongly inhomogeneous elastic media, Moskov. Gos. Univ., Moscow, 1990. (Russian) MR 1115306 (92i:73009)
  • 17. S. A. Nazarov, Homogenization of elliptic systems with periodic coefficients: Weigted $ L^p$ and $ L^{\infty}$ estimates for asymptotic remainders, Algebra i Analiz 18 (2006), no. 2, 117-166; English transl., St. Petersburg Math. J. 18 (2007), no. 2, 269-304. MR 2244938 (2007d:35016)
  • 18. O. A. Ladyzhenskaya and N. N. Ural'tseva, Linear and quasilinear equations of elliptic type, 2nd ed., Nauka, Moscow, 1973; English transl. of 1st ed., Acad. Press, New York-London, 1968. MR 0509265 (58:23009); MR 0244627 (39:5941)
  • 19. 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)
  • 20. -, 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)
  • 21. -, Homogenization with corrector term 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)
  • 22. V. V. Zhikov, On the spectral method in homogenization theory, Tr. Mat. Inst. Steklov. 250 (2005), 95-104; English transl., Proc. Steklov Inst. Math. 2005, no. 3 (250), 85-94. MR 2200910 (2006i:35016)
  • 23. 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. (English) MR 2201316 (2007c:35014)
  • 24. S. E. Pastukhova, On some estimates from the homogenization of problems in plasticity theory, Dokl. Akad. Nauk 406 (2006), no. 5, 604-608. (Russian) MR 2347320 (2008d:35016)
  • 25. S. Agmon, A. Douglis, and L. Nirenberg, Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. II, Comm. Pure Appl. Math. 17 (1964), 35-92. MR 0162050 (28:5252)
  • 26. V. A. Solonnikov, General boundary value problems for systems elliptic in the sense of A. Douglis and L. Nirenberg. I, Izv. Akad. Nauk SSSR Ser. Mat. 28 (1964), 665-706. (Russian); II, Tr. Mat. Inst. Steklov. 92 (1966), 233-297; English transl. in Proc. Steklov Inst. Math. 1968. MR 0211070 (35:1952); MR 0211071 (35:1953)

Similar Articles

Retrieve articles in St. Petersburg Mathematical Journal with MSC (2010): 35J57

Retrieve articles in all journals with MSC (2010): 35J57

Additional Information

G. Cardone
Affiliation: Department of Engineering, University of Sannio, Corso Garibaldi, 107, 84100 Benevento, Italy

A. Corbo Esposito
Affiliation: Department of Automation, Electromagnetism, Information and Industrial Mathematics, University of Cassino, Via G. Di Biasio, 43, 03043 Cassino (FR), Italy

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

Keywords: G{\aa }rding--Korn inequality, homogenization, formally selfadjoint elliptic system, rate of convergence
Received by editor(s): November 24, 2008
Published electronically: May 20, 2010
Additional Notes: S. A. Nazarov was supported by RFBR (grant no. 09-00759).
Article copyright: © Copyright 2010 American Mathematical Society

American Mathematical Society