On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity

Olga A. Oleinik; Gregory Chechkin

Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni (1996)

  • Volume: 7, Issue: 1, page 5-15
  • ISSN: 1120-6330

Abstract

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Boundary value problems for the system of linear elasticity with rapidly alternating boundary conditions are studied and asymptotic behavior of solutions is considered when a small parameter, which defines the oscillation of the boundary conditions, tends to zero. Estimates for the difference between such solutions and solutions of the limit problem are given.

How to cite

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Oleinik, Olga A., and Chechkin, Gregory. "On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity." Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni 7.1 (1996): 5-15. <http://eudml.org/doc/244144>.

@article{Oleinik1996,
abstract = {Boundary value problems for the system of linear elasticity with rapidly alternating boundary conditions are studied and asymptotic behavior of solutions is considered when a small parameter, which defines the oscillation of the boundary conditions, tends to zero. Estimates for the difference between such solutions and solutions of the limit problem are given.},
author = {Oleinik, Olga A., Chechkin, Gregory},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
keywords = {Homogenization; Linear elasticity system; Alternating boundary conditions; boundary homogenization},
language = {eng},
month = {5},
number = {1},
pages = {5-15},
publisher = {Accademia Nazionale dei Lincei},
title = {On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity},
url = {http://eudml.org/doc/244144},
volume = {7},
year = {1996},
}

TY - JOUR
AU - Oleinik, Olga A.
AU - Chechkin, Gregory
TI - On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity
JO - Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni
DA - 1996/5//
PB - Accademia Nazionale dei Lincei
VL - 7
IS - 1
SP - 5
EP - 15
AB - Boundary value problems for the system of linear elasticity with rapidly alternating boundary conditions are studied and asymptotic behavior of solutions is considered when a small parameter, which defines the oscillation of the boundary conditions, tends to zero. Estimates for the difference between such solutions and solutions of the limit problem are given.
LA - eng
KW - Homogenization; Linear elasticity system; Alternating boundary conditions; boundary homogenization
UR - http://eudml.org/doc/244144
ER -

References

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  1. CHECHKIN, G. A., On boundary-value problems for a second order elliptic equation with oscillating boundary conditions. In: V. N. VRAGOV (ed.), Nonclassical Partial Differential Equations. Institute for Mathematics of the Siberian Division of the Russian Academy of Sciences Press, Novosibirsk, 1988, 95-104 (in Russian). Zbl0825.35038MR1126479
  2. LOBO, M. - PEREZ, E., Asymptotic behavior of an elastic body with a surface having small stuck regions. MMAN, 22 (4), 1988, 609-624. Zbl0659.73006MR974290
  3. CHECHKIN, G. A., On the asymptotic properties of a partially fastened membrane. Russian Mathematical Surveys, 44 (4), 1989, 197; UMN, 44 (4), 227. 
  4. BRILLARD, A. - LOBO, M. - PEREZ, E., Homogénéisation des frontières par epiconvergences en élasticité linéaire. MMAN, 24 (1), 1990, 5-26. Zbl0691.73013MR1034896
  5. BRILLARD, A. - LOBO, M. - PEREZ, E., Un problème d'homogénéisation de frontière en élasticité linéaire pour un corps cylindrique. C.R. Acad. Sci. Paris, 311, s. II, 1990, 15-20. Zbl0704.73004MR1068992
  6. CHECHKIN, G. A., Homogenization of boundary-value problems with singular perturbation of the boundary conditions. Mat. Sbornik, 184 (6), 1993, 99-150 (translation Russian Acad. Sci. Sb. Math., (79, 1), 1994, 191-222). Zbl0875.35009MR1234592DOI10.1070/SM1994v079n01ABEH003608
  7. DAMLAMIAN, A. - LI, TA TSIEN, Homogénéisation sur le bord pour des problems elliptiques. C.R. Acad. Sci. Paris, 299 (17), s. I, 1984, 859-862. Zbl0575.35020MR777748
  8. DAMLAMIAN, A. - LI, TA TSIEN, Boundary homogenization for elliptic problems. J. Math. Pure et Appl., 66, 1987, 351-361. Zbl0658.35023MR928268
  9. FICHERA, G., Analisi esistenziale per le soluzioni dei problemi al contorno misti, relativi all'equazione e ai sistemi di equazione del secondo ordine di tipo ellittico, autoaggiunti. Ann Sc. Norm. Sup. Pisa, II, XV, 1946, 75-100. Zbl0035.18603MR35370
  10. KONDRATIEV, V. A. - OLEINIK, O. A., On Korn's inequalities. C.R. Acad. Sci. Paris, 308, s. I, 1989, 483-487. Zbl0698.35067MR995908DOI10.1070/RM1989v044n06ABEH002297
  11. OLEINIK, O. A., Korn's type inequalities and applications to elasticity. In: Convegno Intemazionale in memoria di Vito Volterra (Roma, 8-11 ottobre 1990). Atti dei Convegni Lincei, 92, 1992, 183-209. Zbl0972.35013MR1783034
  12. OLEINIK, O. A., Some mathematical problems of elasticity and Korn's inequalities. In: Partial Differential Equations. Pitman Research Notes in Math, series., vol. 273, Longman, Harlow, 1992, 163-179. Zbl0793.73013
  13. OLEINIK, O. A. - CHECHKIN, G. A., On asymptotics of solutions and eigenvalues of an elliptic problem with rapidly alternating type of boundary conditions. International Center for Theoretical Physics, Trieste, SMR. 719/4, 1993. 
  14. OLEINIK, O. A. - CHECHKIN, G. A., On boundary-value problems for an elliptic equation with rapidly alternating type of boundary conditions. Russian Mathematical Surveys, 48 (6); UMN, 48 (6), 1993, 163-164. Zbl0830.35029MR1264169DOI10.1070/RM1993v048n06ABEH001103
  15. OLEINIK, O. A. - CHECHKIN, G. A., On boundary-value problems for elasticity system with rapidly alternating type of boundary conditions. Russian Mathematical Surveys, 49 (4); UMN, 49 (4), 1994, 114. 
  16. OLEINIK, O. A. - SHAMAEV, A. S. - YOSIFLAN, G. A., Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam1992. Zbl0768.73003MR1195131
  17. SOBOLEV, S. L., Some Applications of Functional Analysis in Mathematical Physics. 3rd ed., AMS Press, Providence1991. Zbl0732.46001
  18. SAMARSKY, A. A., On an influence of fastening to the eigen-frequencies of closed domains. UMN, 5 (3), 1950, 133-134. 
  19. YOSIFLAN, G. A. - OLEINIK, O. A. - SHAMAEV, A. S., On a limit behavior of spectrum of the sequence of operators, which are defined in different Hilbert spaces. UMN, 44 (3), 1989, 157-158. Zbl0684.47002MR1024051DOI10.1070/RM1989v044n03ABEH002116
  20. OLEINIK, O. A., Some Asymptotic Problems of the Theory of Partial Differential Equations. Lezioni Lincee. Accademia Naz. dei Lincei. Cambridge University Press, Cambridge1995. Zbl1075.35500

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