Revista Matemática Complutense

, Volume 31, Issue 1, pp 1–62 | Cite as

On multi-scale asymptotic structure of eigenfunctions in a boundary value problem with concentrated masses near the boundary

  • Sergei A. Nazarov
  • M. Eugenia PérezEmail author


We construct two-term asymptotics \(\lambda ^\varepsilon _k= \varepsilon ^{m-2}(M+\varepsilon \mu _k+ O(\varepsilon ^{3/2}) )\) of eigenvalues of a mixed boundary-value problem in \(\Omega \subset {{\mathbb {R}}}^2\) with many heavy (\(m>2\)) concentrated masses near a straight part \(\Gamma \) of the boundary \(\partial \Omega \). \(\varepsilon \) is a small positive parameter related to size and periodicity of the masses; \(k\in {\mathbb N}\). The main term \(M>0\) is common for all eigenvalues but the correction terms \(\mu _k\), which are eigenvalues of a limit problem with the spectral Steklov boundary conditions on \(\Gamma \), exhibit the effect of asymptotic splitting in the eigenvalue sequence enabling the detection of asymptotic forms of eigenfunctions. The justification scheme implies isolating and purifying singularities of eigenfunctions and leads to a new spectral problem in weighed spaces with a “strongly” singular weight.


Spectral analysis Homogenization problems Concentrated masses Asymptotic splitting of eigenvalues Steklov problem Corner singularities 

Mathematics Subject Classification

35B25 35B27 35P05 35B40 47A75 74H45 



This research work has been partially supported by Spanish MINECO, MTM2013-44883-P. Also, the research work of the first author has been partially supported by Russian Foundation of Basic research (Project 15–01–02175).


