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Elliptic operators with nonstandard growth condition: some results and open problems. (English) Zbl 1439.35173
Kuchment, Peter (ed.) et al., Differential equations, mathematical physics, and applications. Selim Grigorievich Krein centennial. Providence, RI: American Mathematical Society (AMS). Contemp. Math. 734, 277-292 (2019).
Summary: This is a short survey of certain results and open problems on the Dirichlet problem for nonlinear monotone elliptic operators with nonstandard growth condition. In this paper we pay attention to the case when the so-called Lavrentiev phenomenon may occur. Furthermore, as a rule the growth condition accepted in the paper is more general than the well-known \(p(x)\) growth condition. Also we discuss the homogenization result obtained by V. Zhikov and S. Pastukhova in the case of \(p(x)\) growth and its potential extensions.
For the entire collection see [Zbl 1420.35009].

MSC:
35J60 Nonlinear elliptic equations
35J25 Boundary value problems for second-order elliptic equations
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
47H05 Monotone operators and generalizations
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[1] Alkhutov, Yu. A., The Harnack inequality and the H\`‘older property of solutions of nonlinear elliptic equations with a nonstandard growth condition, Differ. Uravn.. Differential Equations, 33 33, 12, 1653-1663 (1998) (1997) · Zbl 0949.35048
[2] Allaire, Gr\'egoire, Homogenization and two-scale convergence, SIAM J. Math. Anal., 23, 6, 1482-1518 (1992) · Zbl 0770.35005
[3] Amaziane, Brahim; Pankratov, Leonid, Homogenization in Sobolev spaces with nonstandard growth: brief review of methods and applications, Int. J. Differ. Equ., Art. ID 693529, 16 pp. (2013) · Zbl 1270.35054
[4] Antontsev, Stanislav; Shmarev, Sergei, Elliptic equations and systems with nonstandard growth conditions: existence, uniqueness and localization properties of solutions, Nonlinear Anal., 65, 4, 728-761 (2006) · Zbl 1245.35033
[5] S. N. Antontsev and S. I. Shmarev, Elliptic equations with anisotropic nonlinearity and nonstandard growth, Handbook of Differential Equations, Stationary Partial Differential Equations, Vol. 3, Elsevier, Amsterdam, 2006, 1-100. · Zbl 1192.35047
[6] Attouch, H., Variational convergence for functions and operators, Applicable Mathematics Series, xiv+423 pp. (1984), Pitman (Advanced Publishing Program), Boston, MA · Zbl 0561.49012
[7] Avci, Mustafa; Pankov, Alexander, Multivalued elliptic operators with nonstandard growth, Adv. Nonlinear Anal., 7, 1, 35-48 (2018) · Zbl 1391.35134
[8] Benkirane, A.; Ould Mohamedhen Val, M., Some approximation properties in Musielak-Orlicz-Sobolev spaces, Thai J. Math., 10, 2, 371-381 (2012) · Zbl 1264.46024
[9] Browder, Felix E., Nonlinear operators and nonlinear equations of evolution in Banach spaces. Nonlinear functional analysis, Proc. Sympos. Pure Math., Vol. XVIII, Part 2, Chicago, Ill., 1968, 1-308 (1976), Amer. Math. Soc., Providence, R. I.
