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Keywords:
homogenization; stochastic-periodic; dynamical system; minimization problem; stochastic two-scale convergence; Orlicz-Sobolev spaces
homogenization; stochastic-periodic; dynamical system; minimization problem; stochastic two-scale convergence; Orlicz-Sobolev spaces
Summary:
We develop the stochastic two-scale convergence method in the framework of Orlicz-Sobolev spaces, in order to deal with the homogenization of coupled stochastic-periodic problems in such spaces. One fundamental in this topic is the extension of compactness results for this method to the Orlicz setting. For the application, we show that the sequence of minimizers of a class of highly oscillatory minimizations problems involving integral functionals with convex and nonstandard growth integrands, converges to the minimizer of a homogenized problem.
References:
[1] Abddaimi, Y., Michaille, G., Licht, C.: Stochastic homogenization for an integral functional of a quasiconvex function with linear growth. Asymptotic Anal. 15 (1997), 183-202. DOI 10.3233/ASY-1997-15203 | MR 1480998 | Zbl 0912.49013
[2] Adams, R. A.: On the Orlicz-Sobolev imbedding theorem. J. Funct. Anal. 24 (1977), 241-257. DOI 10.1016/0022-1236(77)90055-6 | MR 0430770 | Zbl 0344.46077
[3] Allaire, G.: Homogenization and two-scale convergence. SIAM J. Math. Anal. 23 (1992), 1482-1518 \99999DOI99999 10.1137/0523084 . DOI 10.1137/0523084 | MR 1185639 | Zbl 0770.35005
[4] Allaire, G., Briane, M.: Multiscale convergence and reiterated homogenisation. Proc. R. Soc. Edinb., Sect. A 126 (1996), 297-342. DOI 10.1017/S0308210500022757 | MR 1386865 | Zbl 0866.35017
[5] Andrews, K. T., Wright, S.: Stochastic homogenization of elliptic boundary-value problems with $L^p$-data. Asymptotic Anal. 17 (1998), 165-184. DOI 10.3233/ASY-1998-300 | MR 1633112 | Zbl 0954.35051
[6] Baía, M., Fonseca, I.: The limit behavior of a family of variational multiscale problems. Indiana Univ. Math. J. 56 (2007), 1-50. DOI 10.1512/iumj.2007.56.2869 | MR 2305929 | Zbl 1114.35008
[7] Blanc, X., Bris, C. Le, Lions, P.-L.: Stochastic homogenization and random lattices. J. Math. Pures Appl. (9) 88 (2007), 34-63. DOI 10.1016/j.matpur.2007.04.006 | MR 2334772 | Zbl 1129.60055
[8] Bourgeat, A., Mikelić, A., Wright, S.: Stochastic two-scale convergence in the mean and applications. J. Reine Angew. Math. 456 (1994), 19-51. DOI 10.1515/crll.1994.456.19 | MR 1301450 | Zbl 0808.60056
[9] Champion, T., Pascale, L. De: Homogenization of Dirichlet problems with convex bounded constraints on the gradient. Z. Anal. Anwend. 22 (2003), 591-608. DOI 10.4171/ZAA/1164 | MR 2015665 | Zbl 1071.49011
[10] Maso, G. Dal, Modica, L.: Nonlinear stochastic homogenization. Ann. Mat. Pura Appl., IV. Ser. 144 (1986), 347-389. DOI 10.1007/BF01760826 | MR 0870884 | Zbl 0607.49010
[11] Dongho, J., Tachago, J. F., Takougoum, F. Tchinda: Stochastic two-scale convergence in the mean in Orlicz-Sobolev's spaces and applications to the homogenization of an integral functional. Asymptotic Anal. 142 (2025), 352-376. DOI 10.1177/09217134241309718 | MR 4968160
