Advertisement

Journal of Mathematical Sciences

, Volume 159, Issue 1, pp 113–132 | Cite as

Convergence theorems for solutions and energy functionals of boundary value problems in thick multilevel junctions of a new type with perturbed neumann conditions on the boundary of thin rectangles

  • T. A. Mel’nik
  • G. A. Chechkin
  • T. P. Chechkina
Article

Boundary value problems for the Poisson equation are considered in a multilevel thick junction consisting of a junction body and a lot of alternating thin rectangles of two levels depending on their lengths. Rectangles of the first level have a finite length, whereas rectangles of the second level have a length ε α , 0 < α < 1, where ε is the alternation period. On the boundary of thin rectangles, an inhomogeneous Neumann boundary condition involving additional perturbation parameters is imposed. We prove convergence theorems for solutions and energy integrals. Regarding the convergence of solutions of the original problem to solutions of the homogenized problem, we establish some (auxiliary) estimates necessary for obtaining the convergence rate. Bibliography: 48 titles. Illustrations: 3 figures.

Keywords

Integral Identity Homogenize Problem Nonlinear Boundary Condition Energy Integral Perforated Domain 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    V. A. Marchenko and E. Ya. Khruslov, Boundary Value Problems in Domains with a Fine-Grained Boundary [in Russian], Naukova Dumka, Kiev (1974).MATHGoogle Scholar
  2. 2.
    A. Bensoussan, J.-L. Lions, and G. Papanicolau, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).MATHGoogle Scholar
  3. 3.
    N. S. Bakhvalov and G. P. Panasenko, Homogenization: Averaging Processes [in Russian], Nauka, Mowcow (1984); English transl.: Kluwer Academic Publisher, Dordrecht (1990).Google Scholar
  4. 4.
    V. V. Zhikov, S. M. Kozlov, and O. A. Oleinik, Homogenization of Differential Operators and Integral Functions, Springer-verlag, Berlin-New York (1994).Google Scholar
  5. 5.
    D. Cioranescu and J. Saint Jean Paulin, Homogenization of Reticulated Structures, Springer-Verlag, Berlin–New York (1998).Google Scholar
  6. 6.
    S. A. Nazarov, Asymptotic Theory of Thin Plates and Rods [in Russian], Nauchnaya Kniga (IDMI), Novosibirsk (2002).Google Scholar
  7. 7.
    G. A. Chechkin, A. L. Piatnitski, and A. S. Shamaev, Homogenization. Methods and Applications [in Russian], Tamara Rozhkovskaya Publisher, Novosibirsk, (2007); English transl.: Am. Math. Soc., Providence, RI, (2007).MATHGoogle Scholar
  8. 8.
    E. Ya. Khruslov, “On resonance phenomena in a diffraction problem” [in Russian], Teor. Funkts., Funkts. Anal. Pril. 10, 113–120 (1968).Google Scholar
  9. 9.
    V. P. Kotlyarov and E. Ya. Khruslov, “On limit boundary condition in a Neumann problem” [in Russian] Teor. Funkts., Funkts. Anal. Pril. 10, 83–96 (1970).MATHGoogle Scholar
  10. 10.
    T. A. Mel’nik and S. A. Nazarov, “The asymptotic structure of the spectrum in the problem of harmonic oscillations of a hub with heavy spokes” [in Russian], Dokl. Akad. Nauk, Ross. Akad. Nauk 333, no. 1, 13–15 (1993); English transl.: Russ. Acad. Sci., Dokl., Math. 48, no. 3, 428–432 (1994).Google Scholar
  11. 11.
    T. A. Mel’nik and S. A. Nazarov, “The asymptotics of the solution to the Neumann spectral problem in a domain of the “dense-comb” type” [in Russian], Tr. Semin. Im. I. G. Petrovskogo 19, 138–173 (1996); English transl.: J. Math. Sci., New York 85, no. 6, 2326–2346 (1997).Google Scholar
  12. 12.
    T. A. Mel’nyk, “Homogenization of the Poisson equation in a thick periodic junction,” Z. Anal. An. 18, No. 4, 953–975 (1999).MATHMathSciNetGoogle Scholar
  13. 13.
    T. A. Mel’nik and S. A. Nazarov, “Asymptotic analysis of the Neumann problem on the junction of a body and thin heavy rods” [in Russian], Algebra Anal. 12, no. 2, 188–238 (2000); English transl.: St. Petersb. Math. J. 12, no. 2, 317–351 (2001).MathSciNetGoogle Scholar
  14. 14.
    T. A. Mel’nyk, “Homogenization of a singularly perturbed parabolic problem in a thick periodic junction of type 3:2:1,” Ukr. Math. J. 52, No. 11, 1737–1749 (2000).CrossRefMathSciNetGoogle Scholar
