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.
Vengono studiati i problemi ai limiti per il sistema dell'elasticità lineare con condizioni ai limiti rapidamente alternanti. Si considera inoltre il comportamento asintotico della soluzione quando un piccolo parametro, che definisce l'oscillazione delle condizioni al limite, tende a zero. Vengono calcolate stime per la differenza tra tali soluzioni e le soluzioni del problema ai limiti.
@article{RLIN_1996_9_7_1_5_0,
author = {Olga A. Oleinik and Gregory Chechkin},
title = {On asymptotics of solutions and eigenvalues of the boundary value problem with rapidly alternating boundary conditions for the system of elasticity},
journal = {Atti della Accademia Nazionale dei Lincei. Classe di Scienze Fisiche, Matematiche e Naturali. Rendiconti Lincei. Matematica e Applicazioni},
volume = {7},
year = {1996},
pages = {5-15},
zbl = {0876.35012},
mrnumber = {1437647},
language = {en},
url = {http://dml.mathdoc.fr/item/RLIN_1996_9_7_1_5_0}
}
Oleinik, Olga A.; 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, Tome 7 (1996) pp. 5-15. http://gdmltest.u-ga.fr/item/RLIN_1996_9_7_1_5_0/
[1] , On boundary-value problems for a second order elliptic equation with oscillating boundary conditions. In: (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). | MR 1126479 | Zbl 0825.35038
[2] - , Asymptotic behavior of an elastic body with a surface having small stuck regions. MMAN, 22 (4), 1988, 609-624. | MR 974290 | Zbl 0659.73006
[3] , On the asymptotic properties of a partially fastened membrane. Russian Mathematical Surveys, 44 (4), 1989, 197; UMN, 44 (4), 227.
[4] - - , Homogénéisation des frontières par epiconvergences en élasticité linéaire. MMAN, 24 (1), 1990, 5-26. | MR 1034896 | Zbl 0691.73013
[5] - - , Un problème d'homogénéisation de frontière en élasticité linéaire pour un corps cylindrique. C.R. Acad. Sci. Paris, 311, s. II, 1990, 15-20. | MR 1068992 | Zbl 0704.73004
[6] , 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). | MR 1234592 | Zbl 0875.35009
[7] - , Homogénéisation sur le bord pour des problems elliptiques. C.R. Acad. Sci. Paris, 299 (17), s. I, 1984, 859-862. | MR 777748 | Zbl 0575.35020
[8] - , Boundary homogenization for elliptic problems. J. Math. Pure et Appl., 66, 1987, 351-361. | MR 928268 | Zbl 0658.35023
[9] , 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. | MR 35370 | Zbl 0035.18603
[10] - , On Korn's inequalities. C.R. Acad. Sci. Paris, 308, s. I, 1989, 483-487. | MR 995908 | Zbl 0698.35067
[11] , 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. | MR 1783034 | Zbl 0972.35013
[12] , 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. | Zbl 0793.73013
[13] - , 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] - , 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. | MR 1264169 | Zbl 0830.35029
[15] - , 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] - - , Mathematical Problems in Elasticity and Homogenization. North-Holland, Amsterdam 1992. | MR 1195131 | Zbl 0768.73003
[17] , Some Applications of Functional Analysis in Mathematical Physics. 3rd ed., AMS Press, Providence1991. | Zbl 0732.46001
[18] , On an influence of fastening to the eigen-frequencies of closed domains. UMN, 5 (3), 1950, 133-134.
[19] - - , On a limit behavior of spectrum of the sequence of operators, which are defined in different Hilbert spaces. UMN, 44 (3), 1989, 157-158. | MR 1024051 | Zbl 0684.47002
[20] , Some Asymptotic Problems of the Theory of Partial Differential Equations. Lezioni Lincee. Accademia Naz. dei Lincei. Cambridge University Press, Cambridge1995. | Zbl 1075.35500