[10] Chen, Yunmei; Levine, Stacey; Rao, Murali, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66, 4, 1383-1406 (2006) · Zbl 1102.49010
[11] Chiad\`“o Piat, Valeria; Dal Maso, Gianni; Defranceschi, Anneliese, \(G\)-convergence of monotone operators, Ann. Inst. H. Poincar\'”e Anal. Non Lin\'eaire, 7, 3, 123-160 (1990) · Zbl 0731.35033
[12] Chiad\`‘o Piat, Valeria; Defranceschi, Anneliese, Homogenization of monotone operators, Nonlinear Anal., 14, 9, 717-732 (1990) · Zbl 0705.35041
[13] Cioranescu, Doina; Damlamian, Alain; Griso, Georges, Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335, 1, 99-104 (2002) · Zbl 1001.49016
[14] Damlamian, Alain; Meunier, Nicolas; Van Schaftingen, Jean, Periodic homogenization of monotone multivalued operators, Nonlinear Anal., 67, 12, 3217-3239 (2007) · Zbl 1133.35007
[15] Diening, Lars; Harjulehto, Petteri; H\`“ast\'”o, Peter; R\ru\vzi\vcka, Michael, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics 2017, x+509 pp. (2011), Springer, Heidelberg · Zbl 1222.46002
[16] Fan, Xianling, Differential equations of divergence form in Musielak-Sobolev spaces and a sub-supersolution method, J. Math. Anal. Appl., 386, 2, 593-604 (2012) · Zbl 1270.35156
[17] Fan, Xianling; Zhao, Dun, A class of De Giorgi type and H\`‘older continuity, Nonlinear Anal., 36, 3, Ser. A: Theory Methods, 295-318 (1999) · Zbl 0927.46022
[18] Fan, Xianling; Zhao, Dun, The quasi-minimizer of integral functionals with \(m(x)\) growth conditions, Nonlinear Anal., 39, 7, Ser. A: Theory Methods, 807-816 (2000) · Zbl 0943.49029
[19] Francfort, Gilles; Murat, Fran\ccois; Tatar, Luc, Monotone operators in divergence form with \(x\)-dependent multivalued graphs, Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8), 7, 1, 23-59 (2004) · Zbl 1115.35047
[20] Francfort, Gilles A.; Murat, Fran\ccois; Tartar, Luc, Homogenization of monotone operators in divergence form with \(x\)-dependent multivalued graphs, Ann. Mat. Pura Appl. (4), 188, 4, 631-652 (2009) · Zbl 1180.35077
[21] Jikov, V. V.; Kozlov, S. M.; Ole\u\inik, O. A., Homogenization of differential operators and integral functionals, xii+570 pp. (1994), Springer-Verlag, Berlin
[22] Kenmochi, N., Monotonicity and compactness methods for nonlinear variational inequalities. Handbook of differential equations: stationary partial differential equations. Vol. IV, Handb. Differ. Equ., 203-298 (2007), Elsevier/North-Holland, Amsterdam · Zbl 1192.35083
[23] Krasnosel\cprime ski\u\i, M. A.; Ruticki\u\i, Ja. B., Convex functions and Orlicz spaces, Translated from the first Russian edition by Leo F. Boron, xi+249 pp. (1961), P. Noordhoff Ltd., Groningen
[24] Meunier, Nicolas; Van Schaftingen, Jean, Periodic reiterated homogenization for elliptic functions, J. Math. Pures Appl. (9), 84, 12, 1716-1743 (2005) · Zbl 1159.35310
[25] Murat, Fran\ccois, Compacit\'e par compensation, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4), 5, 3, 489-507 (1978) · Zbl 0399.46022
[26] Musielak, Julian, Orlicz spaces and modular spaces, Lecture Notes in Mathematics 1034, iii+222 pp. (1983), Springer-Verlag, Berlin · Zbl 0557.46020
[27] Pankov, Alexander, \(G\)-convergence and homogenization of nonlinear partial differential operators, Mathematics and its Applications 422, xiv+249 pp. (1997), Kluwer Academic Publishers, Dordrecht · Zbl 0883.35001
[28] Papageorgiou, Nikolaos S.; Kyritsi-Yiallourou, Sophia Th., Handbook of applied analysis, Advances in Mechanics and Mathematics 19, xviii+793 pp. (2009), Springer, New York · Zbl 1189.49003
