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The p(.)-obstacle problem for quasilinear elliptic equations in W 1,p(.) (Ω)
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DOI: 10.13140/RG.2.2.24682.62402
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Congr`es International du LEM2I, ENIM-Rabat-Maroc, 12-15 F´evrier 2013
The p(.)-obstacle problem for quasilinear elliptic equations in W1,p(.)(Ω)
A. Qabil ,A. Baalal(1).
1LMACS,Laboratoire de Mod´elisation, Analyse et Controle des Systemes, D´epartement de Math´ematiques et d’Informatique, Facult´e des Sciences Ain Chock, Km 8 Route El Jadida B.P. 5366 Maarif, Casablanca, MAROC
.
∗Corresponding author : qabil79@gmail.com
Abstract: We discuss the existence and uniqueness of p(.)-solutions of the quasilinear elliptic equation in Sobolev spaces with variable exponent,these solutions are obtained by the obstacle problem. We study the comparison principal in W1,p(.)(Ω) for p(.)-supersolution and to prove the existence and uniqueness of the Dirichlet problem related to the sheaf of continuous solutions.
Key word: Quasilinear elliptic equation, the p(.)-obstacle problem, variable exponent,p(.)-Supersolution, p(.)-Subsolution, p(.)-regular set, comparison principal.
Introduction
The equation we have in mind is :
−div (A(x,∇u)) +B(x, u) = 0, (1)
whereA:Rd×Rd→Rd and B:Rd×R→Rare Carath´eodory functions.
An axiomatic potential theory associated with the equation div (A(x,∇u)) = 0 was introduced and discussed in [7]and recently generalized by Hasto and Harjulehto in [?]. These axiomatic setting illustrated by the study of the p(.)-Laplacian problem
∆p(x)u= div(|∇u|p(x)−2∇u). (2)
obtained byA(x,∇u) =|∇u|p(x)−2∇u for all x∈Rd
This paper is divided into four sections: In the second section we introduce the basic notation and after recalling some basic facts about variable exponent spacesin section 3, we move on to elementary properties of (weak) p(.)-solutions and p(.)-supersolutions (resp. p(.)-subsolutions) of equation (1).These results in section 4 follow from the same proofs in fixed exponent case, wa also introduce the p(.)- obstacle problemapparently for the first time in the equation (1).
Inthe section 5 our objectify is to generalized the Comparison Principle in the variable exponent Sobolev spacesW1,p(.)(Ω), for p(.)-supersolutions and p(.)-subsolutions, to prove existence and uniqueness of Dirichlet problem related to the SheafH of continuous solutions of
−div (A(x,∇u)) +B(x, u) = 0
By regularity theory [9], any bounded solution of (1) can be redefined in a set of measure zero so that becomes continuous.
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Congr`es International du LEM2I, ENIM-Rabat-Maroc, 12-15 F´evrier 2013
R´ef´erences
[1] A.Baalal A.Boukricha : Potentail Theory For Quasilinear Elliptic Equations, Electronic Journal of Diferential Equations, Vol. 2001(2001), No. 31, pp. 1-20.
[2] A. Boukricha : Harnack inequality for nonlinear harmonic spaces, Math. Ann.317(2000) 3, 567–583.
[3] D. Zhao, W. J. Qiang and X. L. Fan : On generalized Orlicz spaces Lp(x)(Ω), J. Gansu Sci. 9(2) 1997 1-7.
[4] D. Kinderlehrer and G. Stampacchia :An introduction to variational inequalities and their applica- tions, Academic Press, New York, 1980.
[5] Fumi-Yuki Maeda,Takayori Ono : Properties of harmonic boundary in nonlinear potential theory, Hiroshima Math. J. 30 (2000), 513,523.
[6] J.F.Rodrigues, M.Sanchon and J.M.Urbano : The obstacle problem for nonlinear elliptic equations with variable growth and L1-data arXiv:0802.0378v1 [math.AP] 4 Feb 2008.
[7] J. Heinonen, T. Kilpl¨ainen, and O. Martio : Nonlinear potential theory of degenerate elliptic equations , Clarendon Press, Oxford New York Tokyo, 1993.
[8] J.Chabrowski,Y.Fu : Existence of solutions for p(x)-Laplacian problems on a bounded do- main,J.Math.Anal.Appl.330 (2005).
[9] J. Mal´y and W. P. Ziemmer : Fine regularity of solutions of partial differential equations, Mathe- matical Surveys and monographs, no. 51,American Mathematical Society, 1997.
[10] L.Diening ,P.Harjulehto, P.Hasto, M.Ruzicka : Lebesgue and Sobolev Spaces with Variable Exponents, Academic Press, New York, 2011.
[11] M. Anchon J. M. Urbano : Entropy Solutions for the p(x)-Laplace Equation, Trans. Amer. Math.
Soc, (2000) pp 1-23.
[12] M.B.Benboubker, E.Azroul, A.Barbara : Quasilinear Elliptic Problems With Nonstandard Groweth, Electronic Journal of Diferential Equations, Vol. 2011(2011), No. 62, pp. 1-16.
[13] O. Kov´a˘(c)ik and J. R´akosnik : On spacesLp(x) and W1,p(x), Czechoslovak Math, J. 41(116)(1991), 592-618.
[14] Stanislav Antontsev Sergey Shmarev : Elliptic Equations with Anisotropic Nonlinearity and Non- standard Growth Conditions, Handbook Of Differential Equations Stationary Partial Differential Equations, volume 3 Edited by M. Chipot and P. Quittner c2006 Elsevier.
[15] Takayori Ono : On solutions of quasi-linear partial differential equations
−div (A(x,∇u)) +B(x, u) = 0 RIMS Kokyuroku 1016 (1997), 146-165.
[16] X. L. Fan Q. H. Zhang : Existence for p(x)-Laplacien Dirichlet problem, Non linear Analysis 52 (2003) pp 1843-1852.
[17] Xianling Fan : On the sub-supersolution method for p(x)-Laplacian equations, J.
Math.Anal.Appl.330(2007).
[18] X. L. Fan and D. Zhao : On the generalised Orlicz-Sobolev Space Wk,p(x)(Ω), J. Gansu Educ.
College12(1)(1998) 1-6.
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