Homogenization of multivalued monotone operators with variable growth exponent

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June 2020, 15(2): 281-305. doi: 10.3934/nhm.2020013

Homogenization of multivalued monotone operators with variable growth exponent

Svetlana Pastukhova ^1,, and Valeria Chiadò Piat ^2,

1.	Russian Technological University, prospekt Vernadskogo 78, Moscow 119454, Russia
2.	Politecnico di Torino, Corso Duca degli Abruzzi 24, 10129 Torino, Italy

^* Corresponding author: Svetlana Pastukhova

Received March 2019 Revised April 2020 Published June 2020 Early access May 2020

We consider the Dirichlet problem for an elliptic multivalued maximal monotone operator $ {\mathcal A}_\varepsilon $ satisfying growth estimates of power type with a variable exponent. This exponent $ p_\varepsilon(x) $ and also the symbol of the operator $ {\mathcal A}_\varepsilon $ oscillate with a small period $ \varepsilon $ with respect to the space variable $ x $. We prove a homogenization result for this problem.

Keywords: Homogenization, maximal monotone operators, non-standard growth conditions, variable exponent spaces, compensated compactness Lemma.

Mathematics Subject Classification: Primary: 35J60, 35J57; Secondary: 35J99.

Citation: Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent. Networks and Heterogeneous Media, 2020, 15 (2) : 281-305. doi: 10.3934/nhm.2020013

References:

[1]	M. Avci and A. Pankov, Multivalued elliptic operators with nonstandard growth, Adv. Nonlinear Anal., 7 (2018), 35-48. doi: 10.1515/anona-2016-0043.
[2]	P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calculus of Variations and Partial Differential Equations, 57 (2018), 1-48.
[3]	F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis, Amer. Math. Soc., Providence, R. I., (1976), 1–308.
[4]	A. Braides, Omogeneizzazione di integrali non coercivi, Ricerche Mat., 32 (1983), 347-368.
[5]	M. Bulíček, P. Gwiazda, M. Kalousek and A. Swierczewska-Gwiazda, Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces, Nonlinearity, 32 (2019), 1073-1110. doi: 10.1088/1361-6544/aaf259.
[6]	L. Carbone and C. Sbordone, Some properties of $\Gamma$-limits of integral functionals, Ann. Mat. Pura Appl., 122 (1979), 1-60. doi: 10.1007/BF02411687.
[7]	L. Carbone and C. Sbordone, Some properties of $\Gamma$-limits of integral functionals, Ann. Mat. Pura Appl., 122 (1979), 1-60. doi: 10.1007/BF02411687.
[8]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[9]	V. Chiadò Piat, G. Dal Maso and A. Defranceschi, $G$-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 123-160. doi: 10.1016/S0294-1449(16)30298-0.
[10]	V. Chiadò Piat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Anal., 4 (1990), 717-732. doi: 10.1016/0362-546X(90)90102-M.
[11]	I. Chlebicka, A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces, Nonlinear Analysis, 175 (2018), 1-27. doi: 10.1016/j.na.2018.05.003.
[12]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.
[13]	D. E. Edmunds and J. Rákosník, Density of smooth functions in $W^{k, p}(\Omega)$, Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236. doi: 10.1098/rspa.1992.0059.
[14]	I. Ekeland and R.Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.
[15]	J.-P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24. doi: 10.4064/sm-74-1-17-24.
[16]	V. V. Jikov, S. M. Kozlov and O. A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.
