Asymptotics of simple eigenvalues and eigenfunctions for the Laplace operator in a domain with an oscillating boundary

  • Y. Amirat
  • G. A. Chechkin
  • R. R. Gadyl’shin


The asymptotic behavior of solutions to spectral problems for the Laplace operator in a domain with a rapidly oscillating boundary is analyzed. The leading terms of the asymptotic expansions for eigenelements are constructed, and the asymptotics are substantiated for simple eigenvalues.


oscillating boundary spectrum of the Laplacian asymptotics matching of asymptotic expansions 


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  1. 1.
    Y. Achdou, O. Pironneau, and F. Valentin, “Effective Boundary Conditions for Laminar Flows over Rough Boundaries,” J. Comput. Phys. 147, 187–218 (1998).CrossRefMathSciNetGoogle Scholar
  2. 2.
    Y. Amirat and J. Simon, “Riblets and Drag Minimization,” Optimization Methods in PDE’s: Contemporary Mathematics (Am. Math. Soc., 1997), pp. 9–17.Google Scholar
  3. 3.
    Y. Amirat, D. Bresch, J. Lemoine, et al., “Effect of Rugosity on a Flow Governed by Navier-Stokes Equations,” Q. Appl. Math. 59, 769–785 (2001).MathSciNetGoogle Scholar
  4. 4.
    I. Babuška and R. Vyborny, “Continuous Dependence of Eigenvalues on the Domains,” Czech. Math. J. 15, 169–178 (1965).Google Scholar
  5. 5.
    A. G. Belyaev, “Average of the Third Boundary-Value Problem for the Poisson Equation in a Domain with Rapidly Oscillating Boundary,” Vestn. Mosk. Gos. Univ., Ser. 1: Math. Mech. 6, 63–66 (1988).MATHMathSciNetGoogle Scholar
  6. 6.
    A. G. Belyaev, Dissertation in Mathematics and Physics (Moscow State Univ, Moscow, 1990).Google Scholar
  7. 7.
    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,” Sib. Math. J. 39, 621–644 (1998).Google Scholar
  8. 8.
    G. Bouchitte, A. Lidouh, and P. Suquet, “Homogénéisation de frontière pour la modélisation du contact entre un corps déformable non linéaire un corps rigide,” C. R. Acad. Sci., Ser. I 313, 967–972 (1991).MathSciNetGoogle Scholar
  9. 9.
    R. Brizzi and J. P. Chalot, “Homogénéisation de frontière,” Ric. Mat. 46, 341–387 (1997).MathSciNetGoogle Scholar
  10. 10.
    G. A. Chechkin and T. P. Chechkina, “On Homogenization Problems in Domains of the “Infusorium” Type,” J. Math. Sci. 120, 1470–1482 (2004).CrossRefMathSciNetGoogle Scholar
  11. 11.
    G. A. Chechkin and T. P. Chechkina, “Homogenization Theorem for Problems in Domains of the “Infusorian” Type with Uncoordinated Structure,” J. Math. Sci. 123, 4363–4380 (2004).CrossRefMathSciNetGoogle Scholar
  12. 12.
    G. A. Chechkin and D. Cioranescu, “Vibration of a Thin Plate with a “Rough” Surface,” Nonlinear Partial Differential Equations and Their Applications, Collége de France Seminar, Vol. XIV: Studies in Mathematics and Its Application (Elsevier, Amsterdam, 2002), pp. 147–169.Google Scholar
  13. 13.
    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, 213–234 (1999).CrossRefMathSciNetGoogle Scholar
  14. 14.
    A. Friedman, Hu Bei, and Liu Yong, Preprint No. 1415 (IMA, Univ. Minnesota, Minneapolis, 1996).Google Scholar
  15. 15.
    A. Gaudiello, “Asymptotic Behavior of Nonhomogeneous Neumann Problems in Domains with Oscillating Boundary,” Ric. Math. 43, 239–292 (1994).MATHMathSciNetGoogle Scholar
  16. 16.
    W. Jäger, O. A. Oleinik, and T. A. Shaposhnikova, “On the Averaging of Boundary Value Problems in Domains with Rapidly Oscillating Nonperiodic Boundary,” Differ. Equations 36, 833–846 (2000).Google Scholar
  17. 17.
    W. Jäger and A. Mikelić, “On the Roughness-Induced Effective Boundary Conditions for an Incompressible Viscous Flow,” J. Differ. Equations 170, 96–122 (2001).Google Scholar
  18. 18.
    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
  19. 19.
    V. A. Marchenko and E. Ya. Khruslov, Boundary-Value Problems in Domains with Fine-Grained Boundaries (Naukova Dumka, Kiev, 1974) [in Russian].Google Scholar
  20. 20.
    T. A. Mel’nik and S. A. Nazarov, “Asymptotic Behavior of the Solution of the Neumann Spectral Problem in a Domain of “Tooth Comb” Type,” J. Math. Sci. 85, 2326–2346 (1997).MathSciNetGoogle Scholar
  21. 21.
    S. A. Nazarov and M. V. Olyushin, “Perturbation of the Eigenvalues of the Neumann Problem Due to the Variation of the Domain Boundary,” St. Petersburg Math. J. 5, 371–385 (1994).MathSciNetGoogle Scholar
  22. 22.