  1. 1.
    Arrieta, J.M., Jiménez-Casas, A., Rodríguez-Bernal, A.: Flux terms and Robin boundary conditions as limit of reactions and potentials concentrating at the boundary. Rev. Mat. Iberoam. 24(1), 183–211 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Agmon, S., Douglis, A., Nirenberg, L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Commun. Pure Appl. Math. 12, 623–727 (1959)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Birman, M.S., Solomyak, M.Z.: Spectral Theory of Self-Adjoint Operators in Hilbert Space. Reidel Publ. Company, Dordrecht (1987)zbMATHGoogle Scholar
  4. 4.
    Campillo, M., Dascalu, C., Ionescu, I.: Inestability of a periodic systems of faults. Geophys. J. Int. 159, 212–222 (2004)CrossRefGoogle Scholar
  5. 5.
    Chechkin, G.A.: Asymptotic expansion of the eigenelements of the Laplace operator in a domain with a large number of “light” concentrated masses sparsely located on the boundary: the two-dimensional case. Tr. Mosk. Mat. Obs. 70, 102–182 (2009) [English transl.: Trans. Moscow Math. Soc. 71–134 (2009)]Google Scholar
  6. 6.
    Girouard, A., Polterovich, I.: Spectral geometry of the Steklov problem. J. Spectr. Theory 7(2), 321–359 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Spectral stiff problems in domains surrounded by thin bands: asymptotic and uniform estimates for eigenvalues. J. Math. Pures Appl. 85(4), 598–632 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Gómez, D., Lobo, M., Nazarov, S.A., Pérez, E.: Asymptotics for the spectrum of the Wentzell problem with a small parameter and other related stiff problems. J. Math. Pures Appl. 86(5), 369–402 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Gómez, D., Nazarov, S.A., Pérez, E.: Spectral stiff problems in domains surrounded by thin stiff and heavy bands: local effects for eigenfunctions. Netw. Heterog. Media 6(1), 1–35 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Kondratiev, V.A.: Boundary value problems for elliptic equations in domains with conical or angular points. Tr. Mosk. Mat. Obs. 16, 209–292 (1967) [English transl. Trans. Moscow Math. Soc. 16, 227–313 (1967)]Google Scholar
  11. 11.
    Kondratiev, V.A.: Singularities of the solution of the Dirichlet problem for a second order elliptic equation in the neighborhood of an edge. Differencial’nye Uravnenija 13, 2026–2032 (1977) [English transl. Differ. Equ. 13, 1411–1415 (1977)]Google Scholar
  12. 12.
    Lamberti, P.D., Provenzano, L.: Neumann to Steklov eigenvalues: asymptotic and monotonicity results. Proc. R. Soc. Edinb. Sect. A 147(2), 429–447 (2017)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Lobo, M., Nazarov, S.A., Pérez, E.: Eigen-oscillations of contrasting non-homogeneous elastic bodies: asymptotic and uniform estimates for eigenvalues. IMA J. Appl. Math. 70(3), 419–458 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Lobo, M., Pérez, E.: On vibrations of a body with many concentrated masses near the boundary. Math. Models Methods Appl. Sci. 3(2), 249–273 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Lobo, M., Pérez, E.: Vibrations of a body with many concentrated masses near the boundary: high frequency vibrations. In: Sanchez-Palencia, E. (ed.) Spectral Analysis of Complex Structures, Travaux en Cours 49, Hermann, Paris, pp. 85–101 (1995)Google Scholar
  16. 16.
    Lobo, M., Pérez, E.: Vibrations of a membrane with many concentrated masses near the boundary. Math. Models Methods Appl. Sci. 5(5), 565–585 (1995)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Lobo, M., Pérez, E.: The skin effect in vibrating systems with many concentrated masses. Math. Methods Appl. Sci. 24(1), 59–80 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Lobo, M., Pérez, E.: Local problems in vibrating systems with concentrated masses: a review. C. R. Mec. 331, 303–317 (2003)CrossRefzbMATHGoogle Scholar
  19. 19.
    Lobo, M., Pérez, E.: Long time approximations for solutions of wave equations associated with the Steklov spectral homogenization problems. Math. Methods Appl. Sci. 33, 1356–1371 (2010)MathSciNetzbMATHGoogle Scholar
  20. 20.
    Maz’ja, V.G., Plamenevskii, B.A.: Estimates in \(L_p\) and Hölder classes and the Miranda-Agmon maximum principle for solutions of elliptic boundary value problems in domains with singular points on the boundary. Math. Nachr. 81, 25–82 (1978) [English transl.: Am. Math. Soc. Transl. Ser. 2 123, 1–56 (1984)]Google Scholar
  21. 21.
    Maz’ja, V.G., Plamenevskii, B.A.: On the coefficients in the asymptotics of solutions of elliptic boundary value problems in domains with conical points. Math. Nachr. 76, 29–60 (1977) [English transl.: Am. Math. Soc. Transl. 123, 57–89 (1984)]Google Scholar
  22. 22.
    Mel’nyk, T.A.: Vibrations of a thick periodic junction with concentrated masses. Math. Models Methods Appl. Sci. 11(6), 1001–1027 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  23. 23.
    Mikhlin, S.G.: Variational Methods in Mathematical Physics, International Series of Monographs in Pure and Applied Mathematics, vol. 50. Pergamon Press, Frankfurt (1964)Google Scholar
