RUS  ENG JOURNALS   PEOPLE   ORGANISATIONS   CONFERENCES   SEMINARS   VIDEO LIBRARY   PACKAGE AMSBIB
 General information Latest issue Forthcoming papers Archive Impact factor Subscription Guidelines for authors License agreement Submit a manuscript Search papers Search references RSS Latest issue Current issues Archive issues What is RSS

 Izv. RAN. Ser. Mat.: Year: Volume: Issue: Page: Find

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

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)
References: PDF file   HTML file

English version:
Mathematics of the USSR-Izvestiya, 1987, 29:3, 511–533

Bibliographic databases:

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

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} 

• http://mi.mathnet.ru/eng/izv1568
• http://mi.mathnet.ru/eng/izv/v50/i6/p1156

 SHARE:

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
2. V. G. Maz'Ya, M. Hänler, “Approximation of Solutions of the Neumann Problem in Disintegrating Domains”, Math. Nachr, 162:1 (1993), 261
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
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
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
6. S. A. Nazarov, “Korn inequalities for elastic junctions of massive bodies, thin plates, and rods”, Russian Math. Surveys, 63:1 (2008), 35–107
7. Chechkin, GA, “On the Sapondzhyan-Babuska Paradox”, Applicable Analysis, 87:12 (2008), 1443
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
9. Sweers G., “A Survey on Boundary Conditions for the Biharmonic”, Complex Var. Elliptic Equ., 54:2 (2009), 79–93
10. S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold”, Siberian Math. J., 51:5 (2010), 866–878
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
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
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
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
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
16. José M. Arrieta, P.D.omenico Lamberti, “Spectral stability results for higher-order operators under perturbations of the domain”, Comptes Rendus Mathematique, 2013
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
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
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
•  Number of views: This page: 440 Full text: 138 References: 39 First page: 3