Glasnik Matematicki, Vol. 54, No. 1 (2019), 211-232.

MULTIVALUED ANISOTROPIC PROBLEM WITH FOURIER BOUNDARY CONDITION INVOLVING DIFFUSE RADON MEASURE DATA AND VARIABLE EXPONENTS

Ibrahime Konaté and Stanislas Ouaro

Laboratoire de Mathématiques et Informatique, UFR. Sciences Exactes et Appliquées, Université Joseph Ki Zerbo, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso
e-mail: ibrakonat@yahoo.fr
e-mail: ouaro@yahoo.fr

Abstract.   We study a nonlinear anisotropic elliptic problem under Fourier type boundary condition governed by a general anisotropic operator with variable exponents and diffuse Radon measure data which does not charge the sets of zero p(·)-capacity. We prove an existence and uniqueness result of entropy or renormalized solution.

2010 Mathematics Subject Classification.   35J60, 35J65, 35J20, 35J25

Key words and phrases.   Fourier boundary, generalized Lebesgue-Sobolev spaces, anisotropic Sobolev spaces, weak solution, entropy solution, maximal monotone graph, bounded Radon diffuse measure, Marcinkiewicz spaces

Full text (PDF) (free access)

https://doi.org/10.3336/gm.54.1.09

References:

  1. L. Alvarez, P.-L. Lions, and J.-M. Morel, Image selective smoothing and edge detection by nonlinear diffusion. II, SIAM J. Numer. Anal. 29 (1992), 845-866.
    MathSciNet     CrossRef

  2. M. Bendahmane, M. Langlais and M. Saad, On some anisotropic reaction- diffusion systems with L1-data modeling the propagation of an epidemic disease, Nonlinear Anal. 54 (2003), 617-636.
    MathSciNet     CrossRef

  3. L. Boccardo and T. Gallouët, Nonlinear elliptic and parabolic equations involving measure data, J. Funct. Anal. 87 (1989), 149-169.
    MathSciNet     CrossRef

  4. L. Boccardo and T. Gallouët, Nonlinear elliptic equations with right hand side measures, Comm. Partial Differential Equations 17 (1992), 641-655.
    MathSciNet     CrossRef

  5. L. Boccardo, T. Gallouët and L. Orsina, Existence and uniqueness of entropy solution for nonlinear elliptic equations with measure data, Ann. Inst. Henri Poincaré Anal. Non Linéaire 13 (1996), 539-551.
    MathSciNet     CrossRef

  6. B. Koné, S. Ouaro and F.D.Y. Zongo, Nonlinear elliptic anisotropic problem with Fourier boundary condition, Int. J. Evol. Equ. 8 (2013), 305-328.
    MathSciNet    

  7. M.-M. Boureanu and V. D. Radulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlinear Anal. 75 (2012), 4471-4482.
    MathSciNet     CrossRef

  8. H. Brézis, Analyse fonctionnelle. Théorie et applications, Masson, Paris, 1983.
    MathSciNet    

  9. H. Brézis, Opérateurs maximaux monotones et semi-groupes de contractions dans les espaces de Hilbert, North Holland, Amsterdam, 1973.
    MathSciNet    

  10. Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math. 66 (2006), 1383-1406.
    MathSciNet     CrossRef

  11. G. Dal Maso, F. Murat, L. Orsina and A. Prignet, Renormalized solutions of elliptic equations with general measure data, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 28 (1999), 741-808.
    MathSciNet     CrossRef

  12. L. Diening, Theoretical and numerical results for electrorheological fluids, PhD. thesis, University of Frieburg, Germany, 2002.

  13. G. Dolzmann, N. Hungerbühler and S. Müller, Non-linear elliptic systems with measure-valued right-hand side, Math Z. 226 (1997), 545-574.
    MathSciNet     CrossRef

  14. F. Ettwein and M. Ruzicka, Existence of strong solutions for electrorheological fluids in two dimensions: steady Dirichlet problem, in: Geometric Analysis and Nonlinear Partial Differential Equations, Springer-Verlag, Berlin, 2003, 591-602.
    MathSciNet    

  15. X. Fan, Anisotropic variable exponent Sobolev spaces and p(·)-Laplacian equations, Complex Var. Elliptic Equ. 56 (2011), 623-642.
    MathSciNet     CrossRef

  16. X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and Wm,p(x)(Ω), J. Math. Anal. Appl. 263 (2001), 424-446.
    MathSciNet     CrossRef

  17. I. Ibrango and S. Ouaro, Entropy solutions for anisotropic nonlinear problems with homogeneous Neumann boundary condition, J. Appl. Anal. Comput. 6 (2016), 271-292.
    MathSciNet    

  18. I. Ibrango and S. Ouaro, Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions, Discuss. Math. Differ. Incl. Control Optim. 35 (2015), 123-150.
    MathSciNet     CrossRef

  19. N. Igbida, S. Ouaro and S. Soma, Elliptic problem involving diffuse measure data, J. Differential Equations 253 (2012), 3159-3183.
    MathSciNet     CrossRef

  20. I. Konaté and S. Ouaro, Good Radon measure for anisotropic problems with variable exponent, Electron. J. Differential Equations 2016 (2016), No. 221, 19 pp.
    MathSciNet    

  21. I. Konaté and S. Ouaro, Nonlinear multivalued problems with variable exponent and diffuse measure data in anisotropic space, Gulf J. Math. 6 (2018), 13-30.
    MathSciNet    

  22. B. Koné, S. Ouaro and S. Soma, Weak solutions for anisotropic nonlinear elliptic problem with variable exponent and measure data, Int. J. Evol. Equ. 5 (2010), 327-350.
    MathSciNet    

  23. O. Kovacik and J. Rakosnik, On spaces Lp(x) and Wk,p(x), Czechoslovak Math. J. 41(116) (1991), 592-618.
    MathSciNet    

  24. M. Mihailescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasi-linear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687-698.
    MathSciNet     CrossRef

  25. I. Nyanquini, S. Ouaro and S. Soma, Entropy solution to nonlinear multivalued elliptic problem with variable exponents and measure data, An. Univ. Craiova Ser. Mat. Inform. 40 (2013), 174-198.
    MathSciNet    

  26. I. Nyanquini, S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afr. Mat. 23 (2012), 205-228.
    MathSciNet     CrossRef

  27. S. Ouaro, A. Ouédraogo and S. Soma, Multivalued homogeneous Neumann problem involving diffuse measure data and variable exponent, Nonlinear Dyn. Syst. Theory 16 (2016), 102-114.
    MathSciNet    

  28. C. Pfeiffer, C. Mavroidis, Y. Bar-Cohen, B. Dolgin, Electrorheological fluid based force feedback device, in Proc. SPIE 3840, Telemanipulator and Telepresence Technologies VI, 1999, 88-99.

  29. K. Rajagopal and M. Ružička, Mathematical modelling of electro-rheological fluids, Cont. Mech. Therm. 13 (2001), 59-78.

  30. M. Ružička, Electrorheological fluids: modelling and mathematical theory, Springer, Berlin, 2000.
    MathSciNet     CrossRef

  31. M. Troisi, Teoremi di inclusione per spazi di Sobolev non isotropi, Recherche. Mat 18 (1969), 3-24.
    MathSciNet    
Glasnik Matematicki Home Page