American Institute of Mathematical Sciences

Advanced
  • Previous Article
    Approximate controllability for Navier–Stokes equations in $ \rm3D $ cylinders under Lions boundary conditions by an explicit saturating set
  • EECT Home
  • This Issue
  • Next Article
    Solvability in abstract evolution equations with countable time delays in Banach spaces: Global Lipschitz perturbation
doi: 10.3934/eect.2020067

Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting

Joel Fotso Tachago 1, , Giuliano Gargiulo 2, , Hubert Nnang 3, and  Elvira Zappale 4,,

1. 

Higher Teacher, Training College Mathematics department. University of Bamenda, Faculty of Sciences, P.O. Box 39, Bambili, Cameroon
2. 

Universitá degli Studi del Sannio, Dipartimento di Scienze e Tecnologie, Via De Sanctis, Benevento, 82100, Italy
3. 

University of Yaounde I, École Normale Supérieure de Yaoundé, P.O. Box 47 Yaounde, Cameroon
4. 

Universitá di Salerno, Dipartimento di Ingegneria Industriale, Via Giovanni Paolo Ⅱ, 132 (84084) Fisciano, SA, Italy

* Corresponding author: Elvira Zappale

Received  November 2019 Published  June 2020

Fund Project: The first and the last authors thank ICP-INdAM Research in Pairs programme 2018

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.

Keywords: Convexity, homogenization, reiterated two-scale convergence, Sobolev-Orlicz Spaces.
Mathematics Subject Classification: Primary: 49J45, 46J10; Secondary: 46E30.
Citation: Joel Fotso Tachago, Giuliano Gargiulo, Hubert Nnang, Elvira Zappale. Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting. Evolution Equations & Control Theory, doi: 10.3934/eect.2020067
References:
[1] R. Adams, Sobolev Spaces, Academic Press, New York-London, 1975.   Google Scholar
[2]

R. Adams, On the Orlicz-Sobolev imbedding theorem, J. Functional Analysis, 24 (1977), 241-257.  doi: 10.1016/0022-1236(77)90055-6.  Google Scholar

[3]

G. Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[4]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proc. Royal Soc. Edin., 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[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.  Google Scholar

[14]

I. Fonseca and E. Zappale, Multiscale relaxation of convex functionals, J. Convex Anal., 10 (2003), 325-350.   Google Scholar

[15]

M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Mat. Univ. Trieste, 29 (1997), 141-161.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Fotso Tachago, H. Nnang and E. Zappale, Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands, preprint, 2019, arXiv: math/1901.07217v1. Google Scholar

[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.  Google Scholar

[20]

A. Ioffe, On lower semicontinuity of integral functionals. I, SIAM Journ. Control Optim., 15 (1977), 521-538.  doi: 10.1137/0315035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. Lukkassen, G. Nguetseng, H. 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.  Google Scholar

[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.  Google Scholar

[26]

G. Nguetseng and H. Nnang, Homogenization of nonlinear monotone operators beyong the periodic setting, Electron. J. Differential Equations (2003), No. 36, 1–24.  Google Scholar

[27]

H. Nnang, Homogenéisation déterministe d'opérateurs monotones, Fac. Sc. University of Yaoundé 1, Yaoundé, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1] R. Adams, Sobolev Spaces, Academic Press, New York-London, 1975.   Google Scholar
[2]

R. Adams, On the Orlicz-Sobolev imbedding theorem, J. Functional Analysis, 24 (1977), 241-257.  doi: 10.1016/0022-1236(77)90055-6.  Google Scholar

[3]

G. Allaire, Homogenization and two scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.  doi: 10.1137/0523084.  Google Scholar

[4]

G. Allaire and M. Briane, Multiscale convergence and reiterated homogenization, Proc. Royal Soc. Edin., 126 (1996), 297-342.  doi: 10.1017/S0308210500022757.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

G. Dal Maso, An Introduction to $\Gamma$-Convergence, Birkhäuser Boston, Inc., Boston, MA, 1993. doi: 10.1007/978-1-4612-0327-8.  Google Scholar

