Homogenization of multivalued monotone operators with variable growth exponent

Svetlana Pastukhova; Valeria Chiadò Piat; Svetlana Pastukhova; Valeria Chiadò Piat

doi:10.3934/nhm.2020013

Networks and Heterogeneous Media

2020, Volume 15, Issue 2: 281-305. doi: 10.3934/nhm.2020013

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Homogenization of multivalued monotone operators with variable growth exponent

Svetlana Pastukhova ^{1
,
,},
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

Received: 01 March 2019 Revised: 01 April 2020 Published: 19 May 2020
Primary: 35J60, 35J57; Secondary: 35J99

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.
- Homogenization,
- maximal monotone operators,
- non-standard growth conditions,
- variable exponent spaces,
- compensated compactness Lemma
Citation: Svetlana Pastukhova, Valeria Chiadò Piat. Homogenization of multivalued monotone operators with variable growth exponent[J]. Networks and Heterogeneous Media, 2020, 15(2): 281-305. doi: 10.3934/nhm.2020013

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Abstract

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]	Multivalued elliptic operators with nonstandard growth. Adv. Nonlinear Anal. (2018) 7: 35-48.
[2]	Regularity for general functionals with double phase. Calculus of Variations and Partial Differential Equations (2018) 57: 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]	Omogeneizzazione di integrali non coercivi. Ricerche Mat. (1983) 32: 347-368.
[5]	Homogenization of nonlinear elliptic systems in nonreflexive Musielak-Orlicz spaces. Nonlinearity (2019) 32: 1073-1110.
[6]	Some properties of $\Gamma$-limits of integral functionals. Ann. Mat. Pura Appl. (1979) 122: 1-60.
[7]	Some properties of $\Gamma$-limits of integral functionals. Ann. Mat. Pura Appl. (1979) 122: 1-60.
[8]	C. Castaing and M. Valadier, Convex Analysis and Measurable Multifunctions, Lecture Notes in Mathematics, Vol. 580. Springer-Verlag, Berlin-New York, 1977.
[9]	$G$-convergence of monotone operators. Ann. Inst. H. Poincaré Anal. Non Linéaire (1990) 7: 123-160.
[10]	Homogenization of monotone operators. Nonlinear Anal. (1990) 4: 717-732.
[11]	A pocket guide to nonlinear differential equations in Musielak-Orlicz spaces. Nonlinear Analysis (2018) 175: 1-27.
[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]	Density of smooth functions in $W^{k, p}(\Omega)$. Proc. Roy. Soc. London Ser. A (1992) 437: 229-236.
[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]	Some approximation properties in Orlicz-Sobolev spaces. Studia Math. (1982) 74: 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]	On spaces $L^{p(x)}$ and $W^{k, p(x)}$. Czechoslovak Mathematical Journal (1991) 41: 592-618.
[18]	J.-L. Lions, Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires, Dunod, Gauthier-Villars, Paris, 1969.
[19]	Periodic solutions and homogenization of nonlinear variational problems. Ann. Mat. Pura Appl. (1978) 117: 139-152.
[20]	Regularity for elliptic equations with general growth conditions. Journal of Differential Equations (1993) 105: 296-333.
[21]	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 (2019) 734: 277-292.
[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]	Operator estimates in nonlinear problems of reiterated homogenization. Proceedings of the Steklov Institute of Mathematics (2008) 261: 214-228.
[24]	Galerkin approximations in problems with anisotropic $p(\cdot)$-Laplacian. Applicable Anal. (2019) 98: 345-361.
[25]	On existence and uniqueness classes for the Cauchy problem for parabolic equations of the $p$-Laplace type. Commun. Pure Appl. Anal. (2013) 12: 1783-1812.
[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]	Lavrentiev effect and the averaging of nonlinear variational problem. Differ. Equations (1991) 27: 32-39.
[28]	On the density of smooth functions in Sobolev-Orlich spaces. J. Math. Sci. (2006) 132: 285-294.
[29]	On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (2011) 173: 463-570.
[30]	Homogenization of a Navier-Stokes type system for electrorheological fluid. Complex Variables and Elliptic Equations (2011) 56: 545-558.
[31]	Improved integrability of the gradients of solutions of elliptic equations with variable nonlinearity exponent. Sb. Math. (2008) 199: 1751-1782.
[32]	Homogenization of monotone operators under conditions of coercitivity and growth of variable order. Math. Notes (2011) 90: 48-63.
[33]	On the $\Gamma$-convergence of oscillating integrands with nonstandard coercivity and growth conditions. Sb. Math. (2014) 205: 488-521.
[34]	$\Gamma$-convergence of integrands with nonstandard coercivity and growth conditions. J. Math. Sci. (2014) 196: 535-562.
[35]	Homogenization and two-scale convergence in Sobolev space with oscillating exponent. St. Petersburg Mathematical Journal (2019) 30: 231-251.
[36]	Homogenization of degenerate elliptic equations. Sib. Math. J. (2008) 49: 80-101.
[37]	On the homogenization of degenerate elliptic equations. Dokl. Akad. Nauk (2006) 410: 587-591.

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