- Previous Article
- NHM Home
- This Issue
-
Next Article
Comparative study of macroscopic traffic flow models at road junctions
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 |
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.
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.
|
[1] |
Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks and 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 and 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 and 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 and 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 and 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 and Continuous Dynamical Systems - S, 2022, 15 (3) : 573-586. doi: 10.3934/dcdss.2021152 |
[7] |
Eleonora Messina, Mario Pezzella, Antonia Vecchio. A non-standard numerical scheme for an age-of-infection epidemic model. Journal of Computational Dynamics, 2022, 9 (2) : 239-252. doi: 10.3934/jcd.2021029 |
[8] |
Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks and Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002 |
[9] |
Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181 |
[10] |
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 and Applied Analysis, 2014, 13 (6) : 2253-2272. doi: 10.3934/cpaa.2014.13.2253 |
[11] |
Maria-Magdalena Boureanu. Fourth-order problems with Leray-Lions type operators in variable exponent spaces. Discrete and Continuous Dynamical Systems - S, 2019, 12 (2) : 231-243. doi: 10.3934/dcdss.2019016 |
[12] |
SYLWIA DUDEK, IWONA SKRZYPCZAK. Liouville theorems for elliptic problems in variable exponent spaces. Communications on Pure and Applied Analysis, 2017, 16 (2) : 513-532. doi: 10.3934/cpaa.2017026 |
[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 and Applied Analysis, 2016, 15 (6) : 2135-2160. doi: 10.3934/cpaa.2016031 |
[14] |
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 and Management Optimization, 2017, 13 (1) : 155-185. doi: 10.3934/jimo.2016010 |
[15] |
Gui-Dong Li, Chun-Lei Tang. Existence of positive ground state solutions for Choquard equation with variable exponent growth. Discrete and Continuous Dynamical Systems - S, 2019, 12 (7) : 2035-2050. doi: 10.3934/dcdss.2019131 |
[16] |
Carla Baroncini, Julián Fernández Bonder. An extension of a Theorem of V. Šverák to variable exponent spaces. Communications on Pure and Applied Analysis, 2015, 14 (5) : 1987-2007. doi: 10.3934/cpaa.2015.14.1987 |
[17] |
Giuseppe Da Prato, Alessandra Lunardi. Maximal dissipativity of a class of elliptic degenerate operators in weighted $L^2$ spaces. Discrete and Continuous Dynamical Systems - B, 2006, 6 (4) : 751-760. doi: 10.3934/dcdsb.2006.6.751 |
[18] |
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 and Continuous Dynamical Systems - S, 2013, 6 (5) : 1417-1425. doi: 10.3934/dcdss.2013.6.1417 |
[19] |
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 and Control Theory, 2020, 9 (1) : 219-254. doi: 10.3934/eect.2020004 |
[20] |
Soumia Saïdi. On a second-order functional evolution problem with time and state dependent maximal monotone operators. Evolution Equations and Control Theory, 2021 doi: 10.3934/eect.2021034 |
2020 Impact Factor: 1.213
Tools
Metrics
Other articles
by authors
[Back to Top]