Abstract:
Two groups of singularly perturbed boundary value problems for eigenvalues are considered. The first group contains the following types of perturbations: a small parameter in front of the leading derivatives in the equation or in the boundary condition, a domain with rapidly oscillating boundary, or a thin domain. Problems with perturbations of the data on a set with small diameter ε constitute the second group.
Citation:
S. A. Nazarov, “The two terms asymptotics of the solutions of spectral problems with singular perturbations”, Math. USSR-Sb., 69:2 (1991), 307–340
\Bibitem{Naz90}
\by S.~A.~Nazarov
\paper The two terms asymptotics of the solutions of spectral problems with singular perturbations
\jour Math. USSR-Sb.
\yr 1991
\vol 69
\issue 2
\pages 307--340
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\crossref{https://doi.org/10.1070/SM1991v069n02ABEH001937}
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Linking options:
https://www.mathnet.ru/eng/sm1168
https://doi.org/10.1070/SM1991v069n02ABEH001937
https://www.mathnet.ru/eng/sm/v181/i3/p291
This publication is cited in the following 39 articles:
S. A. Nazarov, J. Taskinen, “Model of a Plane Strain-State of a Two-Dimensional Plate with Small Periodic Areas of Fixed Edge”, J Math Sci, 283:4 (2024), 586
S. A. Nazarov, “Influence of Winkler–Steklov conditions on natural oscillations of elastic weighty body”, Ufa Math. J., 16:1 (2024), 53–79
S. A. Nazarov, “Parasitic eigenvalues of spectral problems for the Laplacian with third-type boundary conditions”, Comput. Math. Math. Phys., 63:7 (2023), 1237–1253
S. A. Nazarov, Ya. Taskinen, “Model ploskogo deformirovannogo sostoyaniya dvumernoi plastiny s melkimi pochti periodicheskimi uchastkami zaschemleniya kraya”, Matematicheskie voprosy teorii rasprostraneniya voln. 51, Zap. nauchn. sem. POMI, 506, POMI, SPb., 2021, 130–174
D. V. Korikov, “Asymptotics of Maxwell system eigenvalues in a domain with small cavities”, St. Petersburg Math. J., 31:1 (2020), 13–51
G. A. Karagulyan, H. Mkoyan, “An exponential estimate for the cubic partial sums of multiple Fourier series”, Izv. Math., 83:2 (2019), 273–286
S. A. Nazarov, “The asymptotics of natural oscillations of a long two-dimensional Kirchhoff plate with variable cross-section”, Sb. Math., 209:9 (2018), 1287–1336
R. R. Gadyl'shin, A. L. Piatnitski, G. A. Chechkin, “On the asymptotic behaviour of eigenvalues of a boundary-value problem
in a planar domain of Steklov sieve type”, Izv. Math., 82:6 (2018), 1108–1135
S. A. Nazarov, “Breakdown of cycles and the possibility of opening spectral gaps
in a square lattice of thin acoustic waveguides”, Izv. Math., 82:6 (2018), 1148–1195
S. A. Nazarov, “The spectra of rectangular lattices of quantum waveguides”, Izv. Math., 81:1 (2017), 29–90
Monique Dauge, Erwan Faou, Zohar Yosibash, Encyclopedia of Computational Mechanics Second Edition, 2017, 1
S. A. Nazarov, “Nonreflection and trapping of elastic waves in a slightly curved isotropic strip”, Dokl. Phys, 59:3 (2014), 139
S. A. Nazarov, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed T-shaped waveguide”, Comput. Math. Math. Phys., 54:5 (2014), 793–814
S. A. Nazarov, “Bounded solutions in a T-shaped waveguide and the spectral properties of the Dirichlet ladder”, Comput. Math. Math. Phys., 54:8 (2014), 1261–1279
S. A. Nazarov, “Structure of the spectrum of a net of quantum waveguides and bounded solutions of a model problem at the threshold”, Dokl. Math, 90:2 (2014), 637
Cardone G., Nazarov S.A., Ruotsalainen K., “Bound States of a Converging Quantum Waveguide”, ESAIM-Math. Model. Numer. Anal.-Model. Math. Anal. Numer., 47:1 (2013), 305–315
Denis Borisov, Giuseppe Cardone, Luisa Faella, Carmen Perugia, “Uniform resolvent convergence for strip with fast oscillating boundary”, Journal of Differential Equations, 2013
Nazarov S.A., “Trapped Surface Waves in a Periodic Layer of a Heavy Liquid”, Pmm-J. Appl. Math. Mech., 75:2 (2011), 235–244
V. A. Kozlov, S. A. Nazarov, “The spectrum asymptotics for the Dirichlet problem in the case of the biharmonic operator in a domain with highly indented boundary”, St. Petersburg Math. J., 22:6 (2011), 941–983
Youcef Amirat, Gregory A. Chechkin, Rustem R. Gadyl’shin, “Spectral boundary homogenization in domains with oscillating boundaries”, Nonlinear Analysis: Real World Applications, 11:6 (2010), 4492
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