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June  2021, 10(2): 297-320. 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. 

Dipartimento di Scienze di Base ed Applicate per l’Ingegneria, Sapienza Università di Roma, Via Antonio Scarpa, 16, 00161 Roma, Italy

* Corresponding author: Elvira Zappale

Received  November 2019 Published  June 2021 Early access  June 2020

Fund Project: The first and the last authors thank ICTP-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 and Control Theory, 2021, 10 (2) : 297-320. doi: 10.3934/eect.2020067
References:
[1] R. Adams, Sobolev 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. 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.

[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.

show all references

References:
[1] R. Adams, Sobolev 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. 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.

[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.

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