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2012 | 10 | 6 | 2019-2032
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Elliptic problems in generalized Orlicz-Musielak spaces

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EN
Abstrakty
EN
We consider a strongly nonlinear monotone elliptic problem in generalized Orlicz-Musielak spaces. We assume neither a Δ2 nor ∇2-condition for an inhomogeneous and anisotropic N-function but assume it to be log-Hölder continuous with respect to x. We show the existence of weak solutions to the zero Dirichlet boundary value problem. Within the proof the L ∞-truncation method is coupled with a special version of the Minty-Browder trick for non-reflexive and non-separable Banach spaces.
Twórcy
  • Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland, pgwiazda@mimuw.edu.pl
  • Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland, minak@mimuw.edu.pl
  • Institute of Applied Mathematics and Mechanics, University of Warsaw, Banacha 2, 02-097, Warszawa, Poland, a.wroblewska@mimuw.edu.pl
Bibliografia
  • [1] Adams R.A., Fournier J.F., Sobolev Spaces, 2nd ed., Pure Appl. Math. (Amst.), 140, Elsevier/Academic Press, Amsterdam, 2003
  • [2] Benkirane A., Douieb J., Ould Mohamedhen Val M., An approximation theorem in Musielak-Orlicz-Sobolev spaces, Comment. Math. Prace Mat., 2011, 51(1), 109–120
  • [3] Benkirane A., Ould Mohamedhen Val M., Some approximation properties in Musielak-Orlicz-Sobolev spaces, Thai J. Math., 2012, 10(2), 371–381
  • [4] Donaldson T, Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces, J. Differential Equations, 1971, 10(3), 507–528 http://dx.doi.org/10.1016/0022-0396(71)90009-X[Crossref]
  • [5] Gossez J.-P., Some approximation properties in Orlicz-Sobolev spaces, Studia Math., 1982, 74(1), 17–24
  • [6] Gossez J.-P., Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients, Trans. Amer. Math. Soc., 1974, 190, 163–205 http://dx.doi.org/10.1090/S0002-9947-1974-0342854-2[Crossref]
  • [7] Gossez J.-P., Orlicz-Sobolev spaces and nonlinear elliptic boundary value problems, In: Nonlinear Analysis, Function Spaces and Applications, Horni Bradlo, 1978, Teubner, Leipzig, 1979, 59–94
  • [8] Gossez J.-P., Mustonen V., Variational inequalities in Orlicz-Sobolev spaces, Nonlinear Anal., 1987, 11(3), 379–392 http://dx.doi.org/10.1016/0362-546X(87)90053-8[Crossref]
  • [9] Gwiazda P., Swierczewska-Gwiazda A., On steady non-Newtonian fluids with growth conditions in generalized Orlicz spaces, Topol. Methods Nonlinear Anal., 2008, 32(1), 103–114
  • [10] Gwiazda P., Swierczewska-Gwiazda A., On non-Newtonian fluids with a property of rapid thickening under different stimulus, Math. Models Methods Appl. Sci., 2008, 18(7), 1073–1092 http://dx.doi.org/10.1142/S0218202508002954[Crossref][WoS]
  • [11] Gwiazda P., Swierczewska-Gwiazda A., Wróblewska A., Monotonicity methods in generalized Orlicz spaces for a class of non-Newtonian fluids, Math. Methods Appl. Sci., 2010, 33(2), 125–137 [WoS]
  • [12] Gwiazda P., Swierczewska-Gwiazda A., Wróblewska A., Generalized Stokes system in Orlicz spaces, Discrete Contin. Dyn. Syst., 2012, 32(6), 2125–2146 http://dx.doi.org/10.3934/dcds.2012.32.2125[Crossref]
  • [13] Gwiazda P., Wittbold P., Wróblewska A., Zimmermann A., Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces, J. Differential Equations, 2012, 253(2), 635–666 http://dx.doi.org/10.1016/j.jde.2012.03.025[Crossref]
  • [14] Gwiazda P., Wittbold P., Wróblewska-Kaminska A., Zimmermann A., Corrigendum to ”Renormalized solutions of nonlinear elliptic problems in generalized Orlicz spaces” [J. Differential Equations 253 (2) (2012) 635–666], J. Differential Equations, 2012, 253(9), 2734–2738 http://dx.doi.org/10.1016/j.jde.2012.07.009[Crossref]
  • [15] Gwiazda P., Swierczewska-Gwiazda A., Parabolic equations in anisotropic Orlicz spaces with general N-functions, In: Progr. Nonlinear Differential Equations Appl., 80, Birkhäuser, Boston, 301–311
  • [16] Krasnosel’skiĭ M.A., Rutickiĭ Ya.B., Convex Functions and Orlicz Spaces, Noordhoff, Groningen, 1961
  • [17] Landes R., Mustonen V., Pseudomonotone mappings in Sobolev-Orlicz spaces and nonlinear boundary value problems on unbounded domains, J. Math. Anal. Appl., 1982, 88(1), 25–36 http://dx.doi.org/10.1016/0022-247X(82)90173-1[Crossref]
  • [18] Musielak J., Orlicz Spaces and Modular Spaces, Lecture Notes in Math., 1034, Springer, Berlin, 1983
  • [19] Mustonen V., Tienari M., On monotone-like mappings in Orlicz-Sobolev spaces, Math. Bohem., 1999, 124(2–3), 255–271
  • [20] Novotný A., Straškraba I., Introduction to the Mathematical Theory of Compressible Flow, Oxford Lecture Ser. Math. Appl., 27, Oxford University Press, Oxford, 2004
  • [21] Rao M.M., Ren Z.D., Theory of Orlicz Spaces, Monogr. Textbooks Pure Appl. Math., 146, Marcel Dekker, New York, 1991
  • [22] Sohr H., The Navier-Stokes Equations, Birkhäuser Adv. Texts Basler Lehrbucher, Birkhäuser, Basel, 2001 http://dx.doi.org/10.1007/978-3-0348-8255-2[Crossref]
  • [23] Tienari M., A Degree Theory for a Class of Mappings of Monotone Type in Orlicz-Sobolev Spaces, Ann. Acad. Sci. Fenn. Ser. A I Math. Dissertationes, 97, Finnish Academy of Science and Letters, Helsinki, 1994
  • [24] Wróblewska A., Steady flow of non-Newtonian fluids - monotonicity methods in generalized Orlicz spaces, Nonlinear Anal., 2010, 72(11), 4136–4147 http://dx.doi.org/10.1016/j.na.2010.01.045[Crossref][WoS]
  • [25] Wróblewska A., Existence results for unsteady flows of nonhomogeneous non-Newtonian incompressible fluids - monotonicity methods in generalized Orlicz spaces, preprint available at: http://mmns.mimuw.edu.pl/preprints/2011-015.pdf
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Bibliografia
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bwmeta1.element.doi-10_2478_s11533-012-0126-3
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