SciELO - Scientific Electronic Library Online

 
vol.14 número2The e -Optimality Conditions for Multiple Objective Fractional Programming Problems for Generalized - Invexity of Higher OrderA New proof of the CR Identity and related Topics índice de autoresíndice de materiabúsqueda de artículos
Home Pagelista alfabética de revistas  

Cubo (Temuco)

versión On-line ISSN 0719-0646

Cubo vol.14 no.2 Temuco  2012

http://dx.doi.org/10.4067/S0719-06462012000200002 

CUBO A Mathematical Journal Vol.14, No02, (15-41). June 2012

Texto completo disponíble en formato PDF

Weak and entropy solutions for a class of nonlinear inhomogeneous Neumann boundary value problem with variable exponent

 

Stanislas Ouaro

Laboratoire d'Analyse Mathématique des Equations (LAME), UFR. Sciences Exactes et Appliquées, Universite de Ouagadougou, 03 BP 7021 Ouaga 03, Ouagadougou, Burkina Faso email: souaro@univ-ouaga.bf, ouaro@yahoo.fr

 


ABSTRACT

We study the existence and uniqueness of weak and entropy solutions for the nonlinear inhomogeneous Neumann boundary value problem involving the Laplace of the form - where is a smooth bounded open domain in . We prove the existence and uniqueness of a weak solution for data , the existence and uniqueness of an entropy solution for data f and ö independent of u and the existence of weak solutions for f dependent on u and .

Keywords and Phrases: Generalized Lebesgue and Sobolev spaces; Weak solution; Entropy solution; Laplace operator.


RESUMEN

Estudiamos la existencia y unicidad de soluciones y entropía debil para el problema no lineal inhomogeneos de Neumann con valores de frontera que involucra el Laplace de la forma en Omega, sobre , donde Omega es en un dominio abierto suave y acotado en para . Probamos la existencia y unicidad de una solución débil para , la existencia y unicidad de una solución de entropía paradata f y ö independiente de u y la existencia de soluciones debiles para f dependiente sobre .

 

2010 AMS Mathematics Subject Classification: 35J20, 35J25, 35D30, 35B38, 35J60.


References

[1] A. Alvino, L. Boccardo, V. Ferone, L. Orsina & G. Trombetti; Existence results for non-linear elliptic equations with degenerate coercivity, Ann. Mat. Pura Appl. 182 (2003), 53-79.         [ Links ]

[2] F. Andreu, N. Igbida, J.M. Mazón & J. Toledo; existence and uniqueness results for quasi-linear elliptic equations with nonlinear boundary conditions, Ann. I.H. Poincare AN., 24 (2007), 61-89.         [ Links ]

[3] F. Andreu, J. M. Mazón, S. Segura De Leon & J. Toledo; Quasi-linear elliptic and parabolic equations in with nonlinear boundary conditions, Adv. Math. Sci. Appl. 7 , no.1 (1997),183-213.         [ Links ]

[4] S.N. Antontsev & J.F. Rodrigues; On stationary thermo-rheological viscous flows. Annal del Univ de Ferrara 52 (2006), 19-36.         [ Links ]

[5] P. Bénilan, L. Boccardo, T. Gallouét, R. Gariepy, M. Pierre, J.L. Vazquez, An theory of existence and uniqueness of nonlinear elliptic equations, Ann Scuola Norm. Sup. Pisa, 22 no.2 (1995), 240-273.         [ Links ]

[6] H. Brezis: Analyse Fonctionnelle: Theorie et Applications, Paris, Masson (1983).         [ Links ]

[7] Y. Chen, S. Levine & M. Rao; Variable exponent, linear growth functionals in image restoration. SIAM. J.Appl. Math., 66 (2006), 1383-1406.         [ Links ]

[8] L. Diening; Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces and , Math. Nachr. 268 (2004), 31-43.         [ Links ]

[9] L. Diening; Theoretical and numerical results for electrorheological fluids, Ph.D. thesis, University of Freiburg, Germany, 2002.         [ Links ]

[10] D. E. Edmunds & J. Rakosnik; Density of smooth functions in , Proc. R. Soc. A 437 (1992), 229-236.         [ Links ]

