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A general homogenization result of spectral problem for linearized elasticity in perforated domains
Applications of Mathematics, Tome 66 (2021) no. 5, pp. 701-724 Cet article a éte moissonné depuis la source Czech Digital Mathematics Library

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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.
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.
DOI : 10.21136/AM.2021.0009-20
Classification : 35B27, 35B40, 47A75, 74B05
Keywords: homogenization; $H$-convergence; perforated domain; linear elasticity; eigenvalue problem 
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     title = {A general homogenization result of spectral problem for linearized elasticity in perforated domains},
     journal = {Applications of Mathematics},
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Ait Yahia, Mohamed Mourad Lhannafi; Haddadou, Hamid. A general homogenization result of spectral problem for linearized elasticity in perforated domains. Applications of Mathematics, Tome 66 (2021) no. 5, pp. 701-724. doi: 10.21136/AM.2021.0009-20

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