Skip to main content

Some obstacle problems in Musielak spaces

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 \(L^1.\)

This is a preview of subscription content, access via your institution.

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 

Download references

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to R. Elarabi.

Ethics declarations

Conflicts of interest

The authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

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

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

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

Keywords

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