Advertisement

Some results on the p(u)-Laplacian problem

  • M. Chipot
  • H. B. de OliveiraEmail author
Article
  • 50 Downloads

Abstract

The p-Laplacian problem with the exponent of nonlinearity p depending on the solution u itself is considered in this work. Both situations when p(u) is a local quantity or when p(u) is nonlocal are studied here. For the associated boundary-value local problem, we prove the existence of weak solutions by using a singular perturbation technique. We also prove the existence of weak solutions to the nonlocal version of the associated boundary-value problem. The issue of uniqueness for these problems is addressed in this work as well, in particular by working out the uniqueness for a one dimensional local problem and by showing that the uniqueness is easily lost in the nonlocal problem.

Mathematics Subject Classification

35J60 35J05 35D30 

Notes

Acknowledgements

We are very greatful to the referees for their constructive remarks. This work was performed when the first author was visiting the USTC in Hefei and during a part time employment at the S. M. Nikolskii Mathematical Institute of RUDN University, 6 Miklukho-Maklay St, Moscow, 117198, supported by the Ministry of Education and Science of the Russian Federation. He is greatful to these institutions for their support. Main part of this work was carried out also during the visit of the second author to the University of Zurich during the first quarter of 2018. Besides the Grant SFRH/BSAB/135242/2017 of the Portuguese Foundation for Science and Technology (FCT), Portugal, which made this visit possible, the second author also wishes to thank to Prof. Michel Chipot who kindly welcomed him at the University of Zurich.

References

  1. 1.
    Andreianov, B., Bendahmane, M., Ouaro, S.: Structural stability for variable exponent elliptic problems. II. The \(p(u)\)-Laplacian and coupled problems. Nonlinear Anal. 72(12), 4649–4660 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Antontsev, S., Shmarev, S.: Evolution PDEs with Nonstandard Growth Conditions. Existence, Uniqueness, Localization, Blow-up. Atlantis Press, Paris (2015)CrossRefzbMATHGoogle Scholar
  3. 3.
    Blomgren, P., Chan, T., Mulet, P., Wong, C.: Total variation image restoration: Numerical methods and extensions. In: Proceedings of the IEEE International Conference on Image Processing, vol. 3, 384–387. IEEE Computer Society Press, Piscataway (1997)Google Scholar
  4. 4.
    Bollt, E., Chartrand, R., Esedoglu, S., Schultz, P., Vixie, K.: Graduated, adaptive image denoising: local compromise between total-variation and isotropic diffusion. Adv. Comput. Math. 31, 61–85 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Chipot, M.: Elements of Nonlinear Analysis. Birkhäuser, Basel (2000)CrossRefzbMATHGoogle Scholar
  6. 6.
    Chipot, M.: Elliptic Equations: An Introductory Course. Birkhäuser, Basel (2009)CrossRefzbMATHGoogle Scholar
  7. 7.
    Chipot, M., Chang, N.-H.: On some mixed boundary value problems with nonlocal diffusion. Adv. Math. Sci. Appl. 14(1), 1–24 (2004)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Chipot, M., Lovat, B.: Existence and uniqueness results for a class of nonlocal elliptic and parabolic problems. Dyn. Contin. Discr. Impuls. Syst. Ser. A Math. Anal. 8(1), 35–51 (2001)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chipot, M., Rodrigues, J.F.: On a class of nonlocal nonlinear elliptic problems. RAIRO: Modél. Math. Anal. Numér 26(3), 447–467 (1992)MathSciNetzbMATHGoogle Scholar
  10. 10.
    Cruz-Uribe, D., Fiorenza, A.: Variable Lebesgue Spaces. Foundations and Harmonic Analysis. Birkhäuser/Springer, Heidelberg (2013)CrossRefzbMATHGoogle Scholar
  11. 11.
    Diening, L., Harjulehto, P., Hästo, P., Ru̇žička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Springer, Heidelberg (2011)CrossRefzbMATHGoogle Scholar
  12. 12.
    Glowinski, R., Marrocco, R.: Sur l’approximation, par éléments finis d’ordre un, et la résolution, par pénalisation-dualité, d’une classe de problèmes de Dirichlet non linéaires. RAIRO Ana. Numer. 9(R–2), 41–76 (1975)zbMATHGoogle Scholar
  13. 13.
    Chan, T., Esedoglu, S., Park, F., Yip, A.: Total Variation Image Restoration: Overview and Recent Developments. In: Paragios, N., Chen, Y., Faugeras, O. (eds.) Handbook of Mathematical Models in Computer Vision, 17–32. Springer, New York (2006)Google Scholar
  14. 14.
    Türola, J.: Image denoising using directional adaptive variable exponents model. J. Math. Imaging. Vis. 57, 56–74 (2017)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Zhikov, V.V.: Averaging of functionals of the calculus of variations and elasticity theory (Russian). Izv. Akad. Nauk SSSR Ser. Mat. 504(4), 675–710 (1986). English translation in Mah. USSR Izvestiya 29 (1987) 33–66Google Scholar
  16. 16.
    Zhikov, V.V.: On the technique for passing to the limit in nonlinear elliptic equations. Funct. Anal. Appl. 43(2), 96–112 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Zhikov, V.V.: On variational problems and nonlinear elliptic equations with nonstandard growth conditions. J. Math. Sci. (N.Y.) 173(5), 463–570 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.IMathUniversität ZürichZurichSwitzerland
  2. 2.FCTUniversidade do AlgarveFaroPortugal
  3. 3.CMAFCIOUniversidade de LisboaLisbonPortugal

Personalised recommendations