  24. 24.
    Nazarov, S.A.: Asymptotics of the solution of a Dirichlet problem in an angular domain with a periodically changing boundary. Mat. Zametki 49(5), 86–96 (1991) [English transl.: Math. Notes 49(5), 502–509 (1991)]Google Scholar
  25. 25.
    Nazarov, S.A.: Interaction of concentrated masses in a harmonically oscillating spatial body with Neumann boundary conditions. RAIRO Modél. Math. Anal. Numér. 27(6), 777–799 (1993)MathSciNetCrossRefzbMATHGoogle Scholar
  26. 26.
    Nazarov, S.A.: Asymptotic Theory of Thin Plates and Rods, vol. 1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga, Novosibirck (2002)Google Scholar
  27. 27.
    Nazarov, S.A.: Uniform estimates of remainders in asymptotic expansions of solutions to the problem on eigen-oscillations of a piezoelectric plate. Probl. Mat. Analiz. 25. Nauchnaya Kniga, Novosibirsk, pp. 99–188 (2003) [English transl.: J. Math. Sci. 114(5), 1657–1725 (2003)]Google Scholar
  28. 28.
    Nazarov, S.A.: Asymptotic behavior of eigenvalues of the Neumann problem for systems with masses concentrated on a thin toroidal set. Vestnik St. Petersburg Univ. Math. 3, 61–71 (2006) [English transl.: Vestnik St. Petersburg Univ. Math. 39(3), 149–157 (2006)]Google Scholar
  29. 29.
    Nazarov, S.A.: The Neumann problem in angular domains with periodic boundaries and parabolic perturbations of the boundaries. Tr. Mosk. Mat. Obs. 69, 182–241 (2008) [English transl. Trans. Moscow Math. Soc. 67, 153–208 (2008)]Google Scholar
  30. 30.
    Nazarov, S.A.: Asymptotics of eigenvalues of boundary value problems for the Laplace operator in a spatial domain with a thin excluded tube. Tr. Mosk. Mat. Obs. 76(1), 1–66 (2015) [English transl.: Trans. Moscow Math. Soc.76(1), 1–53 (2015)]Google Scholar
  31. 31.
    Nazarov, S.A., Plamenevskii, B.A.: Elliptic Problems in Domains with Piecewise Smooth Boundaries. Walter de Gruyter, Berlin (1994)CrossRefGoogle Scholar
  32. 32.
    Nazarov, S.A., Pérez, E.: New asymptotic effects for the spectrum of problems on concentrated masses near the boundary. C. R. Mec. 337, 585–590 (2009)CrossRefGoogle Scholar
  33. 33.
    Nazarov, S.A., Sweers, G.: A hinged plate equation and iterated Dirichlet Laplace operator on domains with concave corners. J. Differ. Equ. 233(1), 151–180 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  34. 34.
    Nikishkin, V.A.: Singularities of the solution to the Dirichlet problem for second-order elliptic equation in a neighborhood of an edge. Vestnik Moskov. Univ. Ser. I. Mat. Mekh. 2, 51–62 (1979) [English transl. Moscow Univ. Math. Bull. 2, 53–64 (1979)]Google Scholar
  35. 35.
    Oleinik, O.A.: Homogenization problems in elasticity. Spectra of singularly perturbed operators. In: Nonclassical Continuum Mechanics (Durham, 1986), London Math. Soc. Lecture Note Ser. 122, pp. 53–95, Cambridge Univ. Press, Cambridge (1987)Google Scholar
  36. 36.
    Pérez, E.: Spectral convergence for vibrating systems containing a part with negligible mass. Math. Methods Appl. Sci. 28(10), 1173–1200 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  37. 37.
    Pérez, E.: On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem. Discrete Contin. Dyn. Syst. Ser. B 7(4), 859–883 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  38. 38.
    Pérez, E.: Long time approximations for solutions of wave equations via standing waves from quasimodes. J. Math. Pures Appl. 90(4), 387–411 (2008)MathSciNetCrossRefzbMATHGoogle Scholar
  39. 39.
    Sanchez-Palencia, E.: Perturbation of eigenvalues in thermoelasticity and vibration of systems with concentrated masses. In: Trends and Applications of Pure Mathematics to Mechanics (Palaiseau, 1983). Lecture Notes in Phys. vol. 195, pp. 346–368. Springer, Berlin (1984)Google Scholar
  40. 40.
    Vanninathan, M.: Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci. Math. Sci. 90(3), 239–271 (1981)MathSciNetCrossRefzbMATHGoogle Scholar
  41. 41.
    Vishik, M.I.: Boundary-value problems for elliptic equations degenerating on the boundary of a region. Mat. Sb. N.S. 35(77), 513–568 (1954) [Engl. Transl.: Trans. Am. Math. Soc. 35, 15–78 (1954)]Google Scholar
  42. 42.
    Vishik, M.I., Lyusternik, L.A.: Regular degeneration and boundary layer for linear differential equations with small parameter. Uspekhi Mat. Nauk. 12(5), 3–122 (1957) [English transl.: Am. Math. Soc. Transl. Ser. 2 20, 239–364 (1962)]Google Scholar

Copyright information

© Universidad Complutense de Madrid 2017

Authors and Affiliations

  1. 1.Department of Elasticity, Mathematics and Mechanics FacultySaint-Petersburg State UniversitySt. PetersburgRussia
  2. 2.Lababoratory of Mechanics of New Nano-MaterialsPeter the Great State Polytechnical UniversitySt. PetersburgRussia
  3. 3.Laboratory of Mathematical Methods in Mechanics of Materials, Institute of Problems of Mechanical EngineeringRASSt. PetersburgRussia
  4. 4.Departamento de Matemática Aplicada y Ciencias de la ComputaciónUniversidad de CantabriaSantanderSpain

Personalised recommendations