Advanced Search
Article Contents
Article Contents

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

  • * Corresponding author: Elvira Zappale

    * Corresponding author: Elvira Zappale
The first and the last authors thank ICTP-INDAM Research in Pairs programme 2018
Abstract Full Text(HTML) Related Papers Cited by
  • The $ \Gamma $-limit of a family of functionals $ u\mapsto \int_{\Omega }f\left( \frac{x}{\varepsilon },\frac{x}{\varepsilon ^{2}},D^{s}u\right) dx $ is obtained for $ s = 1,2 $ and when the integrand $ f = f\left( y,z,v\right) $ is a continous function, periodic in $ y $ and $ z $ and convex with respect to $ v $ with nonstandard growth. The reiterated two-scale limits of second order derivatives are characterized in this setting.

    Mathematics Subject Classification: Primary: 49J45, 46J10; Secondary: 46E30.


    \begin{equation} \\ \end{equation}
  • 加载中
  • [1] R. AdamsSobolev Spaces, Academic Press, New York-London, 1975. 
    [2] R. Adams, On the Orlicz-Sobolev imbedding theorem, J. Functional Analysis, 24 (1977), 241-257.  doi: 10.1016/0022-1236(77)90055-6.
    [3] G. Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.
    [4] G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proc. Royal Soc. Edin., 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.
    [5] M. Baía and I. Fonseca, 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.
    [6] G. Carita, A. M. Ribeiro and E. Zappale, An homogenization result in $W^{1, p}\times L^q$, J. Convex Anal., 18, n. 4, (2011), 1093–1126.
    [7] M. Chmara and J. Maksymiuk, Anisotropic Orlicz-Sobolev spaces of vector valued functions and Lagrange equations, J. Math. Anal. Appl., 456 (2017), 457-475.  doi: 10.1016/j.jmaa.2017.07.032.
    [8] A. Cianchi, Higher-order Sobolev and Poincaré inequalities in Orlicz spaces, Forum Math., 18, (2006), n. 5,745–767. doi: 10.1515/FORUM.2006.037.
    [9] D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method, SIAM J. Math. Anal., 37, n. 5. (2006), 1435–1453. doi: 10.1137/040620898.
    [10] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40, n. 4, (2008), 1585–1620. doi: 10.1137/080713148.
    [11] D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method, Series in Contemporary Mathematics, Vol. 3, Springer, Singapore, 2018. doi: 10.1007/978-981-13-3032-2.
    [12] G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.
    [13] W. Desch and R. Grimmer, On the wellposedness of constitutive laws involving dissipation potentials, Trans. Amer. Math. Soc., 353 (2001), 5095-5120.  doi: 10.1090/S0002-9947-01-02847-1.
    [14] I. Fonseca and E. Zappale, Multiscale relaxation of convex functionals, J. Convex Anal., 10 (2003), 325-350. 
    [15] M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Mat. Univ. Trieste, 29 (1997), 141-161. 
    [16] J. F. Tachago and H. Nnang, 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.
    [17] J. F. Tachago, H. Nnang and E. Zappale, Relaxation of periodic and nonstandard growth integrands by means of two scale convergence, in Integral Methods in Science and Engineering–Analytic Treatment and Numerical Approximations, Birkhäuser/Springer, Cham, 2019, 123–132. doi: 10.1007/978-3-030-16077-7.
    [18] J. Fotso Tachago, H. Nnang and E. Zappale, Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands, Opuscula Math., 41 (2021), 113-143. doi: 10.7494/OpMath.2021.41.1.113.
    [19] A. Gaudiello and O. Guibé, Homogenization of an evolution problem with $ L\log L$ data in a domain with oscillating boundary, Ann. Mat. Pura Appl., 197 (2018), 153-169.  doi: 10.1007/s10231-017-0673-0.
    [20] A. Ioffe, On lower semicontinuity of integral functionals. I, SIAM Journ. Control Optim., 15 (1977), 521-538.  doi: 10.1137/0315035.
    [21] R. K. Bogning and H. Nnang, Periodic homogenization of parabolic nonstandard monotone operators, Acta Appl. Math., 125 (2013), 209-229.  doi: 10.1007/s10440-012-9788-x.
    [22] P. A. Kozarzewski and E. Zappale, Orlicz equi-integrability for scaled gradients, J. Elliptic Parabol. Equ., 3 (2017), 1-13.  doi: 10.1007/s41808-017-0001-2.
    [23] P. A. Kozarzewski and E. Zappale, A note on optimal design for thin structures in the Orlicz-Sobolev setting, Integral Methods in Science and Engineering, Vol. 1, (2017), Birkhäuser Basel, 161–171.
    [24] D. LukkassenG. NguetsengH. Nnang and P. Wall, Reiterated homogenization of nonlinear monotone operators in a general deterministic setting, J. Funct. Spaces Appl., 7 (2009), 121-152.  doi: 10.1155/2009/102486.
    [25] G. Nguetseng, A general convergent result for functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.  doi: 10.1137/0520043.
    [26] G. Nguetseng and H. Nnang, Homogenization of nonlinear monotone operators beyong the periodic setting, Electron. J. Differential Equations (2003), No. 36, 1–24.
    [27] H. Nnang, Homogenéisation déterministe d'opérateurs monotones, Fac. Sc. University of Yaoundé 1, Yaoundé, 2004.
    [28] H. Nnang, Deterministic Homogenization of Nonlinear Degenerated Elliptic Operators with Nonstandard Growth, Act. Math. Sin., 30 (2014), 1621-1654.  doi: 10.1007/s10114-014-2131-x.
    [29] M. M. Rao and Z. D. Ren, Theory of Orlicz spaces, Monographs and Textbooks in Pure and Applied Mathematics, Vol. 146, Marcel Dekker, Inc., New York, 1991.
    [30] E. Zappale, A note on dimension reduction for unbounded integrals with periodic microstructure via the unfolding method for slender domains, Evol. Equ. Control Theory, 6 (2017), 299-318.  doi: 10.3934/eect.2017016.
  • 加载中

Article Metrics

HTML views(799) PDF downloads(238) Cited by(0)

Access History



    DownLoad:  Full-Size Img  PowerPoint