This paper is focused on the asymptotic behavior of sequences of functions, whose partial derivatives estimates in one or more directions are highly contrasted with respect to the periodic parameter ε. In particular, a direct application for the homogenization of a homogeneous Dirichlet problem defined on an anisotropic structure is presented. In general, the obtained results can be applied to thin structures where the behavior is different according to the observed direction.
B.Cabarrubias and P.Donato, Homogenization of some evolution problems in domains with small holes, Electron. J. Diff. Equ.169 (2016), 1–26.
2.
D.Cioranescu, A.Damlamian and G.Griso, Periodic unfolding and homogenization, Comptes Rendus Mathématique335(1) (2002), 99–104. doi:10.1016/S1631-073X(02)02429-9.
3.
D.Cioranescu, A.Damlamian and G.Griso, The Stokes problem in perforated domains by the periodic unfolding method, in: Proceedings Conference New Trends in Continuum Mechanics, M.Suliciu, ed., Theta, Bucarest, 2005, pp. 67–80.
4.
D.Cioranescu, A.Damlamian and G.Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal.40(4) (2008), 1585–1620. doi:10.1137/080713148.
5.
D.Cioranescu, A.Damlamian and G.Griso, The Periodic Unfolding Method. Theory and Applications to Partial Differential Problems, Springer, Singapore, 2018. doi:10.1007/978-981-13-3032-2.
6.
D.Cioranescu, A.Damlamian and J.Orlik, Homogenization via unfolding in periodic elasticity with contact on closed and open cracks, Asymptotic Analysis82 (2013), 201–232. doi:10.3233/ASY-2012-1141.
7.
D.Cioranescu, P.Donato and R.Zaki, The periodic unfolding method in perforated domains, Portugaliae Mathematica4 (2006), 467–496.
8.
A.Damlamian, An elementary introduction to periodic unfolding, in: Proc. of the Narvik Conference 2004, A.Damlamian, D.Lukkassen, A.Meidell and A.Piatnitski, eds, Gakuto Int. Series, Math. Sci. Appl., Vol. 24, Gakkotosho, 2006, pp. 119–136.
9.
A.Damlamian and N.Meunier, The “strange term” in the periodic homogenization for multivalued Leray–Lions operators in perforated domains, Ricerche mat.59 (2010), 281–312. doi:10.1007/s11587-010-0087-4.
10.
A.Damlamian, N.Meunier and J.Van Schaftingen, Periodic homogenization for monotone multivalued operators, Nonlinear Analysis67 (2007), 3217–3239. doi:10.1016/j.na.2006.10.007.
11.
P.Donato, H.Le Nguyen and R.Tardieu, The periodic unfolding method for a class of imperfect transmission problems, Journal of Math. Sci.176(6) (2011), 891–927. doi:10.1007/s10958-011-0443-2.
12.
P.Donato and Z.Yang, The periodic unfolding method for the heat equation in perforated domains, Sci. China Math59 (2016), 891–906. doi:10.1007/s11425-015-5103-4.
13.
R.Falconi, G.Griso and J.Orlik, Periodic unfolding for lattice structures (submitted). Preprint at HAL, https://hal.archives-ouvertes.fr/hal-03532516.
14.
R.Falconi, G.Griso, J.Orlik and S.Wackerle, Asymptotic behavior for textiles with loose contact (submitted).
15.
G.Griso, M.Hauck and J.Orlik, Asymptotic analysis for periodic perforated shells, ESAIM: M2AN55(1) (2021), 1–36. doi:10.1051/m2an/2020067.
16.
G.Griso, L.Khilkova and J.Orlik, Asymptotic behavior of unstable structures made of beams (submitted).
17.
G.Griso, L.Khilkova, J.Orlik and O.Sivak, Homogenization of perforated elastic structures, J. Elast.141 (2020), 181–225. doi:10.1007/s10659-020-09781.
18.
G.Griso, L.Khilkova, J.Orlik and O.Sivak, Asymptotic behavior of stable structures made of beams, J. Elast (2021). doi:10.1007/s10659-021-09816-w.
19.
G.Griso, A.Migunova and J.Orlik, Homogenization via unfolding in periodic layer with contact, Asymptotic Analysis99(1–2) (2016), 23–52. doi:10.3233/ASY-161374.
20.
G.Griso, A.Migunova and J.Orlik, Asymptotic analysis for domains separated by a thin layer made of periodic vertical beams, J. Elast.128(2) (2017), 291–331. doi:10.1007/s10659-017-9628-3.
21.
G.Griso, J.Orlik and S.Wackerle, Asymptotic behavior for textiles in von-Kármán regime, J. Math. Pures Appl144 (2020), 164–193. doi:10.1016/j.matpur.2020.10.002.
22.
G.Griso, J.Orlik and S.Wackerle, Homogenization of textiles, SIAM J. Math. Anal.52(2) (2020), 1639–1689. doi:10.1137/19M1288693.
23.
A.Ould Hammouda, Homogenization of a class of Neumann problems in perforated domains, Asymptotic Analysis71(1–2) (2011), 33–57. doi:10.3233/ASY-2010-1010.