    J. Nevard and J. B. Keller, “Homogenization of Rough Boundaries and Interfaces,” SIAM J. Appl. Math. 57, 1660–1686 (1997).CrossRefMathSciNetGoogle Scholar
  23. 23.
    E. Sánchez-Palencia, Homogenization Techniques for Composite Media (Springer-Verlag, Berlin, 1987).Google Scholar
  24. 24.
    O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization (North-Holland, Amsterdam, 1992).Google Scholar
  25. 25.
    A. M. II’in, Matching of Asymptotic Expansions of Solutions of Boundary Value Problems (Am. Math. Soc., Providence, 1992).Google Scholar
  26. 26.
    M. Lobo-Hidalgo and E. Sánchez-Palencia, “Sur certaines propriétés spectrales des perturbations du domaine dans les problémes aux limites,” Commun. Partial Differ. Equations 4, 1085–1098 (1979).Google Scholar
  27. 27.
    S. L. Sobolev, Some Applications of Functional Analysis in Mathematical Physics (Am. Math. Soc., Providence, 1991).Google Scholar
  28. 28.
    S. L. Sobolev, Selected Problems in the Theory of Function Spaces and Generalized Functions (Nauka, Moscow, 1989) [in Russian].Google Scholar
  29. 29.
    R. R. Gadyl’shin, “Characteristic Frequencies of Bodies with Thin Spikes. I: Convergence and Estimates,” Math. Notes 54, 1192–1199 (1993).MathSciNetGoogle Scholar
  30. 30.
    R. R. Gadyl’shin, “Concordance Method of Asymptotic Expansions in a Singularly-Perturbed Boundary-Value Problem for the Laplace Operator,” J. Math. Sci. 125, 579–609 (2005).MathSciNetGoogle Scholar
  31. 31.
    R. R. Gadyl’shin, “Asymptotic Properties of an Eigenvalue of a Problem for a Singularly Perturbed Self-Adjoint Elliptic Equation with a Small Parameter in the Boundary Conditions,” Differ. Equations 22, 474–483 (1986).MathSciNetGoogle Scholar
  32. 32.
    M. Yu. Planida, “On the Convergence of Solutions of Singularly Perturbed Boundary-Value Problems for the Laplace Operator,” Math. Notes 71, 794–803 (2002).CrossRefMATHMathSciNetGoogle Scholar
  33. 33.
    A. M. Il’in, “A Boundary-Value Problem for an Elliptic Equation of Second Order in a Domain with a Narrow Slit. I: The Two-Dimensional Case,” Math. Sib. (N.S.) 99, 514–537 (1976).Google Scholar
  34. 34.
    A. M. Il’in, “Boundary-Value Problem for an Elliptic Equation of Second Order in a Domain with a Narrow Slit. II: Domain with a Small Opening,” Math. Sib. (N.S.) 103, 265–284 (1977).Google Scholar
  35. 35.
    A. M. Il’in, “Study of the Asymptotic Behavior of the Solution of an Elliptic Boundary-Value Problem in a Domain with a Small Hole,” Tr. Semin. I.G. Petrovskogo 6, 57–82 (1981).Google Scholar
  36. 36.
    R. R. Gadyl’shin, “Asymptotics of the Minimum Eigenvalue for a Circle with Fast Oscillating Boundary Conditions,” C. R. Acad. Sci., Sér. I 323, 319–323 (1996).MathSciNetGoogle Scholar
  37. 37.
    R. R. Gadyl’shin, “Boundary-Value Problem for the Laplacian with Rapidly Oscillating Boundary Conditions,” Dokl. Math. 58, 293–296 (1998).Google Scholar
  38. 38.
    R. R. Gadyl’shin, “On the Eigenvalue Asymptotics for Periodically Clamped Membranes,” St. Petersburg Math. J. 10, 1–14 (1999).MathSciNetGoogle Scholar
  39. 39.
    G. Allaire and M. Amar, “Boundary Layer Tails in Periodic Homogenization,” ESAIM Control Optim. Calculus Variations 4, 209–243 (1999).MathSciNetGoogle Scholar
  40. 40.
    E. M. Landis and G. P. Panasenko, “A Theorem on the Asymptotics of Solutions of Elliptic Equations with Coefficients Periodic in All Variables Except One,” Sov. Math. Dokl. 18, 1140–1143 (1977).Google Scholar
  41. 41.
    J.-L. Lions, Some Methods in the Mathematical Analysis of Systems and Their Control (Gordon & Breach, New York, 1981).Google Scholar
  42. 42.
    S. A. Nazarov, Binomial Asymptotic Behavior of Solutions of Spectral Problems with Singular Perturbations, Mat. Sb. 181, 291–320 (1990).Google Scholar

Copyright information

© MAIK “Nauka/Interperiodica” 2006

Authors and Affiliations

  • Y. Amirat
    • 1
  • G. A. Chechkin
    • 2
  • R. R. Gadyl’shin
    • 3
    • 4
  1. 1.Laboratoire de MathématiquesUniversité Blaise PascalAubière cedexFrance
  2. 2.Department of Differential Equations, Faculty of Mechanics and MathematicsMoscow State UniversityMoscowRussia
  3. 3.Institute of Mathematics (with Computing Center)Russian Academy of SciencesUfaRussia
  4. 4.Department of Mathematical Analysis, Faculty of Physics and MathematicsBashkir State Pedagogical UniversityUfaRussia

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