全部 标题 作者
关键词 摘要

OALib Journal期刊
ISSN: 2333-9721
费用:99美元

查看量下载量

相关文章

更多...

An Approximation of Hedberg’s Type in Sobolev Spaces with Variable Exponent and Application

DOI: 10.1155/2014/549051

Full-Text   Cite this paper   Add to My Lib

Abstract:

The aim of this paper is to extend the usual framework of PDE with to include a large class of cases with , whose coefficient satisfies conditions (including growth conditions) which guarantee the solvability of the problem . This new framework is conceptually more involved than the classical one includes many more fundamental examples. Thus our main result can be applied to various types of PDEs such as reaction-diffusion equations, Burgers type equation, Navier-Stokes equation, and p-Laplace equation. 1. Introduction This paper is motivated by the study of the unilateral problem associated with the following equation: We show the existence of variational solutions of this elliptic boundary value problem for strongly elliptic systems of order on a domain in in generalized divergence form as follows: The function satisfies a sign condition but has otherwise completely unrestricted growth with respect to . Equations of type (1) were first considered by Browder [1] as an application to the theory of not everywhere defined mapping of monotone type. For , that is, of second order, their solvability under fairly general and natural assumptions was proved by Hess [2]. The treatment of the case is more involved due to the lack of a simple truncation operator in higher order Sobolev spaces. Webb [3] observed that rather delicate approximation procedure introduced in nonlinear potential theory by Hedberg [4] could be used in place of truncation. This yielded the solvability of (1) for . Brezis and Browder [5] then used this approximation procedure to solve a question which they had considered earlier [6] about the action of some distribution. They also showed that their result on the action of some distributions could itself be used in place of truncation in the study the problem (1). In a more general case, Boccardo et al. studied inequations associated with (1), see [7]. The functional setting in all the results mentioned above is that of the usual Sobolev spaces , and the functions in (2) are supposed to satisfy polynomial growth conditions with respect to and its derivatives. Benkirane and Gossez established this result in the Orlicz-Sobolev spaces , see [8–10]. It is our purpose in this paper to study these problems in this setting of Sobolev spaces with variable exponent of the harder higher order case . We consider problem (1) as well as Hedberg’s approximation theorem and Brezis-Browder’s question on the action of some distributions. The paper is structured as follows. After some necessary preliminaries, in Section 3, we give the proof of the

References

[1]  F. E. Browder, “Existence theory for boundary value problems for quasi linear elliptic systems with strongly nonlinear lower order terms,” in Proceedings of the Symposia in Pure Mathematics, vol. 23, pp. 269–286, The American Mathematical Society, 1971.
[2]  P. Hess, “A strongly nonlinear elliptic boundary value problem,” Journal of Mathematical Analysis and Applications, vol. 43, no. 1, pp. 241–249, 1973.
[3]  J. R. L. Webb, “Boundary value problems for strongly nonlinear elliptic equations,” Journal of the London Mathematical Society, vol. 2, no. 21, pp. 123–132, 1980.
[4]  L. I. Hedberg, “Two approximation problems in function spaces,” Arkiv f?r Matematik, vol. 16, no. 1, pp. 51–81, 1978.
[5]  H. Brezis and F. E. Browder, “Some properties of higher order Sobolev spaces,” Journal of Mathematical Analysis and Applications, vol. 245, pp. 0021–7824, 1982.
[6]  H. Brezis and F. E. Browder, “A property of sobolev spaces,” Communications in Partial Differential Equations, vol. 4, pp. 1077–1083, 1979.
[7]  L. Boccardo, D. Giachetti, and F. Murat, “A generalization of a theorem of H. Brezis and F. E. Browder and applications to some unilateral problems,” Annales de l'Institut Henri Poincaré C, vol. 7, pp. 367–384, 1990.
[8]  A. Benkirane, “Approximation de type hedberg dans les espaces et applications,” Annales de la Faculté des Sciences de Toulouse, vol. 11, pp. 67–78, 1990.
[9]  A. Benkirane and G. P. Gossez, “An approximation theorem for higer order Orlicz-Sobolev spaces,” Studia Mathematica, vol. 92, pp. 231–255, 1989.
[10]  A. Benkirane, “A theorem of H. Brezis and F. E. Browder type in Orlicz spaces and application,” Pitman Research Notes in Mathematics Series, vol. 343, pp. 10–16, 1996.
[11]  L. Diening, P. Harjulehto, P. H?st?, and M. Ru?i?ka, Lebesgue and Sobolev Spaces with Variable Exponents, vol. 2017 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2011.
[12]  P. Gurka, P. Harjulehto, and A. Nekvinda, “Bessel potential spaces with variable exponent,” Mathematical Inequalities and Applications, vol. 10, no. 3, pp. 661–676, 2007.
[13]  D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “Corrections to the maximal operator on variable Lpspaces,” Annales Academi? Scientiarum Fennic?, vol. 29, pp. 247–249, 2004.
[14]  D. Cruz-Uribe, A. Fiorenza, and C. J. Neugebauer, “The maximal function on variable Lp spaces,” Annales Academiae Scientiarum Fennicae Mathematica, vol. 28, no. 1, pp. 223–238, 2003.
[15]  L. Diening, “Maximal function on generalized lebesgue spaces ,” Mathematical Inequalities and Applications, vol. 7, no. 2, pp. 245–253, 2004.
[16]  A. Nekvinda, “Hardy-Little-Wood maximal operator on ,” Mathematical Inequalities and Applications, vol. 7, no. 2, pp. 255–266, 2004.
[17]  D. R. Adams and L. I. Hedberg, Function Spaces and Potential Theory, Springer, New York, NY, USA, 1996.
[18]  E. Stein, Singular Integrals and Differentiability Properties of Functions, Princeton University Press, Princeton, NJ, USA, 1970.
[19]  X. L. Fan and D. Zhao, “On the spaces and ,” Journal of Mathematical Analysis and Applications, vol. 236, pp. 424–446, 2001.

Full-Text

Contact Us

service@oalib.com

QQ:3279437679

WhatsApp +8615387084133