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Asymptotic expansion of eigenvalues of the neumann problem in a domain with a thin bridge

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References

  1. A. A. Arsen'ev, “On singularities of analytic continuation and resonance properties of the scattering problem for the Helmholtz equation,” Zhurn. Vichislit. Matematiki i Mat. Fiziki, No. 12, 112–138 (1972).

    Google Scholar 

  2. J. T. Beale, “Scattering frequencies of resonator,” Comm. Pure Appl. Math.,26, 549–563 (1973).

    Google Scholar 

  3. S. Jimbo, “Characterization of the eigenfunctions on the singularly perturbed domain,” Proc. Jap. Acad.63, No. 8, 285–288 (1987); Ibid. S. Jimbo, “Characterization of the eigenfunctions on the singularly perturbed domain,”, Proc. Jap. Acad.64, No. 1, 14–16 (1988).

    Google Scholar 

  4. C. Anne, “Spectre du laplacien et ecrasement d'anses,” Ann. Scient. Ec. Norm. Sup., 4 ser.,20, 271–280 (1987).

    Google Scholar 

  5. C. Anne, “Perturbation du spectre X−TUBeY (conditions de Neumann),” Seminaire de Theorie Spectrale et Geometrie de l'Institut Fourier,4, 17–23 (1986).

    Google Scholar 

  6. I. Chavel and E. A. Feldman, “Spectra of Manifolds Less a Small Domain,” Duke Math. J.,56, No. 2, 399–414 (1988).

    Google Scholar 

  7. A. L. Gol'denveîzer, “Constructing the approximate theory of membrane bending by method of asymptotic integration of elasticity theory equations,” Prikladnaya Matematika i Mekhanika,27, No. 6, 1057–1074 (1962).

    Google Scholar 

  8. A. L. Gol'denveîzer and A. V. Kolos, “On constructing two-dimensional equations for elastic thin membranes,” Prikladnaya Matematika i Mekhanika,29, No. 1, 141–161 (1965).

    Google Scholar 

  9. M. G. Dzhavadov, “Asymptotic expansions of the solution to the boundary value problem for elliptic equations of second order in thin domains,” Differents. Uravnenia,5, No. 10, 1901–1909 (1968).

    Google Scholar 

  10. I. E. Zino and E. A. Tropp, Asymptotic Methods in Problems of Heat Conduction and Thermoelasticity [in Russian], Leningrad University Press, Leningrad (1978).

    Google Scholar 

  11. S. A. Nazarov, “Structure of solutions to elliptic boundary value problems in thin domains,” Vest. LGU, No. 7, 65–68 (1982).

    Google Scholar 

  12. S. N. Leora, S. A. Nazarov, and A. V. Proskura, “Deriving limit equations for elliptic problems in thin domains on a computer,” Zhurn. Vichislit. Matematiki i Mat. Fiziki,26, No. 7, 1032–1048 (1986).

    Google Scholar 

  13. I. S. Gohberg and M. G. Kreîn, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space [in Russian], Nauka, Moscow (1965).

    Google Scholar 

  14. S. Agmon and L. Nirenberg, “Properties of solutions of ordinary differential equations in Banach space,” Comm. Pure Appl. Math.,16, 121–239 (1963).

    Google Scholar 

  15. V. A. Kondrat'ev, “boundary value problems for elliptic equations in domains with conic or angular points,” Tr. Mosk. Mat. Obshch,16, 209–292 (1967).

    Google Scholar 

  16. V. G. Maz'ya and B. A. Plamenevskiî, “On the coefficients of the asymptotic expansions for solutions of elliptic boundary value problems in a domain with conic points,” Math. Nachz.,76, 29–60 (1977).

    Google Scholar 

  17. S. A. Nazarov and P. K. Chern'yaev, “Antiplane shift of a domain with two nearly situated cracks,” Prikladnaya Matematika i Mekhanika,50, No. 5, 815–825 (1986).

    Google Scholar 

  18. E. M. Landis, Second-Order Equations of Elliptic and Parabolic Types [in Russian], Nauka, Moscow (1971).

    Google Scholar 

  19. M. I. Vishik and L. A. Lyusternik, “Solutions to some perturbation problems in the case of matrices and selfadjoint and nonselfadjoint differential equations,” Usp. Mat. Nauk,15, No. 3, 3–80 (1960).

    Google Scholar 

  20. M. M. Vaînberg and V. A. Trenogin, The Theory of Branching of Solutions of Nonlinear Equations, Noordhoff, Leyden (1974).

    Google Scholar 

  21. M. I. Vishik and L. A. Lyusternik, “Regular degeneration and boundary layer for linear differential equations with a small parameter,” Usp. Mat. Nauk,12, No. 5, 3–122 (1957).

    Google Scholar 

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St. Petersburg. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 80–96, July–August, 1992.

Translated by G. V. Dyatlov

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Nazarov, S.A., Polyakova, O.R. Asymptotic expansion of eigenvalues of the neumann problem in a domain with a thin bridge. Sib Math J 33, 618–633 (1992). https://doi.org/10.1007/BF00971127

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  • DOI: https://doi.org/10.1007/BF00971127

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