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Some obstacle problems in Musielak spaces

Ricerche di Matematica (2022)Cite this article

Abstract

In this paper we use the penalization method to prove the existence of solution for variational inequalities of Leray–Lions type, in the setting of Musielak spaces and where the Musielak function doesn’t satisfy the \(\Delta _2\)-condition. Here the right-hand side is in \(L^1.\)

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References

  1. Aharouch, L., Azroul, E., Rhoudaf, M.: Existence of solutions for unilateral problems in \(L^1\) involving lower order terms in divergence form in Orlicz spaces. J. Appl. Anal. 13, 151–181 (2007)

    MathSciNet  Article  Google Scholar 

  2. Ait Khellou, M., Benkirane, A., Douiri, S.M.: Existence of solutions for elliptic equations having natural growth terms in Musielak–Orlicz spaces. J. Math. Comput. Sci. 4(4), 665–688 (2014)

    Google Scholar 

  3. Ait Khellou, M., Benkirane, A., Douiri, S.M.: Some properties of Musielak spaces with only the log-Hölder continuity condition and application. Ann. Funct. Anal. 11, 1062–1080 (2020)

    MathSciNet  Article  Google Scholar 

  4. Benkirane, A., Elmahi, A., Meskine, D.: On the limit of some penalized problems involving increasing powers. Asymptot. Anal. 36, 303–317 (2003)

    MathSciNet  MATH  Google Scholar 

  5. Benkirane, A., Sidi El Vally, M.: Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21, 787–811 (2014)

    MathSciNet  Article  Google Scholar 

  6. Benkirane, A., Sidi El Vally, M.S.: Variational inequalities in Musielak–Orlicz–Sobolev spaces. Bull. Belg. Math. Soc. Simon Stevin 21(5), 787–811 (2014)

    MathSciNet  Article  Google Scholar 

  7. Benkirane, A., Sidi El Vally, M.S.: Some approximation properties in Musielak–Orlicz–Sobolev spaces. Thai J. Math. 10(2), 371–381 (2012)

    MathSciNet  Google Scholar 

  8. Dall’aglio, A., Orsina, L.: On the limit of some nonlinear elliptic equations involving increasing powers. Asympt. Anal. 14, 49–71 (1997)

    MathSciNet  MATH  Google Scholar 

  9. Elarabi, R., Rhoudaf, M., Sabiki, H.: Entropy solution for a nonlinear elliptic problem with lower order term in Musielak–Orlicz spaces. Ric. Mat., 1–31 (2017)

  10. Gossez, J.-P.: Some approximation properties in Orlicz–Sobolev. Studia Math. 74, 17–24 (1982)

    MathSciNet  Article  Google Scholar 

  11. Musielak, J.: Modular Spaces and Orlicz Spaces. Lecture Notes in Mathematics, vol. 1034. Springer Verlag, Berlin (1983)

    Book  Google Scholar 

  12. Rajagopal, K.R., Ru̇z̃ic̃ka, M.: Mathematical modeling of electrorheological materials. Contin. Mech. Thermodyn. 13, 59–78 (2001)

    Article  Google Scholar 

  13. Ru̇žic̆ka, M.: Electrorheological Fluids, Modeling and Mathematical Theory. Lecture Notes in Mathematics. Springer, Berlin (2000)

    Book  Google Scholar 

  14. Perona, P., Malik, J.: Scale-space and edge detection using anisotropic diffusion. IEEE Trans. Pattern Anal. Mach. Intell. 12, 629–639 (1990)

    Article  Google Scholar 

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Authors and Affiliations

  1. Faculté des Sciences de Meknès, Equipe: EDPs et Calculs Scientifiques, Université Moulay-Ismail-Meknès, Meknes, Morocco

    R. Elarabi & M. Rhoudaf

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  1. R. Elarabi
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  2. M. Rhoudaf
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Correspondence to R. Elarabi.

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Elarabi, R., Rhoudaf, M. Some obstacle problems in Musielak spaces. Ricerche mat (2022). https://doi.org/10.1007/s11587-021-00679-w

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  • DOI: https://doi.org/10.1007/s11587-021-00679-w

Keywords

  • Nonlinear elliptic problems
  • Musielak–Sobolev spaces
  • Variational inequalities
  • Bilateral problems