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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
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    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    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  mathnet  crossref  mathscinet  zmath  elib  elib
    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.  crossref  mathscinet  zmath  isi  elib  scopus
    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  crossref  mathscinet  zmath  isi
    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  crossref  mathscinet  zmath  isi  elib  scopus
    8. D Borisov, G Cardone, “Homogenization of the planar waveguide with frequently alternating boundary conditions”, J Phys A Math Theor, 42:36 (2009), 365205  crossref  mathscinet  zmath  isi  elib  scopus
    9. Olendski O., Mikhailovska L., “Theory of a curved planar waveguide with Robin boundary conditions”, Phys. Rev. E, 81:3 (2010), 036606, 14 pp.  crossref  adsnasa  isi  elib  scopus
    10. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “On a Waveguide with Frequently Alternating Boundary Conditions: Homogenized Neumann Condition”, Ann Henri Poincaré, 2010  crossref  mathscinet  isi  scopus
    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  crossref  mathscinet  zmath  adsnasa  isi  scopus
    12. Denis Borisov, Renata Bunoiu, Giuseppe Cardone, “Waveguide with non-periodically alternating Dirichlet and Robin conditions: homogenization and asymptotics”, Z. Angew. Math. Phys, 2012  crossref  mathscinet  isi  scopus
    13. D.I. Borisov, “Discrete spectrum of thin $\mathcal{PT}$-symmetric waveguide”, Ufa Math. J., 6:1 (2014), 29–55  mathnet  crossref  isi  elib
    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  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    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  crossref  mathscinet  zmath  isi  elib  scopus
    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  mathnet  crossref  elib
  • Известия Российской академии наук. Серия математическая Izvestiya: Mathematics
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