[17]	O. Kováčik and J. Rákosníík, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.
[18]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[19]	P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152. doi: 10.1007/BF02417888.
[20]	P. Marcellini, Regularity for elliptic equations with general growth conditions, Journal of Differential Equations, 105 (1993), 296-333. doi: 10.1006/jdeq.1993.1091.
[21]	A. Pankov, Elliptic operators with nonstandard growth condition: Some results and open problems, Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemp. Math., Amer. Math. Soc., Providence, RI, 734 (2019), 277-292. doi: 10.1090/conm/734/14777.
[22]	D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff Publishers, The Hague, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978.
[23]	S. E. Pastukhova, Operator estimates in nonlinear problems of reiterated homogenization, Proceedings of the Steklov Institute of Mathematics, 261 (2008), 214-228. doi: 10.1134/S0081543808020168.
[24]	S. E. Pastukhova and D. A. Yakubovich, Galerkin approximations in problems with anisotropic $p(\cdot)$-Laplacian, Applicable Anal., 98 (2019), 345-361. doi: 10.1080/00036811.2018.1451641.
[25]	M. D. Surnachev and V. V. Zhikov, On existence and uniqueness classes for the Cauchy problem for parabolic equations of the $p$-Laplace type, Commun. Pure Appl. Anal., 12 (2013), 1783-1812. doi: 10.3934/cpaa.2013.12.1783.
[26]	V. V. Zhikov, Averaging of functionals of the calculus of vatiations and elasticity theory, Izvestiya Acad. Nauk of SSSR. Ser. Math., 50 (1986), 675–710,877.
[27]	V. V. Zhikov, Lavrentiev effect and the averaging of nonlinear variational problem, Differ. Equations, 27 (1991), 32-39.
[28]	V. V. Zhikov, On the density of smooth functions in Sobolev-Orlich spaces, J. Math. Sci., 132 (2006), 285-294. doi: 10.1007/s10958-005-0497-0.
[29]	V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci., 173 (2011), 463-570. doi: 10.1007/s10958-011-0260-7.
[30]	V. V. Zhikov, Homogenization of a Navier-Stokes type system for electrorheological fluid, Complex Variables and Elliptic Equations, 56 (2011), 545-558. doi: 10.1080/17476933.2010.487214.
[31]	V. V. Zhikov and S. E. Pastukhova, Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent, Sb. Math., 199 (2008), 1751-1782. doi: 10.1070/SM2008v199n12ABEH003980.
[32]	V. V. Zhikov and S. E. Pastukhova, Homogenization of monotone operators under conditions of coercitivity and growth of variable order, Math. Notes, 90 (2011), 48-63. doi: 10.1134/S0001434611070078.
[33]	V. V. Zhikov and S. E. Pastukhova, On the $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions, Sb. Math., 205 (2014), 488-521. doi: 10.1070/sm2014v205n04abeh004385.
[34]	V. V. Zhikov and S. E. Pastukhova, $\Gamma$-convergence of integrands with nonstandard coercivity and growth conditions, J. Math. Sci., 196 (2014), 535-562. doi: 10.1007/s10958-014-1674-9.
[35]	V. V. Zhikov and S. E. Pastukhova, Homogenization and two-scale convergence in Sobolev space with oscillating exponent, St. Petersburg Mathematical Journal, 30 (2019), 231-251. doi: 10.1090/spmj/1540.
[36]	V. V. Zhikov and S. E. Pastukhova, Homogenization of degenerate elliptic equations, Sib. Math. J., 49 (2008), 80-101. doi: 10.1007/s11202-008-0008-x.
[37]	V. V. Zhikov, S. E. Pastukhova and S. V. Tikhomirova, On the homogenization of degenerate elliptic equations, Dokl. Akad. Nauk, 410 (2006), 587-591.