  15. 15.
    S. A. Nazarov, “Junctions of singularly degenerating domains with different limit dimensions I, II” [in Russian], Tr. Semin. Im. I. G. Petrovskogo 18, 1–18 (1995); 20, 155–196 (1997); English transl.: J. Math. Sci., New York 80, no. 5, 1989–2034 (1996); 97, no. 3, 4085–4108 (1999).Google Scholar
  16. 16.
    T. A. Mel’nyk, “Homogenization of a boundary value problem with a nonlinear boundary condition in a thick junction of type 3:2:1,” Math. Models Meth. Appl. Sci. 31, No. 9, 1005–1027 (2008).MATHMathSciNetGoogle Scholar
  17. 17.
    U. De Maio, T. Durante, and T.A. Mel’nyk, “Asymptotic Approximation for the Solution to the Robin Problem in a Thick Multi-Level Junction,” Math. Models Meth. Appl. Sci. 15, No. 12, 1897–1921 (2005).MATHCrossRefGoogle Scholar
  18. 18.
    T. A. Mel’nyk, “Asymptotic behavior of eigenvalues and eigenfunctions of the Fourier problem in a thick multi-level junction,” Ukr. Math. J. 58, No. 2, 220–243 (2006).CrossRefMathSciNetGoogle Scholar
  19. 19.
    T. Durante, T. A. Mel’nyk, and P. S. Vashchuk, “Asymptotic approximation for the solution to a boundary value problem with varying type of boundary conditions in a thick two-level junction,” Nonlinear Oscillations 9, No. 3, 336–355 (2006).CrossRefMathSciNetGoogle Scholar
  20. 20.
    T. A. Mel’nik and P. S. Vashchuk, “Homogenization of a boundary value problem with mixed type of boundary conditions in a thick junction” [in Russian], Differ. Uravn. 43, no. 5, 677–784 (2007); English transl.: Differ. Equ. 43, no. 5, 696–703 (2007).MathSciNetGoogle Scholar
  21. 21.
    T. Durante and T. A. Mel’nyk, “Asymptotic analysis of a parabolic problem in a thick two-level junction,” J. Math. Phys., Anal., Geometry 3, No. 3, 313–341 (2007).MATHMathSciNetGoogle Scholar
  22. 22.
    T. A. Mel’nyk and G. A. Chechkin, “ Homogenization of a boundary value problem in a thick cascade junction” [in Russian], Probl. Mat. Anal. 37, 47–72 (2008); English transl.: J. Math. Sci., New York 154, No. 1, 50–77 (2008).Google Scholar
  23. 23.
    T. A. Mel’nyk and G. A. Chechkin, “Asymptotic analysis of boundary value problems in thick cascade junctions”[in Russian], Dok. NAN Urk. 9, 16–22 (2008).MathSciNetGoogle Scholar
  24. 24.
    T. A. Mel’nyk and G. A. Chechkin, “ Aymptotic analysis of boundary value problems in three-dimensional thick multilevel juncitons” [in Russian], Mat. Sb. 200, No. 3, 49–74 (2009).Google Scholar
  25. 25.
    D. Blanchard, A. Gaudiello, and T. A. Mel’nyk, “Boundary homogenization and reduction of dimension in a Kirchhoff-Love plate,” SIAM J. Math. Anal. 39, No. 6, 1764–1787 (2008).MATHCrossRefMathSciNetGoogle Scholar
  26. 26.
    D. Blanchard, A. Gaudiello, and J. Mossino, “Highly oscillating boundaries and reduction of dimension in the critical case,” Anal. Appl. 5, 137–163 (2007).MATHCrossRefMathSciNetGoogle Scholar
  27. 27.
    D. Blanchard, A. Gaudiello, and G. Griso, “Junction of a periodic family of elastic rods with 3d plate Part I II,” J. Math. Pures Appl. 88, No. 9, 1–33 (2007); 88, No. 9, 149–190 (2007).MATHMathSciNetGoogle Scholar
  28. 28.
    C. D’Apice, U. De Maio, and T. A. Mel’nyk, “Asymptotic analysis of a perturbed parabolic problem in a thick junction of type 3:2:2,” Networks Heterogen. Media 2, 255–277 (2007).MATHGoogle Scholar
  29. 29.
    Yu. A. Kazmerchuk and T. A. Mel’nyk, “Homogenization of the Signorini boundary value problem in a thick plane junction,” Nonlinear Oscillations 12, No. 1 (2009).Google Scholar
  30. 30.
    Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics of the Solution of a Dirichlet Spectral Problem in a Junction with Highly Oscillating Boundary,” CR Mécanique 336, No. 9. 693–698 (2008).MATHCrossRefGoogle Scholar
  31. 31.
    Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics of Simple Eigenvalues and Eigenfunctions for the Laplace Operator in a Domain with Oscillating Boundary” Zh. Vych. Mat. Mat. Phyz. 46, no. 1, 102–115 (2006); English edition: Comput. Math. Math. Phys. 46, no. 1, 97–110 (2006).MATHMathSciNetGoogle Scholar
  32. 32.
