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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  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 & 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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

A. Braides, Omogeneizzazione di integrali non coercivi, Ricerche Mat., 32 (1983), 347-368.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. Kováčik and J. Rákosníík, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.   Google Scholar

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[19]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152.  doi: 10.1007/BF02417888.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

V. V. Zhikov, Lavrentiev effect and the averaging of nonlinear variational problem, Differ. Equations, 27 (1991), 32-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[4]

A. Braides, Omogeneizzazione di integrali non coercivi, Ricerche Mat., 32 (1983), 347-368.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[8]

C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[17]

O. Kováčik and J. Rákosníík, On spaces $L^{p(x)}$ and $W^{k, p(x)}$, Czechoslovak Mathematical Journal, 41 (1991), 592-618.   Google Scholar

[18]

J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.  Google Scholar

[19]

P. Marcellini, Periodic solutions and homogenization of nonlinear variational problems, Ann. Mat. Pura Appl., 117 (1978), 139-152.  doi: 10.1007/BF02417888.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[27]

V. V. Zhikov, Lavrentiev effect and the averaging of nonlinear variational problem, Differ. Equations, 27 (1991), 32-39.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

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