Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions
Idrissa Ibrango ; Stanislas Ouaro
Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 35 (2015), p. 123-150 / Harvested from The Polish Digital Mathematics Library

The goal of this paper is to study nonlinear anisotropic problems with Fourier boundary conditions. We first prove, by using the technic of monotone operators in Banach spaces, the existence of weak solutions, and by approximation methods, we prove a result of existence and uniqueness of entropy solution.

Publié le : 2015-01-01
EUDML-ID : urn:eudml:doc:276704
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     author = {Idrissa Ibrango and Stanislas Ouaro},
     title = {Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions},
     journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},
     volume = {35},
     year = {2015},
     pages = {123-150},
     language = {en},
     url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1175}
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Idrissa Ibrango; Stanislas Ouaro. Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions. Discussiones Mathematicae, Differential Inclusions, Control and Optimization, Tome 35 (2015) pp. 123-150. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_7151_dmdico_1175/

[000] [1] S.N. Antontsev and J.F. Rodrigues, On stationary thermorheological viscous flows, Annal. del Univ. de Ferrara 52 (2006), 19-36. doi: 10.1007/s11565-006-0002-9 | Zbl 1117.76004

[001] [2] B.K. Bonzi and S. Ouaro, Entropy solution for a doubly nonlinear elliptic problem with variable exponent, J. Math. Anal. Appl. 370 (2) (2010), 392-405. doi: 10.1016/j.jmaa.2010.05.022 | Zbl 1200.35109

[002] [3] B.K. Bonzi, S. Ouaro and F.D.Y. Zongo, Nonlinear elliptic anisotropic problem with Fourier boundary condition, Int. J. Evol. Equ. 8 (4) (2013), 305-328. | Zbl 1303.35014

[003] [4] B.K. Bonzi, S. Ouaro and F.D.Y. Zongo, Entropy solutions to nonlinear elliptic anisotropic problem with Robin boundary condition, Matematiche 68 (2013), 53-76. | Zbl 1310.35128

[004] [5] B.K. Bonzi, S. Ouaro and F.D.Y. Zongo, Entropy solutions for nonlinear elliptic anisotropic homogeneous Neumann problem, Int. J. Differ. Equ. Article 476781 (2013), pp. 14. doi: 10.1155/2013/476781 | Zbl 1271.35013

[005] [6] M. Boureanu and V. D. Radulescu, Anisotropic Neumann problems in Sobolev spaces with variable exponent, Nonlin. Anal. 75 (2012), 4471-4482. doi: 10.1016/j.na.2011.09.033 | Zbl 1262.35090

[006] [7] Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM. J. Appl. Math. 66 (2006), 1383-1406. doi: 10.1137/050624522 | Zbl 1102.49010

[007] [8] X. Fan, Anisotropic variable exponent Sobolev spaces and p(·)-Laplacian equations, Complex Var. Elliptic Equ. 55 (2010), 1-20. doi: 10.1080/17476930902999082

[008] [9] X. Fan and D. Zhao, On the spaces Lp(x)(Ω) and W1,p(x)(Ω), J. Math. Appl. 263 (2001), 424-446.

[009] [10] B. Koné, S. Ouaro and S. Traoré, Weak solutions for anisotropic nonlinear elliptic equations with variable exponents, Electron. J. Diff. Equ. 144 (2009), 1-11. | Zbl 1182.35092

[010] [11] O. Kovacik and J. Rakosnik, On spaces Lp(x) and W1,p(x), Czech. Math. J. 41 (1991), 592-618.

[011] [12] M. Mihailescu, P. Pucci and V. Radulescu, Eigenvalue problems for anisotropic quasilinear elliptic equations with variable exponent, J. Math. Anal. Appl. 340 (2008), 687-698. doi: 10.1016/j.jmaa.2007.09.015 | Zbl 1135.35058

[012] [13] M. Mihailescu and V. Radulescu, A multiplicity result for a nonlinear degenerate problem arising in the theory of electrorheological fluids, Proc. R. Soc. A 462 (2006), 2625-2641. doi: 10.1098/rspa.2005.1633 | Zbl 1149.76692

[013] [14] I. Nyanquini and S. Ouaro, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afr. Mat. 23 (2012), 205-228. doi: 10.1007/s13370-011-0030-1 | Zbl 1292.35105

[014] [15] S. Ouaro, Well-posedness results for anisotropic nonlinear elliptic equations with variable exponent and L¹-data, Cubo J. 12 (2010), 133-148. doi: 10.4067/S0719-06462010000100012 | Zbl 1218.35092

[015] [16] R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical surveys and monographs, vol. 49, American Mathematical Society. | Zbl 0870.35004

[016] [17] M. Troisi, Theoremi di inclusione per spazi di Sobolev non isotropi, Recherche. Mat. 18 (1969), 3-24. | Zbl 0182.16802