[12] Gambin, B., Telega, J. J.: Effective properties of elastic solids with randomly distributed microcracks. Mech. Res. Commun. 27 (2000), 697-706. DOI 10.1016/S0093-6413(00)00143-9 | MR 1808554 | Zbl 1058.74612
[13] Kalousek, M.: Homogenization of incompressible generalized Stokes flows through a porous medium. Nonlinear Anal., Theory Methods Appl., Ser. A 136 (2016), 1-39. DOI 10.1016/j.na.2016.01.025 | MR 3474401 | Zbl 1398.35173
[14] Mingione, G., Rădulescu, V.: Recent developments in problems with nonstandard growth and nonuniform ellipticity. J. Math. Anal. Appl. 501 (2021), Article ID 125197, 41 pages. DOI 10.1016/j.jmaa.2021.125197 | MR 4258810 | Zbl 1467.49003
[15] Nguetseng, G.: A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989), 608-623. DOI 10.1137/0520043 | MR 0990867 | Zbl 0688.35007
[16] Nguetseng, G., Sango, M., Woukeng, J. L.: Reiterated ergodic algebras and applications. Commun. Math. Phys. 300 (2010), 835-876. DOI 10.1007/s00220-010-1127-3 | MR 2736964 | Zbl 1228.46049
[17] Sango, M., Woukeng, J. L.: Stochastic two-scale convergence of an integral functional. Asymptotic Anal. 73 (2011), 97-123. DOI 10.3233/ASY-2011-1038 | MR 2841226 | Zbl 1228.60073
[18] Tachago, J. F., Gargiulo, G., Nnang, H., Zappale, E.: Multiscale homogenization of integral convex functionals in Orlicz-Sobolev setting. Evol. Equ. Control Theory 10 (2021), 297-320. DOI 10.3934/eect.2020067 | MR 4218104 | Zbl 1479.49024
[19] Tachago, J. F., Gargiulo, G., Nnang, H., Zappale, E.: Some convergence results on the periodic unfolding operator in Orlicz setting. Integral Methods in Science and Engineering Birkhäuser, Cham (2023), 361-371. DOI 10.1007/978-3-031-34099-4_29
[20] Tachago, J. F., Nnang, H.: Two-scale convergence of integral functionals with convex, periodic and nonstandard growth integrands. Acta Appl. Math. 121 (2012), 175-196. DOI 10.1007/s10440-012-9702-6 | MR 2966971 | Zbl 1258.35018
[21] Tachago, J. F., Nnang, H.: Stochastic-periodic homogenization of Maxwell's equations with linear and periodic conductivity. Acta Math. Sin., Engl. Ser. 33 (2017), 117-152. DOI 10.1007/s10114-016-5597-x | MR 3581610 | Zbl 1367.35024
[22] Tachago, J. F., Nnang, H., Tchinda, F., Zappale, E.: (Two-scale) $W^1 L^{\Phi}$-gradient Young measures and homogenization of integral functionals in Orlicz-Sobolev spaces. J. Elliptic Parabol. Equ. 10 (2024), 1275-1299. DOI 10.1007/s41808-024-00294-4 | MR 4813680 | Zbl 1552.49011
[23] Tachago, J. F., Nnang, H., Zappale, E.: Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands. Opusc. Math. 41 (2021), 113-143. DOI 10.7494/OpMath.2021.41.1.113 | MR 4302444 | Zbl 1469.35023
[24] Tachago, J. F., Nnang, H., Zappale, E.: Reiterated homogenization of nonlinear degenerate elliptic operators with nonstandard growth. Differ. Integral Equ. 37 (2024), 717-752. DOI 10.57262/die037-0910-717 | MR 4753939 | Zbl 7893415
[25] Zander, V.: Fubini theorems for Orlicz spaces of Lebesgue-Bochner measurable functions. Proc. Am. Math. Soc. 32 (1972), 102-110. DOI 10.1090/S0002-9939-1972-0291791-4 | MR 0291791 | Zbl 0256.28006
[26] Zhikov, V. V., Kozlov, S. M., Olejnik, 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
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