ENTROPY SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS NEUMANN PROBLEMS INVOLVING THE GENERALIZED <i>P</i>(<i>X</i>)-LAPLACE OPERATOR

E. Azroul; M. B. Benboubker; S. Ouaro

doi:10.11948/2013009

2013 Volume 3 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2013 > 3(2): 105-121

Next Article Previous Article

E. Azroul, M. B. Benboubker, S. Ouaro. ENTROPY SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS NEUMANN PROBLEMS INVOLVING THE GENERALIZED P(X)-LAPLACE OPERATOR[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 105-121. doi: 10.11948/2013009

Citation:

E. Azroul, M. B. Benboubker, S. Ouaro. ENTROPY SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS NEUMANN PROBLEMS INVOLVING THE GENERALIZED P(X)-LAPLACE OPERATOR[J]. Journal of Applied Analysis & Computation, 2013, 3(2): 105-121. doi: 10.11948/2013009

ENTROPY SOLUTIONS FOR NONLINEAR NONHOMOGENEOUS NEUMANN PROBLEMS INVOLVING THE GENERALIZED P(X)-LAPLACE OPERATOR

1 Laboratoire d'Analyse Mathématique et Applications(LAMA), Département de Mathématiques, Faculté des Sciences Dhar El Mahraz, B. P 1796 Atlas Fès, Morocco;
2 LAME, UFR SEA, Université de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso

Abstract

Abstract

In this work we investigate a class of nonlinear p(x) Laplace problems with Neumann nonhomogeneous boundary conditions and L¹ data. The techniques of entropy solutions for elliptic equations are used to prove the existence of a solution.
- Generalized Sobolev spaces /
- Neumann boundary conditions /
- Entropy solution /
- noncoercive operator
MSC: 35J20;35J25;35D30;35B38;35J60

FullText(HTML)

References

[1]	E. Azroul and M. B. Benboubker & M. Rhoudaf, Entropy solution for some p(x)-Quasilinear problems with right-hand side measure, Afr. diaspora J. Math., 13(2012), 23-44. Google Scholar
[2]	E. Azroul, H. Redwane and C. Yazough, Strongly nonlinear nonhomogeneous elliptic unilateral problems with L¹ data and no sign conditions, Electron. J. Diff. Equ.,79(2012), 1-20. Google Scholar
[3]	F. Andreu, N. Igbida, J. M. Mazón and J. Toledo, L¹ existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions, Ann. Ins. H. Poincaré Anal. Non Linéaire, 20(2007), 61-89. Google Scholar
[4]	F. Andreu, J. M. Mazón, S. Segura De Léon and J. Toledo, Quasi-linear elliptic and parabolic equations in L¹ with nonlinear boundary conditions, Adv. Math. Sci. Appl., 7(1997), 183-213. Google Scholar
[5]	S. Antontsev and J. F. Rodrigues, On stationary thermorheological viscous flows. Ann. Univ. Ferrara Sez. VⅡ Sci. Mat., 52(2006), 19-36. Google Scholar
[6]	M. B. Benboubker, E. Azroul and A. Barbara, Quasilinear elliptic problems with nonstandard growth, Electron. J. Diff. Equ., 62(2011), 1-16. Google Scholar
[7]	M. Bendahmane and P. Wittbold, Renormalized solutions for nonlinear elliptic equations with variable exponents and L¹-data, Nonlinear Anal., 70(2009), 567-583. Google Scholar
[8]	P. Bénilan, L. Boccardo, T. Gallouët, R. Gariepy, M. Pierre and J.L. Vazquez, An L¹ theory of existence and uniqueness of nonlinear elliptic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci., 22(1995), 241-273. Google Scholar
[9]	Y. Chen, S. Levine and M. Rao, Variable exponent, linear growth functionals in image restoration, SIAM J. Appl. Math., 66(2006), 1383-1406. Google Scholar
[10]	L. Diening, P. Harjulehto, P. Hästö and M. Ružička, Lebesgue and Sobolev spaces with variable exponents, Lecture Notes in Mathematics, Springer-Verlag, Heidelberg, 2017(2011). Google Scholar
[11]	X.L. Fan and D. Zhao, On the generalised Orlicz-Sobolev Space W^k.p(x)(Ω), J. Gansu Educ. College, 12(1998), 1-6. Google Scholar
[12]	P. Halmos, Measure Theory, D. Van Nostrand, New York, 1950. Google Scholar
[13]	S. Ouaro and S. Soma, Weak and entropy solutions to nonlinear Neumann boundary problems with variable exponent, Complex Var. Elliptic Equ., 56(2011), 829-851. Google Scholar
[14]	I. Nyanquini and S. Ouar, Entropy solution for nonlinear elliptic problem involving variable exponent and Fourier type boundary condition, Afr. Mat., 23(2012), 205-228. Google Scholar
[15]	S. Ouaro and A. Tchousso, Well-posedness result for a nonlinear elliptic problem involving variable exponent and Robin type boundary condition, Afr. Diaspora J. Math., 11(2011), 36-64. Google Scholar
[16]	A. Prignet, Conditions aux limites non homogènes pour des problèmes elliptiques avec second membre mesure, Ann. Fac. Sc, tome 6, 2(1997), 297-318. Google Scholar
[17]	M. Sanchon and J. M. Urbano, Entropy solutions for the p(x)-Laplace equation, Trans. Amer. Math. Soc., 361(2009), 6387-6405. Google Scholar
[18]	M. Ružička, Electrorheological fluids:modeling and mathematical theory, lecture Notes in Mathematics, Springer-Verlag, Berlin, 1748(2000). Google Scholar
[19]	L. Wang, Y. Fan and W. Ge, Existence and multiplicity of solutions for a Neumann problem involving the p(x)-Laplace operator. Nonlinear Anal., 71(2009), 4259-4270. Google Scholar
[20]	J. Yao, Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal., 68(2008), 1271-1283. Google Scholar
[21]	V. Zhikov, Averaging of functionals of the calculus of variations and elasticity theory, Math. USSR Izvestiya, 29(1987), 33-66. Google Scholar
[22]	V. Zhikov, Meyer-type estimates for solving the nonlinear Stokes system. Differential Equations., 33(1997), 108-115. Google Scholar
[23]	D. Zhao, W.J. Qiang and X.L. Fan, On generalized Orlicz spaces Lp(x)(Ω), J. Gansu Sci., 9(1997), 1-7. Google Scholar