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 Izv. RAN. Ser. Mat., 2003, Volume 67, Issue 6, Pages 23–70 (Mi izv459)

This article is cited in 16 scientific papers (total in 16 papers)

Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating non-periodic boundary conditions

D. I. Borisov

Abstract: We consider a singularly perturbed spectral boundary-value problem for the Laplace operator in a two-dimensional domain with frequently alternating non-periodic boundary conditions. Under certain very weak restrictions on the alternation structure of the boundary conditions, we obtain the first terms of the asymptotic expansions of the eigenelements of this problem. Under still weaker restrictions, we obtain estimates for the rate of convergence of the eigenvalues.

DOI: https://doi.org/10.4213/im459

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English version:
Izvestiya: Mathematics, 2003, 67:6, 1101–1148

Bibliographic databases:

UDC: 517.956
MSC: 35P05, 35J05, 35B25, 35J25, 35B27
Received: 12.09.2002

Citation: D. I. Borisov, “Asymptotics and estimates for the eigenelements of the Laplacian with frequently alternating non-periodic boundary conditions”, Izv. RAN. Ser. Mat., 67:6 (2003), 23–70; Izv. Math., 67:6 (2003), 1101–1148

Citation in format AMSBIB
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Citing articles on Google Scholar: Russian citations, English citations
Related articles on Google Scholar: Russian articles, English articles

This publication is cited in the following articles:
1. G. A. Chechkin, “Splitting of a multiple eigenvalue in a problem on concentrated masses”, Russian Math. Surveys, 59:4 (2004), 790–791
2. G. A. Chechkin, “Asymptotic expansions of eigenvalues and eigenfunctions of an elliptic operator in a domain with many “light” concentrated masses situated on the boundary. Two-dimensional case”, Izv. Math., 69:4 (2005), 805–846
3. M. Yu. Planida, “Asymptotics of the eigenelements of the Laplacian with singular perturbations of boundary conditions on narrow and thin sets”, Sb. Math., 196:5 (2005), 703–741
4. D. I. Borisov, “On a problem with nonperiodic frequent alternation of boundary conditions imposed on fast oscillating sets”, Comput. Math. Math. Phys., 46:2 (2006), 271–281
5. Chechkin G.A., Koroleva Yu.O., Persson L.-E., “On the precise asymptotics of the constant in Friedrich's inequality for functions vanishing on the part of the boundary with microinhomogeneous structure”, J. Inequal. Appl., 2007, 34138, 13 pp.
6. Amirat Y., Chechkin G.A., Gadyl'shin R.R., “Asymptotics for eigenelements of Laplacian in domain with oscillating boundary: multiple eigenvalues”, Appl. Anal., 86:7 (2007), 873–897
7. Pérez E., “On periodic Steklov type eigenvalue problems on half-bands and the spectral homogenization problem”, Discrete Contin. Dyn. Syst. Ser. B, 7:4 (2007), 859–883
8. D Borisov, G Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J Phys A Math Theor, 42:36 (2009), 365205
9. Olendski O., Mikhailovska L., “Theory of a curved planar waveguide with Robin boundary conditions”, Phys. Rev. E, 81:3 (2010), 036606, 14 pp.
10. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition”, Ann Henri Poincaré, 2010
11. H Najar, O Olendski, “Spectral and localization properties of the Dirichlet wave guide with two concentric Neumann discs”, J. Phys. A: Math. Theor, 44:30 (2011), 305304
12. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys, 2012
13. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55
14. T. F. Sharapov, “On the resolvent of multidimensional operators with frequently changing boundary conditions in the case of the homogenized Dirichlet condition”, Sb. Math., 205:10 (2014), 1492–1527
15. Borisov D.I., “on the Band Spectrum of a Schrodinger Operator in a Periodic System of Domains Coupled By Small Windows”, Russ. J. Math. Phys., 22:2 (2015), 153–160
16. T. F. Sharapov, “On resolvent of multi-dimensional operators with frequent alternation of boundary conditions: critical case”, Ufa Math. J., 8:2 (2016), 65–94
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