Homogenization of multivalued monotone operators with variable growth exponent

Svetlana Pastukhova; Valeria Chiadò Piat

doi:10.3934/nhm.2020013

Previous Article
NHM Home
This Issue
Next Article
Comparative study of macroscopic traffic flow models at road junctions

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 & 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

[1]	Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
[2]	Mostafa Abounouh, Hassan Al-Moatassime, Sabah Kaouri. Non-standard boundary conditions for the linearized Korteweg-de Vries equation. Discrete & Continuous Dynamical Systems - S, 2021, 14 (8) : 2625-2654. doi: 10.3934/dcdss.2021066
[3]	Maksym Berezhnyi, Evgen Khruslov. Non-standard dynamics of elastic composites. Networks & Heterogeneous Media, 2011, 6 (1) : 89-109. doi: 10.3934/nhm.2011.6.89
[4]	Dag Lukkassen, Annette Meidell, Peter Wall. Multiscale homogenization of monotone operators. Discrete & Continuous Dynamical Systems, 2008, 22 (3) : 711-727. doi: 10.3934/dcds.2008.22.711
[5]	Christopher Cox, Renato Feres. Differential geometry of rigid bodies collisions and non-standard billiards. Discrete & Continuous Dynamical Systems, 2016, 36 (11) : 6065-6099. doi: 10.3934/dcds.2016065
[6]	Natália Martins. A non-standard class of variational problems of Herglotz type. Discrete & Continuous Dynamical Systems - S, 2021 doi: 10.3934/dcdss.2021152
[7]	Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks & Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002
[8]	Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks & Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181
[9]	P. Cerejeiras, U. Kähler, M. M. Rodrigues, N. Vieira. Hodge type decomposition in variable exponent spaces for the time-dependent operators: the Schrödinger case. Communications on Pure & Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253
[10]	Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete & Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016
[11]	SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure & Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026
[12]	Jiangyan Peng, Dingcheng Wang. Asymptotics for ruin probabilities of a non-standard renewal risk model with dependence structures and exponential Lévy process investment returns. Journal of Industrial & Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010
[13]	Dachun Yang, Sibei Yang. Maximal function characterizations of Musielak-Orlicz-Hardy spaces associated to non-negative self-adjoint operators satisfying Gaussian estimates. Communications on Pure & Applied Analysis, 2016, 15 (6) : 2135-2160. doi: 10.3934/cpaa.2016031
[14]	Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete & Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131
[15]	Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure & Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987
[16]	Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete & Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751
[17]	Aneta Wróblewska-Kamińska. Local pressure methods in Orlicz spaces for the motion of rigid bodies in a non-Newtonian fluid with general growth conditions. Discrete & Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417
[18]	Dalila Azzam-Laouir, Warda Belhoula, Charles Castaing, M. D. P. Monteiro Marques. Multi-valued perturbation to evolution problems involving time dependent maximal monotone operators. Evolution Equations & Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004
[19]	Soumia Saïdi. On a second-order functional evolution problem with time and state dependent maximal monotone operators. Evolution Equations & Control Theory, 2021 doi: 10.3934/eect.2021034
[20]	Pavel Jirásek. On Compactness Conditions for the $p$-Laplacian. Communications on Pure & Applied Analysis, 2016, 15 (3) : 715-726. doi: 10.3934/cpaa.2016.15.715

American Institute of Mathematical Sciences

Homogenization of multivalued monotone operators with variable growth exponent

References:

References:

Tools

Metrics

Other articles
by authors

Tools

Tools

Metrics

Other articles
by authors

American Institute of Mathematical Sciences

Homogenization of multivalued monotone operators with variable growth exponent

References:

References:

Tools

Metrics

Other articlesby authors

Tools

Tools

Metrics

Other articlesby authors

Other articles
by authors

Other articles
by authors