# Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions

Idrissa Ibrango; Stanislas Ouaro

Discussiones Mathematicae, Differential Inclusions, Control and Optimization (2015)

- Volume: 35, Issue: 2, page 123-150
- ISSN: 1509-9407

## Access Full Article

top## Abstract

top## How to cite

topIdrissa Ibrango, and Stanislas Ouaro. "Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions." Discussiones Mathematicae, Differential Inclusions, Control and Optimization 35.2 (2015): 123-150. <http://eudml.org/doc/276704>.

@article{IdrissaIbrango2015,

abstract = {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.},

author = {Idrissa Ibrango, Stanislas Ouaro},

journal = {Discussiones Mathematicae, Differential Inclusions, Control and Optimization},

keywords = {anisotropic Sobolev spaces; variable exponent; monotone operator; Fourier boundary conditions; entropy solutions},

language = {eng},

number = {2},

pages = {123-150},

title = {Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions},

url = {http://eudml.org/doc/276704},

volume = {35},

year = {2015},

}

TY - JOUR

AU - Idrissa Ibrango

AU - Stanislas Ouaro

TI - Entropy solution for doubly nonlinear elliptic anisotropic problems with Fourier boundary conditions

JO - Discussiones Mathematicae, Differential Inclusions, Control and Optimization

PY - 2015

VL - 35

IS - 2

SP - 123

EP - 150

AB - 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.

LA - eng

KW - anisotropic Sobolev spaces; variable exponent; monotone operator; Fourier boundary conditions; entropy solutions

UR - http://eudml.org/doc/276704

ER -

## References

top- [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 Zbl1117.76004
- [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 Zbl1200.35109
- [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. Zbl1303.35014
- [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. Zbl1310.35128
- [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 Zbl1271.35013
- [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 Zbl1262.35090
- [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 Zbl1102.49010
- [8] X. Fan, Anisotropic variable exponent Sobolev spaces and $\overrightarrow{p}\left(\xb7\right)$-Laplacian equations, Complex Var. Elliptic Equ. 55 (2010), 1-20. doi: 10.1080/17476930902999082
- [9] X. Fan and D. Zhao, On the spaces ${L}^{p\left(x\right)}\left(\Omega \right)$ and ${W}^{1,p\left(x\right)}\left(\Omega \right)$, J. Math. Appl. 263 (2001), 424-446.
- [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. Zbl1182.35092
- [11] O. Kovacik and J. Rakosnik, On spaces ${L}^{p\left(x\right)}$ and ${W}^{1,p\left(x\right)}$, Czech. Math. J. 41 (1991), 592-618.
- [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 Zbl1135.35058
- [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 Zbl1149.76692
- [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 Zbl1292.35105
- [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 Zbl1218.35092
- [16] R.E. Showalter, Monotone operators in Banach space and nonlinear partial differential equations, Mathematical surveys and monographs, vol. 49, American Mathematical Society. Zbl0870.35004
- [17] M. Troisi, Theoremi di inclusione per spazi di Sobolev non isotropi, Recherche. Mat. 18 (1969), 3-24. Zbl0182.16802

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.