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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

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