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Izv. Akad. Nauk SSSR Ser. Mat., 1984, Volume 48, Issue 2, Pages 347–371 (Mi izv1449)  

This article is cited in 51 scientific papers (total in 51 papers)

Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes

V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevskii


Abstract: Complete asymptotic expansions are found for the first eigenvalues and eigenfunctions of classical problems for the Laplace operator in two- and three-dimensional domains with small holes.
Bibliography: 17 titles.

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English version:
Mathematics of the USSR-Izvestiya, 1985, 24:2, 321–345

Bibliographic databases:

UDC: 517.9
MSC: 35J05, 35P15
Received: 23.11.1982

Citation: V. G. Maz'ya, S. A. Nazarov, B. A. Plamenevskii, “Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes”, Izv. Akad. Nauk SSSR Ser. Mat., 48:2 (1984), 347–371; Math. USSR-Izv., 24:2 (1985), 321–345

Citation in format AMSBIB
\Bibitem{MazNazPla84}
\by V.~G.~Maz'ya, S.~A.~Nazarov, B.~A.~Plamenevskii
\paper Asymptotic expansions of the eigenvalues of boundary value problems for the Laplace operator in domains with small holes
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1984
\vol 48
\issue 2
\pages 347--371
\mathnet{http://mi.mathnet.ru/izv1449}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=740795}
\zmath{https://zbmath.org/?q=an:0566.35031}
\transl
\jour Math. USSR-Izv.
\yr 1985
\vol 24
\issue 2
\pages 321--345
\crossref{https://doi.org/10.1070/IM1985v024n02ABEH001237}


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    1. Yu. D. Golovatyj, S. A. Nazarov, O. A. Oleinik, “The asymptotic behaviour of eigenvalues and eigenfunctions in problems on vibrations of a medium with singular perturbation of the density”, Russian Math. Surveys, 43:5 (1988), 229–230  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. S. A. Nazarov, “The two terms asymptotics of the solutions of spectral problems with singular perturbations”, Math. USSR-Sb., 69:2 (1991), 307–340  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    3. S. A. Nazarov, “Asymptotic solution of a variational inequality modelling a friction”, Math. USSR-Izv., 37:2 (1991), 337–369  mathnet  crossref  mathscinet  zmath  adsnasa
    4. R. R. Gadyl'shin, “Influence of the position of the opening and its shape on the properties of a Helmholtz resonator”, Theoret. and Math. Phys., 93:1 (1992), 1151–1159  mathnet  crossref  mathscinet  isi
    5. R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. I. Convergence and estimates”, Math. Notes, 54:6 (1993), 1192–1199  mathnet  crossref  mathscinet  zmath  isi
    6. G. A. Chechkin, “Averaging of boundary value problems with a singular perturbation of the boundary conditions”, Russian Acad. Sci. Sb. Math., 79:1 (1994), 191–222  mathnet  crossref  mathscinet  zmath  isi
    7. R. R. Gadyl'shin, “On the Poles of an Acoustic Resonator”, Funct. Anal. Appl., 27:4 (1993), 229–239  mathnet  crossref  mathscinet  zmath  isi
    8. R. R. Gadyl'shin, “Asymptotic behavior of the second eigenfrequency for a system of two bodies connected by a thin rod”, Theoret. and Math. Phys., 97:1 (1993), 1154–1159  mathnet  crossref  mathscinet  zmath  isi
    9. R. R. Gadyl'shin, “On scattering by bodies with narrow channels”, Russian Acad. Sci. Sb. Math., 82:2 (1995), 293–313  mathnet  crossref  mathscinet  zmath  isi
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    11. A. G. Belyaev, G. A. Chechkin, “Homogenization of a mixed boundary-value problem for the Laplace operator in the case of an insoluble 'limit' problem”, Sb. Math., 186:4 (1995), 511–525  mathnet  crossref  mathscinet  zmath  isi
    12. R. R. Gadyl'shin, “On the perturbation of the Laplacian spectrum when the boundary condition type changes on a small part of the boundary”, Comput. Math. Math. Phys., 36:7 (1996), 889–898  mathnet  mathscinet  zmath  isi
    13. R. R. Gadyl'shin, “Characteristic frequencies of bodies with thin spikes. III. Frequency splitting”, Math. Notes, 61:4 (1997), 409–416  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    14. R. R. Gadyl'shin, “Existence and asymptotics of poles with small imaginary part for the Helmholtz resonator”, Russian Math. Surveys, 52:1 (1997), 1–72  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    15. A. R. Danilin, “Asymptotic behaviour of bounded controls for a singular elliptic problem in a domain with a small cavity”, Sb. Math., 189:11 (1998), 1611–1642  mathnet  crossref  crossref  mathscinet  zmath  isi
    16. S. A. Nazarov, A. S. Slutskij, “Asymptotic behaviour of solutions of boundary-value problems for equations with rapidly oscillating coefficients in a domain with a small cavity”, Sb. Math., 189:9 (1998), 1385–1422  mathnet  crossref  crossref  mathscinet  zmath  isi
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    18. M. Yu. Planida, “On the Convergence of Solutions of Singularly Perturbed Boundary-Value Problems for the Laplace Operator”, Math. Notes, 71:6 (2002), 794–803  mathnet  crossref  crossref  mathscinet  zmath  isi
    19. H. Ammari, S. Moskow, “Asymptotic expansions for eigenvalues in the presence of small inhomogeneities”, Math Meth Appl Sci, 26:1 (2003), 67  crossref  mathscinet  zmath  isi
    20. H AMMARI, A KHELIFI, “Electromagnetic scattering by small dielectric inhomogeneities1”, Journal de Mathématiques Pures et Appliqués, 82:7 (2003), 749  crossref
    21. Marina Yu. Planida, “Asymptotics for eigenvalues of the Laplacian with a Neumann boundary condition on a thin cut-out tube”, Comptes Rendus Mécanique, 331:8 (2003), 531  crossref
    22. Serguei A. Nazarov, Jan Sokolowski, “Self adjoint extensions of differential operators in application to shape optimization”, Comptes Rendus Mécanique, 331:10 (2003), 667  crossref
    23. M. Yu. Planida, “Asymptotics of the Eigenelements of the Laplace Operator when the Boundary-Condition Type Changes on a Narrow Flattened Strip”, Math. Notes, 75:2 (2004), 213–228  mathnet  crossref  crossref  mathscinet  zmath  isi  elib
    24. A. R. Bikmetov, D. I. Borisov, “Discrete Spectrum of the Schrodinger Operator Perturbed by a Narrowly Supported Potential”, Theoret. and Math. Phys., 145:3 (2005), 1691–1702  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib  elib
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    50. S. A. Nazarov, “Asimptotika sobstvennykh chisel vnutri lakun spektra periodicheskikh volnovodov s malymi singulyarnymi vozmuscheniyami”, Matematicheskie voprosy teorii rasprostraneniya voln. 48, Posvyaschaetsya pamyati Aleksandra Pavlovicha KAChALOVA, Zap. nauchn. sem. POMI, 471, POMI, SPb., 2018, 168–210  mathnet
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