[11] D. E. Edmunds & J. Rakosnik; Sobolev embeddings with variable exponent, Sudia Math. 143 (2000), 267-293.         [ Links ]

[12] D. E. Edmunds & J. Rakosnik; Sobolev embeddings with variable exponent, II, Math. Nachr. 246-247 (2002), 53-67.         [ Links ]

[13] X. Fan & S. Deng; Multiplicity of Positive Solutions for a Class of Inhomogeneous Neumann Problems Involving the Laplacian, Nonlinear differ.equ.appl. 16 (2009), 255-271.         [ Links ]

[14] X. Fan & Q. Zhang; Existence of solutions for Laplacian Dirichlet problem, Nonlinear Anal. 52 (2003), 1843-1852.         [ Links ]

[15] X. Fan & D. Zhao; On the spaces and , J. Math. Anal. Appl. 263 (2001), 424-446.         [ Links ]

[16] P. Halmos: Measure Theory, D. Van Nostrand, New York (1950).         [ Links ]

[17] O. Kovacik & J. Rakosnik; On spaces and , Czech. Math. J. 41 (1991), 592-618.         [ Links ]

[18] M. Krasnosel'skii; Topological methods in the theory of nonlinear integral equations, Perga-mon Press, New York, 1964.         [ Links ]

[19] M. Mihailescu & 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.         [ Links ]

[20] C. B. Jr. Morrey; Multiple Intóegrals in the Calculus of Variations, Springer-Verlag, 1966.         [ Links ]

[21] J. Musielak; Orlicz Spaces and modular spaces. Lecture Notes in Mathematics, vol. 1034 (1983), springer, Berlin.         [ Links ]

[22] H. Nakano; Modulared semi-ordered linear spaces. Maruzen Co., Ltd., Tokyo, 1950.         [ Links ]

[23] J. Necas; Les móethodes Directes en Thóeorie des Equations Elliptiques, Masson et Cie, Paris, 1967.         [ Links ]

[24] S. Ouaro & S. Soma; Weak and entropy solutions to nonlinear Neumann boundary value problem with variable exponent. Complex var. Elliptic Equ, 56, No. 7-9, 829-851 (2011).         [ Links ]

[25] S. Ouaro & S. Traoré; Existence and uniqueness of entropy solutions to nonlinear elliptic problems with variable growth. Int. J. Evol. Equ. 4 (2010), no. 4, 451-471.         [ Links ]

[26] S. Ouaro & S. Traoré; Weak and entropy solutions to nonlinear elliptic problems with variable exponent. J. Convex Anal. 16 , No. 2 (2009), 523-541.         [ Links ]

[27] K.R. Rajagopal & M. Ruzicka; Mathematical Modeling of Electrorheological Materials, Contin. Mech. Thermodyn. 13 (2001), 59-78.         [ Links ]

[28] B. Ricceri; On three critical points theorem, Arch. Math. (Basel)75 (2000), 220-226.         [ Links ]

[29] M. Ruzicka; Electrorheological fluids: modelling and mathematical theory, Lecture Notes in Mathematics 1748, Springer-Verlag, Berlin, 2002.         [ Links ]

[30] M. Sanchon & J. M. Urbano; Entropy solutions for the Laplace Equation, Trans. Amer. Math. Soc. 361 (2009), no. 12, 6387-6405.         [ Links ]

[31] I. Sharapudinov; On the topology of the space , Math. Zametki 26 (1978), 613-632.         [ Links ]

[32] I. Tsenov; Generalization of the problem of best approximation of a function in the space , Uch. Zap. Dagestan Gos. Univ. 7 (1961), 25-37.         [ Links ]

[33] L. Wang, Y. Fan & W. Ge; Existence and multiplicity of solutions for a Neumann problem involving the Laplace operator. Nonlinear Anal. 71 (2009), 4259-4270.         [ Links ]

[34] J. Yao; Solutions for Neumann boundary value problems involving p(x)-Laplace operators, Nonlinear Anal. 68 (2008), 1271-1283.         [ Links ]

[35] V. Zhikov; On passing to the limit in nonlinear variational problem. Math. Sb. 183 (1992),47-84.         [ Links ]


Received: January 2011. Revised: September 2011.