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Izv. Akad. Nauk SSSR Ser. Mat., 1986, Volume 50, Issue 6, Pages 1156–1177 (Mi izv1568)  

This article is cited in 19 scientific papers (total in 19 papers)

Paradoxes of limit passage in solutions of boundary value problems involving the approximation of smooth domains by polygonal domains

V. G. Maz'ya, S. A. Nazarov


Abstract: The Sapondzhyan–Babuska paradox consists in the fact that, when thin circular plates are approximated by regular polygons with freely supported edges, the limit solution does not satisfy the conditions of free support on the circle. In this article, new effects of the same nature are found. In particular, plates with convex holes are considered. Here, in contrast to the case of convex plates, the boundary conditions on the polygon are not preserved in the limit. Methods of approximating a smooth contour leading to passage to the limit from conditions of free support to conditions of rigid support are discussed.
Bibliography: 20 titles.

Full text: PDF file (2032 kB)
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English version:
Mathematics of the USSR-Izvestiya, 1987, 29:3, 511–533

Bibliographic databases:

UDC: 517.946:539.3
MSC: 74C35, 73K10, 35J67
Received: 10.12.1984

Citation: V. G. Maz'ya, S. A. Nazarov, “Paradoxes of limit passage in solutions of boundary value problems involving the approximation of smooth domains by polygonal domains”, Izv. Akad. Nauk SSSR Ser. Mat., 50:6 (1986), 1156–1177; Math. USSR-Izv., 29:3 (1987), 511–533

Citation in format AMSBIB
\Bibitem{MazNaz86}
\by V.~G.~Maz'ya, S.~A.~Nazarov
\paper Paradoxes of limit passage in solutions of boundary value problems involving the approximation of smooth domains by polygonal domains
\jour Izv. Akad. Nauk SSSR Ser. Mat.
\yr 1986
\vol 50
\issue 6
\pages 1156--1177
\mathnet{http://mi.mathnet.ru/izv1568}
\mathscinet{http://www.ams.org/mathscinet-getitem?mr=883157}
\zmath{https://zbmath.org/?q=an:0635.73062}
\transl
\jour Math. USSR-Izv.
\yr 1987
\vol 29
\issue 3
\pages 511--533
\crossref{https://doi.org/10.1070/IM1987v029n03ABEH000981}


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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. S. A. Nazarov, “Asymptotics of the solution of the Dirichlet problem for an equation with rapidly oscillating coefficients in a rectangle”, Math. USSR-Sb., 73:1 (1992), 79–110  mathnet  crossref  mathscinet  zmath  adsnasa  isi
    2. V. G. Maz'Ya, M. Hänler, “Approximation of Solutions of the Neumann Problem in Disintegrating Domains”, Math. Nachr, 162:1 (1993), 261  crossref
    3. Hamid Bellout, Sheryl L. Wills, “Perturbation of the domain and regularity of the solutions of the bipolar fluid flow equations in polygonal domains”, International Journal of Non-Linear Mechanics, 30:3 (1995), 235  crossref
    4. S. A. Nazarov, M. V. Olyushin, “Approximation of smooth contours by polygonal ones. Paradoxes in problems for the Lame system”, Izv. Math., 61:3 (1997), 619–646  mathnet  crossref  crossref  mathscinet  zmath  isi
    5. S. A. Nazarov, “Asymptotics of solutions and modelling the problems of elasticity theory in domains with rapidly oscillating boundaries”, Izv. Math., 72:3 (2008), 509–564  mathnet  crossref  crossref  mathscinet  zmath  adsnasa  isi  elib
    6. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107  mathnet  crossref  crossref  mathscinet  zmath  isi  elib  elib
    7. Chechkin, GA, “On the Sapondzhyan-Babuska Paradox”, Applicable Analysis, 87:12 (2008), 1443  crossref  isi
    8. Nazarov, SA, “Scenarios for the quasistatic growth of a slightly curved and kinked crack”, Pmm Journal of Applied Mathematics and Mechanics, 72:3 (2008), 347  crossref  isi
    9. Sweers G., “A Survey on Boundary Conditions for the Biharmonic”, Complex Var. Elliptic Equ., 54:2 (2009), 79–93  crossref  mathscinet  zmath  isi
    10. S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold”, Siberian Math. J., 51:5 (2010), 866–878  mathnet  crossref  mathscinet  isi  elib
    11. 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  mathnet  crossref  mathscinet  zmath  isi
    12. S. A. Nazarov, “Asymptotics of trapped modes and eigenvalues below the continuous spectrum of a waveguide with a thin shielding obstacle”, St. Petersburg Math. J., 23:3 (2012), 571–601  mathnet  crossref  mathscinet  zmath  isi  elib  elib
    13. Nazarov S.A., Svirs G.Kh., Stilyanou A., “O paradoksakh v zadachakh izgiba mnogougolnykh plastin s sharnirno zakreplennym kraem”, Doklady Akademii nauk, 439:4 (2011), 476–480  elib
    14. Nazarov S.A., Sweers G., Stylianou A., “Paradoxes in Problems on Bending of Polygonal Plates with a Hinged/Supported Edge”, Dokl. Phys., 56:8 (2011), 439–443  crossref  adsnasa  isi
    15. Ibrahima Dione, Cristian Tibirna, José Urquiza, “Stokes equations with penalised slip boundary conditions”, International Journal of Computational Fluid Dynamics, 27:6-7 (2013), 283  crossref
    16. José M. Arrieta, P.D.omenico Lamberti, “Spectral stability results for higher-order operators under perturbations of the domain”, Comptes Rendus Mathematique, 2013  crossref
    17. Dione I., Urquiza J.M., “Finite Element Approximations of the Lame System with Penalized Ideal Contact Boundary Conditions”, Appl. Math. Comput., 223 (2013), 115–126  crossref  isi
    18. José M. Urquiza, André Garon, Marie-Isabelle Farinas, “Weak imposition of the slip boundary condition on curved boundaries for Stokes flow”, Journal of Computational Physics, 256 (2014), 748  crossref
    19. S. A. Nazarov, “Asymptotics of eigenvalues of the Dirichlet problem in a skewed $\mathcal{T}$-shaped waveguide”, Comput. Math. Math. Phys., 54:5 (2014), 793–814  mathnet  crossref  crossref  mathscinet  isi  elib  elib
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