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.
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”, Math. USSR-Izv., 29:3 (1987), 511–533
\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 Math. USSR-Izv.
\yr 1987
\vol 29
\issue 3
\pages 511--533
\mathnet{http://mi.mathnet.ru/eng/im1568}
\crossref{https://doi.org/10.1070/IM1987v029n03ABEH000981}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=883157}
\zmath{https://zbmath.org/?q=an:0635.73062}
Linking options:
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https://doi.org/10.1070/IM1987v029n03ABEH000981
https://www.mathnet.ru/eng/im/v50/i6/p1156
This publication is cited in the following 31 articles:
A. A. Chernyaev, “Geometricheskoe modelirovanie formy parallelogrammnykh plastin v zadache svobodnykh kolebanii s ispolzovaniem konformnykh radiusov”, Vestn. Tomsk. gos. un-ta. Matem. i mekh., 2021, no. 70, 143–159
D. Gomes, S. A. Nazarov, M.-E. Peres, “Tochechnoe kreplenie plastiny Kirkhgofa vdol ee kromki”, Matematicheskie voprosy teorii rasprostraneniya voln. 50, Posvyaschaetsya devyanostoletiyu Vasiliya Mikhailovicha BABIChA, Zap. nauchn. sem. POMI, 493, POMI, SPb., 2020, 107–137
Alexandre Kawano, Luís Yamaoka, “Uniqueness in the determination of vibration sources in polygonal plates observing the displacement over a space‐time open set”, Z Angew Math Mech, 99:8 (2019)
Francesco Ferraresso, Pier Domenico Lamberti, “On a Babuška Paradox for Polyharmonic Operators: Spectral Stability and Boundary Homogenization for Intermediate Problems”, Integr. Equ. Oper. Theory, 91:6 (2019)
José M. Arrieta, Francesco Ferraresso, Pier Domenico Lamberti, “Boundary homogenization for a triharmonic intermediate problem”, Math Methods in App Sciences, 41:3 (2018), 979
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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
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
Hamid Bellout, Frederick Bloom, Incompressible Bipolar and Non-Newtonian Viscous Fluid Flow, 2014, 137
Ibrahima Dione, Cristian Tibirna, José Urquiza, “Stokes equations with penalised slip boundary conditions”, International Journal of Computational Fluid Dynamics, 27:6-7 (2013), 283
José M. Arrieta, P.D.omenico Lamberti, “Spectral stability results for higher-order operators under perturbations of the domain”, Comptes Rendus Mathematique, 2013
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
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
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
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
S. A. Nazarov, “Variational and asymptotic methods for finding eigenvalues below the continuous spectrum threshold”, Siberian Math. J., 51:5 (2010), 866–878
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
Sweers G., “A Survey on Boundary Conditions for the Biharmonic”, Complex Var. Elliptic Equ., 54:2 (2009), 79–93
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
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