References
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).
J. T. Beale, “Scattering frequencies of resonator,” Comm. Pure Appl. Math.,26, 549–563 (1973).
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).
C. Anne, “Spectre du laplacien et ecrasement d'anses,” Ann. Scient. Ec. Norm. Sup., 4 ser.,20, 271–280 (1987).
C. Anne, “Perturbation du spectre X−TUBeY (conditions de Neumann),” Seminaire de Theorie Spectrale et Geometrie de l'Institut Fourier,4, 17–23 (1986).
I. Chavel and E. A. Feldman, “Spectra of Manifolds Less a Small Domain,” Duke Math. J.,56, No. 2, 399–414 (1988).
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).
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).
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).
I. E. Zino and E. A. Tropp, Asymptotic Methods in Problems of Heat Conduction and Thermoelasticity [in Russian], Leningrad University Press, Leningrad (1978).
S. A. Nazarov, “Structure of solutions to elliptic boundary value problems in thin domains,” Vest. LGU, No. 7, 65–68 (1982).
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).
I. S. Gohberg and M. G. Kreîn, Introduction to the Theory of Linear Nonselfadjoint Operators in Hilbert Space [in Russian], Nauka, Moscow (1965).
S. Agmon and L. Nirenberg, “Properties of solutions of ordinary differential equations in Banach space,” Comm. Pure Appl. Math.,16, 121–239 (1963).
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).
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).
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).
E. M. Landis, Second-Order Equations of Elliptic and Parabolic Types [in Russian], Nauka, Moscow (1971).
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).
M. M. Vaînberg and V. A. Trenogin, The Theory of Branching of Solutions of Nonlinear Equations, Noordhoff, Leyden (1974).
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).
Additional information
St. Petersburg. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 4, pp. 80–96, July–August, 1992.
Translated by G. V. Dyatlov
Rights and permissions
About this article
Cite this article
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
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF00971127