[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.  Google Scholar

[14]

I. Fonseca and E. Zappale, Multiscale relaxation of convex functionals, J. Convex Anal., 10 (2003), 325-350.   Google Scholar

[15]

M. Focardi, Semicontinuity of vectorial functionals in Orlicz-Sobolev spaces, Rend. Istit. Mat. Univ. Trieste, 29 (1997), 141-161.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

J. Fotso Tachago, H. Nnang and E. Zappale, Reiterated periodic homogenization of integral functionals with convex and nonstandard growth integrands, preprint, 2019, arXiv: math/1901.07217v1. Google Scholar

[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.  Google Scholar

[20]

A. Ioffe, On lower semicontinuity of integral functionals. I, SIAM Journ. Control Optim., 15 (1977), 521-538.  doi: 10.1137/0315035.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[24]

D. Lukkassen, G. Nguetseng, H. 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.  Google Scholar

[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.  Google Scholar

[26]

G. Nguetseng and H. Nnang, Homogenization of nonlinear monotone operators beyong the periodic setting, Electron. J. Differential Equations (2003), No. 36, 1–24.  Google Scholar

[27]

H. Nnang, Homogenéisation déterministe d'opérateurs monotones, Fac. Sc. University of Yaoundé 1, Yaoundé, 2004. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[1]

Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete & Continuous Dynamical Systems - A, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485
[2]

Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223
[3]

Jean Louis Woukeng. $\sum $-convergence and reiterated homogenization of nonlinear parabolic operators. Communications on Pure & Applied Analysis, 2010, 9 (6) : 1753-1789. doi: 10.3934/cpaa.2010.9.1753
[4]

Jie Zhao. Convergence rates for elliptic reiterated homogenization problems. Communications on Pure & Applied Analysis, 2013, 12 (6) : 2787-2795. doi: 10.3934/cpaa.2013.12.2787
[5]

Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks & Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353
[6]

Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control & Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016
[7]

Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure & Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757
[8]

Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete & Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058
[9]

Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete & Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151
[10]

Vy Khoi Le. On the existence of nontrivial solutions of inequalities in Orlicz-Sobolev spaces. Discrete & Continuous Dynamical Systems - S, 2012, 5 (4) : 809-818. doi: 10.3934/dcdss.2012.5.809
[11]

Duchao Liu, Beibei Wang, Peihao Zhao. On the trace regularity results of Musielak-Orlicz-Sobolev spaces in a bounded domain. Communications on Pure & Applied Analysis, 2016, 15 (5) : 1643-1659. doi: 10.3934/cpaa.2016018
[12]

Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic & Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707
[13]

Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks & Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143
[14]

Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic & Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251
[15]

Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic & Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65
[16]

Wen-ming He, Jun-zhi Cui. The estimate of the multi-scale homogenization method for Green's function on Sobolev space $W^{1,q}(\Omega)$. Communications on Pure & Applied Analysis, 2012, 11 (2) : 501-516. doi: 10.3934/cpaa.2012.11.501
[17]

Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827
[18]

Piotr Gwiazda, Agnieszka Świerczewska-Gwiazda, Aneta Wróblewska. Generalized Stokes system in Orlicz spaces. Discrete & Continuous Dynamical Systems - A, 2012, 32 (6) : 2125-2146. doi: 10.3934/dcds.2012.32.2125
[19]

Piotr Gwiazda, Piotr Minakowski, Agnieszka Świerczewska-Gwiazda. On the anisotropic Orlicz spaces applied in the problems of continuum mechanics. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1291-1306. doi: 10.3934/dcdss.2013.6.1291
[20]

Huilian Jia, Lihe Wang, Fengping Yao, Shulin Zhou. Regularity theory in Orlicz spaces for the poisson and heat equations. Communications on Pure & Applied Analysis, 2008, 7 (2) : 407-416. doi: 10.3934/cpaa.2008.7.407

2019 Impact Factor: 0.953

Tools

Article outline

[Back to Top]

Copyright © 2020 American Institute of Mathematical Sciences