The asymptotics of the solution to the neumann spectral problem in a domain of the "dense-comb" type

Journal of Mathematical Sciences

Volumn 85, Issue 6, 1997, Pages 2326-2346

  Melnik, T A ^a   Nazarov, S A ^a  

^a NONE

Author keywords

[No Author keywords available]

Indexed keywords

EID: 53249107920     PISSN: 10723374     EISSN: 15738795     Source Type: Journal    
DOI: 10.1007/BF02355841     Document Type: Article

References (27)

1. R. Courant and D. Hilbert, Methods of Mathematical Physics, Vol. 1, Interscience, New York (1962).
2. I. Babushka and R. Vyborny, "Continuous dependence of the eigenvalues on the domains," Czech. Math. J., 15, 169-178 (1965).
3. V. P. Kotlyarov and E. Ya. Khruslov, "On the limit boundary condition of the Neumann problem," Teor. Funkts., Funkts. Anal. Prilozhen., 10, 83-96 (1970).
4. I. E. Egorova and E. Ya. Khruslov, "Asymptotic behavior of solutions of the second boundary-value problem in domains with random thin gaps," Teor. Funkts., Funkts. Anal. Prilozhen., 52, 91-103 (1989).
5. F. Fleury and E. Sanchez-Palencia, "Asymptotics and spectral properties of the acoustic vibrations of a body perforated by narrow channels," 'Bull. Sci. Math., 110, 149-176 (1986).
6. P. G. Ciarlet, Plate Junctions in Elastic Microstructures, Masson, Paris-New York (1990).
7. S. A. Nazarov and B. A. Plamenevskii, "Asymptotics of the spectrum of the Neumann problem in singularly degenerating thin domains," Algebra Analiz, 2, No. 2, 85-111 (1990).
8. Jimbo Shuichi, "Remarks on the behavior on a singularly perturbed domain with several thin channels," Commun. Part. Diff. Equat., 17, No. 3-4, 523-552 (1992).
9. T. A. Melnik, "Homogenization of elliptic equations that describe processes in highly inhomogeneous thin perforated domains with rapidly changing thickness," Dokl. Akad. Nauk Ukr., 10, 15-18 (1991).
10. S. A. Nazarov, "Structure of solutions of boundary-value problems for elliptic equations in thin domains," Vestn. Leningr. Gos. Univ., 7, 65-68 (1982).
11. A. L. Goldenveiser, Theory of Thin Elastic Envelopes [in Russian], Nauka, Moscow (1976).
12. O. A. Oleinik, A. S. Shamaev, and G. A. Yosifian, Mathematical Problems in Elasticity and Homogenization, North-Holland, Amsterdam (1992).
13. A. Bensoussan, J. L. Lions, and G. Papanicolaou, Asymptotic Analysis for Periodic Structures, North-Holland, Amsterdam (1978).
14. N. S. Bakhvalov and G. P. Panasenko, Homogenization of Processes in Periodic Media [in Russian], Nauka, Moscow (1984).
15. E. Sanchez-Palencia, Nonhomogenuous Media and Vibration Theory, Springer, New York (1980).
16. S. A. Nazarov and B. A. Plamenevskii, Elliptic Problems in Domains with Piecewise Smooth Boundary [in Russian], Nauka, Moscow (1991).
17. W. G. Mazia, S. A. Nazarov, and B. A. Plamenevskii, Asymptotische Theorie Elliptischer Randwertaufgaben in Singular Gestorten Gebieten, Vol. 1, 2, Akademie-Verlag, Berlin (1991).
18. A. M. Il'in, Matching of Asymptotic Expansions for Solutions of Boundary Value Problems [in Russian], Nauka, Moscow (1989).
19. F. Riesz and B. Sz.-Nagy, Leçons d'Analyse Fonctionnelle, Akadémiai Kiadó, Budapest (1972).
20. Yu. Sh. Abramov, Variational Methods in the Theory of Operator Pencils [in Russian], Leningrad University Press (1983).
21. T. A. Melnik, "Spectral properties of self-adjoint discontinuous operator-functions," Dokl. Akad. Nauk Ukr., 12, 33-36 (1994).
22. O. A. Ladyzhenskaya and N. N. Ural'ceva, Linear and Quasilinear Elliptic Equations [in Russian], Nauka, Moscow (1973).
23. T. A. Melnik, "Asymptotic expansions for eigenvalues and eigenfunctions of elliptic boundary-value problems with rapidly oscillating coefficients in a perforated cube," Trudy Seminara im. I. G. Petrovskogo, 17, 51-88 (1994).
24. M. I. Vishik and L. A. Lyusternik, "Regular degeneration and boundary layer for linear differential equations with a parameter," Usp. Mat. Nauk, 12, No. 5, 3-192 (1957).
25. O. A. Oleinik and G. A. Yosifian, "On homogenization of the elasticity theory system with rapidly oscillating coefficients in a perforated domain," in: N. E. Kochin and the Development of Mechanics [in Russian], Nauka, Moscow (1984), pp. 237-249.
26. D. Cioranescu and J. Saint Jean Paulin, "Homogenization in open sets with holes," J. Math. Anal. Appl., 71, No. 2, 590-607 (1979).
27. A. S. Il'insky and G. Ya. Slepyan, Vibration and Waves in Electrodynamical Systems with Losses [in Russian], Moscow University Press (1983).