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

Homogenization of multivalued monotone operators with variable growth exponent

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.

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. BaroniM. 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. BulíčekP. GwiazdaM. 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 PiatG. 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. ZhikovS. 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. BaroniM. 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. BulíčekP. GwiazdaM. 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 PiatG. 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. ZhikovS. E. Pastukhova and S. V. Tikhomirova, On the homogenization of degenerate elliptic equations, Dokl. Akad. Nauk, 410 (2006), 587-591. 

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