    Y. Amirat, G. A. Chechkin, and R. R. Gadyl’shin, “Asymptotics for Eigenelements of Laplacian in Domain with Oscillating Boundary:. Multiple Eigenvalues,” Appl. Anal. 86, No. 7, 873–897 (2007).MATHCrossRefMathSciNetGoogle Scholar
  33. 33.
    A. G. Belyaev, On Singular Perturbations of Boundary Value Problems, Thesis, Moscow State University, Moscow (1990).Google Scholar
  34. 34.
    A. G. Belyaev, A. L. Piatnitski, and G. A. Chechkin, “Asymptotic behavior of a solution to a boundary value problem in a perforated domain with oscillating boundary” [in Russian], Sib. Mat. Zh. 39, no. 4, 730–754 (1998); English transl.: Sib. Math. J. 39, no. 4, 621–644 (1998).MATHGoogle Scholar
  35. 35.
    G. Bouchitté, A. Lidouh, and P. Suquet, “Homogénéisation de fronti`ere pour la modélisation du contact entre un corps déformable non linéaire et un corps rigide,” C. R. Acad. Sc. Paris. Sér. I 313, 967–972 (1991).MATHGoogle Scholar
  36. 36.
    G. A. Chechkin and D. Cioranescu, “Vibration of a Thin Plate with a “Rough” Surface,” In: Nonlinear Partial Differential Equations and their Applications. Coll`ege de France Seminar. Volume XIV. Studies in Mathematics and its Applications, Elsevier (2002), pp. 147–169.Google Scholar
  37. 37.
    G. A. Chechkin, A. Friedman, and A. L. Piatnitski, “The Boundary Value Problem in Domains with Very Rapidly Oscillating Boundary,” J. Math. Anal. Appl. 231, No. 1, 213–234 (1999).MATHCrossRefMathSciNetGoogle Scholar
  38. 38.
    E. N. Dancer and D. Daners, “Domain Perturbation for Elliptic Equations Subject to Robin Boundary Conditions,” J. Differ. Equ. 138, No. 1, 86–132 (1997).MATHCrossRefMathSciNetGoogle Scholar
  39. 39.
    A. Gaudiello, “Asymptotic behavior of non-homogeneous Neumann problems in domains with oscillating boundary,” Ric. Math 43, 239–292 (1994).MATHMathSciNetGoogle Scholar
  40. 40.
    W. Jäger and A. Mikelić, “On the roughness-induced boundary conditions for an incompressible viscous flow,” J. Differ. Equ. 170, No. 1, 96–122 (2001).MATHCrossRefGoogle Scholar
  41. 41.
    W. Kohler, G. Papanicolaou, and S. Varadhan, “Boundary and Interface Problems in Regions with Very Rough Boundaries,” In: Multiple Scattering and Waves in Random Media, North-Holland, Amsterdam (1981), pp. 165–197.Google Scholar
  42. 42.
    J. Nevard and J. B. Keller, “Homogenization of Rough Boundaries and Interfaces,” SIAM J. Appl. Math, 57, No. 6, 1660–1686 (1997).MATHCrossRefMathSciNetGoogle Scholar
  43. 43.
    T. A. Mel’nik, “Averaging of elliptic equations describing processes in strongly inhomogeneous thin perforated domains with rapidly changing thickness” [in Russian], Akad. Nauk Ukr. SSR 10, 15–18 (1991).MathSciNetGoogle Scholar
  44. 44.
    G. A. Chechkin and T. P. Chechkina, “On homogenization of problems in domains of the “infusorium” type” [in Rissian], Tr. Semin. Im. I. G. Petrovskogo 23, 386–607 (2003); English transl.: J. Math. Sci., New York 120, no. 3, 1470–1482 (2004).Google Scholar
  45. 45.
    G. A. Chechkin and T. P. Chechkina, “An averaging theorem for problems in domains of “infusoria” type with inconsistent structure” [in Russian], In: Sovrem. Mat. Prilozh. No. 2, Differ. Uravn. Chast. Proizvod., 139–954 (2003); Englsih transl.: J. Math. Sci. New York 123, no. 5, 4363–4380 (2004).Google Scholar
  46. 46.
    O. A. Ladyzhenskaya, Boundary Value Problems of Mathematical Physics [in Russian], Nauka, Moscow (1973).Google Scholar
  47. 47.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics [in Russian], 3rd ed., Nauka, Moscow (1988); English transl.: Am. Math. Soc., Providence, RI (1991).Google Scholar
  48. 48.
    S. L. Sobolev, Selected Topics in the Theory of Function Spaces and Distributions [in Russian], Nauka, Moscow (1989).Google Scholar

Copyright information

© Springer Science+Business Media, Inc. 2009

Authors and Affiliations

  • T. A. Mel’nik
    • 1
  • G. A. Chechkin
    • 2
    • 3
  • T. P. Chechkina
    • 4
  1. 1.National Taras Shevchenko University of KyivKyivUkraine
  2. 2.Lomonosov Moscow State UniversityMoscowRussia
  3. 3.Narvik University CollegeNarvikNorway
  4. 4.Moscow Engineering Physical Institute (State University)MoscowRussia

Personalised recommendations