show all references

References:

[1]	M. Avci and A. Pankov, Multivalued elliptic operators with nonstandard growth, Adv. Nonlinear Anal., 7 (2018), 35-48. doi: 10.1515/anona-2016-0043.
[2]	P. Baroni, M. Colombo and G. Mingione, Regularity for general functionals with double phase, Calculus of Variations and Partial Differential Equations, 57 (2018), 1-48.
[3]	F. E. Browder, Nonlinear operators and nonlinear equations of evolution in Banach spaces, Nonlinear Functional Analysis, Amer. Math. Soc., Providence, R. I., (1976), 1–308.
[4]	A. Braides, Omogeneizzazione di integrali non coercivi, Ricerche Mat., 32 (1983), 347-368.
[5]	M. Bulíček, P. Gwiazda, M. Kalousek and A. Swierczewska-Gwiazda, Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces, Nonlinearity, 32 (2019), 1073-1110. doi: 10.1088/1361-6544/aaf259.
[6]	L. Carbone and C. Sbordone, Some properties of $\Gamma$-limits of integral functionals, Ann. Mat. Pura Appl., 122 (1979), 1-60. doi: 10.1007/BF02411687.
[7]	L. Carbone and C. Sbordone, Some properties of $\Gamma$-limits of integral functionals, Ann. Mat. Pura Appl., 122 (1979), 1-60. doi: 10.1007/BF02411687.
[8]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[9]	V. Chiadò Piat, G. Dal Maso and A. Defranceschi, $G$-convergence of monotone operators, Ann. Inst. H. Poincaré Anal. Non Linéaire, 7 (1990), 123-160. doi: 10.1016/S0294-1449(16)30298-0.
[10]	V. Chiadò Piat and A. Defranceschi, Homogenization of monotone operators, Nonlinear Anal., 4 (1990), 717-732. doi: 10.1016/0362-546X(90)90102-M.
[11]	I. Chlebicka, A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces, Nonlinear Analysis, 175 (2018), 1-27. doi: 10.1016/j.na.2018.05.003.
[12]	L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Lecture Notes in Mathematics, 2017. Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.
[13]	D. E. Edmunds and J. Rákosník, Density of smooth functions in $W^{k, p}(\Omega)$, Proc. Roy. Soc. London Ser. A, 437 (1992), 229-236. doi: 10.1098/rspa.1992.0059.
[14]	I. Ekeland and R.Temam, Convex Analysis and Variational Problems, Studies in Mathematics and its Applications, Vol. 1. North-Holland Publishing Co., Amsterdam-Oxford, American Elsevier Publishing Co., Inc., New York, 1976.
[15]	J.-P. Gossez, Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 74 (1982), 17-24. doi: 10.4064/sm-74-1-17-24.
[16]	V. V. Jikov, S. M. Kozlov and O. A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994. doi: 10.1007/978-3-642-84659-5.
[17]	O. Kováčik and J. Rákosníík, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.
[18]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[19]	P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152. doi: 10.1007/BF02417888.
[20]	P. Marcellini, Regularity for elliptic equations with general growth conditions, Journal of Differential Equations, 105 (1993), 296-333. doi: 10.1006/jdeq.1993.1091.
[21]	A. Pankov, Elliptic operators with nonstandard growth condition: Some results and open problems, Differential Equations, Mathematical Physics, and Applications: Selim Grigorievich Krein Centennial, Contemp. Math., Amer. Math. Soc., Providence, RI, 734 (2019), 277-292. doi: 10.1090/conm/734/14777.
[22]	D. Pascali and S. Sburlan, Nonlinear Mappings of Monotone Type, Martinus Nijhoff Publishers, The Hague, Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn, 1978.
[23]	S. E. Pastukhova, Operator estimates in nonlinear problems of reiterated homogenization, Proceedings of the Steklov Institute of Mathematics, 261 (2008), 214-228. doi: 10.1134/S0081543808020168.
[24]	S. E. Pastukhova and D. A. Yakubovich, Galerkin approximations in problems with anisotropic $p(\cdot)$-Laplacian, Applicable Anal., 98 (2019), 345-361. doi: 10.1080/00036811.2018.1451641.
[25]	M. D. Surnachev and V. V. Zhikov, On existence and uniqueness classes for the Cauchy problem for parabolic equations of the $p$-Laplace type, Commun. Pure Appl. Anal., 12 (2013), 1783-1812. doi: 10.3934/cpaa.2013.12.1783.
[26]	V. V. Zhikov, Averaging of functionals of the calculus of vatiations and elasticity theory, Izvestiya Acad. Nauk of SSSR. Ser. Math., 50 (1986), 675–710,877.
[27]	V. V. Zhikov, Lavrentiev effect and the averaging of nonlinear variational problem, Differ. Equations, 27 (1991), 32-39.
[28]	V. V. Zhikov, On the density of smooth functions in Sobolev-Orlich spaces, J. Math. Sci., 132 (2006), 285-294. doi: 10.1007/s10958-005-0497-0.
[29]	V. V. Zhikov, On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci., 173 (2011), 463-570. doi: 10.1007/s10958-011-0260-7.
[30]	V. V. Zhikov, Homogenization of a Navier-Stokes type system for electrorheological fluid, Complex Variables and Elliptic Equations, 56 (2011), 545-558. doi: 10.1080/17476933.2010.487214.
[31]	V. V. Zhikov and S. E. Pastukhova, Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent, Sb. Math., 199 (2008), 1751-1782. doi: 10.1070/SM2008v199n12ABEH003980.
[32]	V. V. Zhikov and S. E. Pastukhova, Homogenization of monotone operators under conditions of coercitivity and growth of variable order, Math. Notes, 90 (2011), 48-63. doi: 10.1134/S0001434611070078.
[33]	V. V. Zhikov and S. E. Pastukhova, On the $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions, Sb. Math., 205 (2014), 488-521. doi: 10.1070/sm2014v205n04abeh004385.
[34]	V. V. Zhikov and S. E. Pastukhova, $\Gamma$-convergence of integrands with nonstandard coercivity and growth conditions, J. Math. Sci., 196 (2014), 535-562. doi: 10.1007/s10958-014-1674-9.
[35]	V. V. Zhikov and S. E. Pastukhova, Homogenization and two-scale convergence in Sobolev space with oscillating exponent, St. Petersburg Mathematical Journal, 30 (2019), 231-251. doi: 10.1090/spmj/1540.
[36]	V. V. Zhikov and S. E. Pastukhova, Homogenization of degenerate elliptic equations, Sib. Math. J., 49 (2008), 80-101. doi: 10.1007/s11202-008-0008-x.
[37]	V. V. Zhikov, S. E. Pastukhova and S. V. Tikhomirova, On the homogenization of degenerate elliptic equations, Dokl. Akad. Nauk, 410 (2006), 587-591.

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