[29] Pascali, Dan; Sburlan, Silviu, Nonlinear mappings of monotone type, x+341 pp. (1978), Martinus Nijhoff Publishers, The Hague; Sijthoff & Noordhoff International Publishers, Alphen aan den Rijn · Zbl 0423.47021
[30] Pastukhova, S. E.; Khripunova, A. S., Some versions of the compensated compactness principle, Mat. Sb.. Sb. Math., 202 202, 9-10, 1387-1412 (2011) · Zbl 1246.46027
[31] Pastukhova, S. E.; Khripunova, A. S., Gamma-closure of some classes of nonstandard convex integrands, J. Math. Sci. (N.Y.), 177, 1, 83-108 (2011) · Zbl 1290.49027
[32] R\uadulescu, Vicen\ctiu D.; Repov\vs, Du\vsan D., Partial differential equations with variable exponents, Monographs and Research Notes in Mathematics, xxi+301 pp. (2015), CRC Press, Boca Raton, FL · Zbl 1343.35003
[33] R\ru\vzi\vcka, Michael, Electrorheological fluids: modeling and mathematical theory, Lecture Notes in Mathematics 1748, xvi+176 pp. (2000), Springer-Verlag, Berlin · Zbl 0962.76001
[34] Zeidler, Eberhard, Nonlinear functional analysis and its applications. IV, xxiv+975 pp. (1988), Springer-Verlag, New York · Zbl 0648.47036
[35] Zeidler, Eberhard, Nonlinear functional analysis and its applications. II/B, i-xvi and 469-1202 (1990), Springer-Verlag, New York
[36] Zhikov, V. V., The Lavrent\cprime ev effect and averaging of nonlinear variational problems, Differentsial\cprime nye Uravneniya. Differential Equations, 27 27, 1, 32-39 (1991) · Zbl 0749.49022
[37] Zhikov, Vasili\u\i V., On Lavrentiev’s phenomenon, Russian J. Math. Phys., 3, 2, 249-269 (1995) · Zbl 0910.49020
[38] Zhikov, V. V., Solvability of the three-dimensional thermistor problem, Tr. Mat. Inst. Steklova. Proc. Steklov Inst. Math., 261 261, 1, 98-111 (2008) · Zbl 1237.35058
[39] Zhikov, V. V., On the technique of the passage to the limit in nonlinear elliptic equations, Funktsional. Anal. i Prilozhen.. Funct. Anal. Appl., 43 43, 2, 96-112 (2009) · Zbl 1271.35033
[40] Zhikov, V. V., On variational problems and nonlinear elliptic equations with nonstandard growth conditions, J. Math. Sci. (N.Y.), 173, 5, 463-570 (2011) · Zbl 1279.49005
[41] Zhikov, V. V.; Pastukhova, S. E., On the improved integrability of the gradient of solutions of elliptic equations with a variable nonlinearity exponent, Mat. Sb.. Sb. Math., 199 199, 11-12, 1751-1782 (2008) · Zbl 1172.35024
[42] Zhikov, V. V.; Pastukhova, S. E., The compensated compactness principle, Dokl. Akad. Nauk. Dokl. Math., 433 82, 1, 590-595 (2010) · Zbl 1211.35075
[43] Zhikov, V. V.; Pastukhova, S. E., Lemmas on compensated compactness in elliptic and parabolic equations, Tr. Mat. Inst. Steklova. Proc. Steklov Inst. Math., 270 270, 1, 104-131 (2010) · Zbl 1216.35045
[44] Zhikov, V. V.; Pastukhova, S. E., Homogenization of monotone operators under conditions of coercitivity and growth of variable order, Mat. Zametki. Math. Notes, 90 90, 1-2, 48-63 (2011) · Zbl 1237.35012
[45] Zhikov, V. V.; Pastukhova, S. E., Uniform convexity and variational convergence, Trans. Moscow Math. Soc., 205-231 (2014) · Zbl 1321.49026
[46] Zhikov, V. V.; Pastukhova, S. E., Homogenization and two-scale convergence in a Sobolev space with an oscillating exponent, Algebra i Analiz, 30, 2, 114-144 (2018)
[47] Zhikov, V. V.; Yosifian, G. A., Introduction to the theory of two-scale convergence, J. Math. Sci. (N.Y.), 197, 3, 325-357 (2014) · Zbl 1328.46029
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