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On the approximate controllability of neutral integro-differential inclusions of Sobolev-type with infinite delay
Multiscale homogenization of integral convex functionals in Orlicz Sobolev setting
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 |
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.
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. 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. |
[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. 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. |
[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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