Article
Full entry |
PDF
(0.3 MB)
Feedback
PDF
(0.3 MB)
Feedback
Keywords:
homogenization; $H$-convergence; perforated domain; linear elasticity; eigenvalue problem
homogenization; $H$-convergence; perforated domain; linear elasticity; eigenvalue problem
Summary:
The goal of this paper is to establish a general homogenization result for linearized elasticity of an eigenvalue problem defined over perforated domains, beyond the periodic setting, within the framework of the $H^0$-convergence theory. Our main homogenization result states that the knowledge of the fourth-order tensor $A^0$, the $H^0$-limit of $A^{\varepsilon }$, is sufficient to determine the homogenized eigenvalue problem and preserve the structure of the spectrum. This theorem is proved essentially by using Tartar's method of test functions, and some general arguments of spectral analysis used in the literature on the homogenization of eigenvalue problems. Moreover, we give a result on a particular case of a simple eigenvalue of the homogenized problem. We conclude our work by some comments and perspectives.
References:
[1] Briane, M.: Homogenization in general periodically perforated domains by a spectral approach. Calc. Var. Partial Differ. Equ. 15 (2002), 1-24. DOI 10.1007/s005260100115 | MR 1920712 | Zbl 1028.35018
[2] Briane, M., Damlamian, A., Donato, P.: $H$-convergence in perforated domains. Nonlinear Partial Differential Equations and Their Applications Pitman Research Notes in Mathematics Series 391. Longman, Harlow (1998), 62-100. MR 1773075 | Zbl 0943.35005
[3] Cancedda, A.: Spectral homogenization for a Robin-Neumann problem. Boll. Unione Mat. Ital. 10 (2017), 199-222. DOI 10.1007/s40574-016-0075-z | MR 3655025 | Zbl 1377.35092
[4] Cioranescu, D., Paulin, J. Saint Jean: Homogenization of Reticuled Structures. Applied Mathematical Sciences 136. Springer, New York (1999). DOI 10.1007/978-1-4612-2158-6 | MR 1676922 | Zbl 0929.35002
[5] Damlamian, A., Donato, P.: Which sequences of holes are admissible for periodic homogenization with Neumann boundary condition?. ESAIM, Control Optim. Calc. Var. 8 (2002), 555-585. DOI 10.1051/cocv:2002046 | MR 1932963 | Zbl 1073.35020
[6] Douanla, H.: Homogenization of Steklov spectral problems with indefinite density function in perforated domains. Acta Appl. Math. 123 (2013), 261-284. DOI 10.1007/s10440-012-9765-4 | MR 3010234 | Zbl 1263.35022
[7] Hajji, M. El: Homogenization of linearized elasticity systems with traction condition in perforated domains. Electron. J. Differ. Equ. 1999 (1999), Article ID 41, 11 pages. MR 1713600 | Zbl 0952.74060
[8] Hajji, M. El, Donato, P.: $H^0$-convergence for the linearized elasticity system. Asymptotic Anal. 21 (1999), 161-186. MR 1723547 | Zbl 0942.74057
[9] Francfort, G. A., Murat, F.: Homogenization and optimal bounds in linear elasticity. Arch. Ration. Mech. Anal. 94 (1986), 307-334. DOI 10.1007/BF00280908 | MR 0846892 | Zbl 0604.73013
[10] Georgelin, C.: Contribution à l'étude de quelques problèmes en élasticité tridimensionnelle: Thèse de Doctorat. Université de Paris IV, Paris (1989), French.
[11] Haddadou, H.: Iterated homogenization for the linearized elasticity by $H^{0}_{e}$-convergence. Ric. Mat. 54 (2005), 137-163. MR 2290210 | Zbl 1387.35034
[12] Haddadou, H.: A property of the $H$-convergence for elasticity in perforated domains. Electron. J. Differ. Equ. 137 (2006), Article ID 137, 11 pages. MR 2276562 | Zbl 1128.35017
[13] Jikov, V. V., Kozlov, S. M., Oleinik, O. A.: Homogenization of Differential Operators and Integral Functionals. Springer, Berlin (1994). DOI 10.1007/978-3-642-84659-5 | MR 1329546 | Zbl 0838.35001
[14] Kesavan, S.: Homogenization of elliptic eigenvalue problems I. Appl. Math. Optim. 5 (1979), 153-167. DOI 10.1007/BF01442551 | MR 0533617 | Zbl 0415.35061
[15] Léné, F.: Comportement macroscopique de matériaux élastiques comportant des inclusions rigides ou des trous répartis périodiquement. C. R. Acad. Sci., Paris, Sér. A 286 (1978), 75-78 French. MR 0486458 | Zbl 0372.73001
[16] Murat, F., Tartar, L.: $H$-convergence. Topics in the Mathematical Modelling of Composite Materials Progress in Nonlinear Differential Equations and Their Applications 31. Birkhäuser, Boston (1997), 21-43. DOI 10.1007/978-1-4612-2032-9_3 | MR 1493039 | Zbl 0920.35019
[17] Nandakumar, A. K.: Homogenization of eigenvalue problems of elasticity in perforated domains. Asymptotic Anal. 9 (1994), 337-358. DOI 10.3233/ASY-1994-9403 | MR 1301169 | Zbl 0814.35135
[18] Oleinik, O. A., Shamaev, A. S., Yosifian, G. A.: Mathematical Problems in Elasticity and Homogenization. Studies in Mathematics and Its Applications 26. North-Holland, Amsterdam (1992). DOI 10.1016/s0168-2024(08)x7009-2 | MR 1195131 | Zbl 0768.73003
[19] Spagnolo, S.: Sulla convergenza di soluzioni di equazioni paraboliche ed ellittiche. Ann. Sc. Norm. Super. Pisa, Sci. Fis. Mat., III. Ser. 22 (1968), 571-597 Italian. MR 0240443 | Zbl 0174.42101
[20] Suslina, T. A.: Spectral approach to homogenization of elliptic operators in a perforated space. Rev. Math. Phys. 30 (2018), Article ID 1840016, 57 pages. DOI 10.1142/S0129055X18400160 | MR 3846431 | Zbl 1411.35029
[21] Tartar, L.: Problèmes d'homogénéisation dans les équations aux dérivées partielles. Cours Peccot, Collège de France, Paris (1977), French.
[22] Tartar, L.: The General Theory of Homogenization: A Personalized Introduction. Lecture Notes of the Unione Matematica Italiana 7. Springer, Berlin (2009). DOI 10.1007/978-3-642-05195-1 | MR 2582099 | Zbl 1188.35004
[23] Vanninathan, M.: Homogenization of eigenvalue problems in perforated domains. Proc. Indian Acad. Sci., Math. Sci. 90 (1981), 239-271. DOI 10.1007/BF02838079 | MR 0635561 | Zbl 0486.35063
Search
Partner of
