- Research Article
- Open Access
© Fengping Yao. 2010
- Received: 29 December 2009
- Accepted: 23 March 2010
- Published: 30 May 2010
- Weak Solution
- Global Condition
- Orlicz Space
- Type Estimate
- Quasilinear Elliptic Equation
As usual, the solutions of (1.1) are taken in a weak sense. We now state the definition of weak solutions.
Orlicz spaces have been studied as a generalization of spaces since they were introduced by Orlicz  (see [10–16]). The theory of Orlicz spaces plays a crucial role in a very wide spectrum (see ). Here for the reader's convenience, we will give some definitions on the general Orlicz spaces. We denote by the function class that consists of all functions which are increasing and convex.
Moreover, we give the following lemma.
Now we are set to state the main result.
Our approach is based on the paper . Recently Acerbi and Mingione  obtained local , , gradient estimates for the degenerate parabolic -Laplacian systems which are not homogeneous if . There, they invented a new iteration-covering approach, which is completely free from harmonic analysis, in order to avoid the use of the maximal function operator.
This paper will be organized as follows. In Section 2, we give a new normalization method and the iteration-covering procedure, which are very important to obtain the main result. We finish the proof of Theorem 1.7 in Section 3.
2.1. New Normalization
Lemma 2.1 (new normalization).
Thus we complete the proof.
2.2. The Iteration-Covering Procedure
From the argument above we know that for a.e. there exists a ball constructed as above. Therefore, applying Vitali's covering lemma, we can find a family of disjoint balls with so that (2.12) and (2.13) hold truely.
Thus we obtain the desired estimate (2.14). This completes our proof.
In the following it is sufficient to consider the proof of Theorem 1.7 as an a priori estimate, therefore assuming a priori that . This assumption can be removed in a standard way via an approximation argument as the one in [12, 15, 18].
We first state the definition of the global weak solutions.
From the definition above we can easily obtain the following lemma.
and then finish the proof.
If the conclusion (3.21) is true, then the conclusion (3.20) can follow from [20, Lemma ].
This completes our proof.
Furthermore, from the new normalization in Lemma 2.1, we can easily obtain the following corollary of Lemma 3.4.
Now we are ready to prove the main result, Theorem 1.7.
Then by an elementary scaling argument, we can finish the proof of the main result.
This work is supported in part by Tianyuan Foundation (10926084) and Research Fund for the Doctoral Program of Higher Education of China (20093108120003). Moreover, the author wishes to the department of mathematics at Shanghai university which was supported by the Shanghai Leading Academic Discipline Project (J50101) and Key Disciplines of Shanghai Municipality (S30104).
- DiBenedetto E, Manfredi J: On the higher integrability of the gradient of weak solutions of certain degenerate elliptic systems. American Journal of Mathematics 1993, 115(5):1107–1134. 10.2307/2375066MathSciNetView ArticleMATHGoogle Scholar
- Iwaniec T: Projections onto gradient fields and -estimates for degenerated elliptic operators. Studia Mathematica 1983, 75(3):293–312.MathSciNetMATHGoogle Scholar
- Acerbi E, Mingione G: Gradient estimates for the -Laplacean system. Journal fur die Reine und Angewandte Mathematik 2005, 584: 117–148.MathSciNetView ArticleMATHGoogle Scholar
- Byun S-S, Wang L: Quasilinear elliptic equations with BMO coefficients in Lipschitz domains. Transactions of the American Mathematical Society 2007, 359(12):5899–5913. 10.1090/S0002-9947-07-04238-9MathSciNetView ArticleMATHGoogle Scholar
- Kinnunen J, Zhou S: A local estimate for nonlinear equations with discontinuous coefficients. Communications in Partial Differential Equations 1999, 24(11–12):2043–2068. 10.1080/03605309908821494MathSciNetView ArticleMATHGoogle Scholar
- Boccardo L, Gallouët T: Nonlinear elliptic and parabolic equations involving measure data. Journal of Functional Analysis 1989, 87(1):149–169. 10.1016/0022-1236(89)90005-0MathSciNetView ArticleMATHGoogle Scholar
- Boccardo L, Gallouët T: Nonlinear elliptic equations with right-hand side measures. Communications in Partial Differential Equations 1992, 17(3–4):641–655.MathSciNetView ArticleMATHGoogle Scholar
- Byun S-S, Wang L: -estimates for general nonlinear elliptic equations. Indiana University Mathematics Journal 2007, 56(6):3193–3221. 10.1512/iumj.2007.56.3034MathSciNetView ArticleMATHGoogle Scholar
- Orlicz W: Üeber eine gewisse Klasse von Räumen vom Typus . Bulletin International de l'Académie Polonaise Série A 1932, 8: 207–220.MATHGoogle Scholar
- Adams RA, Fournier JJF: Sobolev Spaces, Pure and Applied Mathematics. Volume 140. 2nd edition. Academic Press, New York, NY, USA; 2003:xiv+305.Google Scholar
- Benkirane A, Elmahi A: An existence theorem for a strongly nonlinear elliptic problem in Orlicz spaces. Nonlinear Analysis: Theory, Methods & Applications 1999, 36(1):11–24. 10.1016/S0362-546X(97)00612-3MathSciNetView ArticleMATHGoogle Scholar
- Byun S-S, Yao F, Zhou S: Gradient estimates in Orlicz space for nonlinear elliptic equations. Journal of Functional Analysis 2008, 255(8):1851–1873.MathSciNetView ArticleMATHGoogle Scholar
- Cianchi A: Hardy inequalities in Orlicz spaces. Transactions of the American Mathematical Society 1999, 351(6):2459–2478. 10.1090/S0002-9947-99-01985-6MathSciNetView ArticleMATHGoogle Scholar
- Kokilashvili V, Krbec M: Weighted Inequalities in Lorentz and Orlicz Spaces. World Scientific, River Edge, NJ, USA; 1991:xii+233.View ArticleMATHGoogle Scholar
- Wang L, Yao F, Zhou S, Jia H: Optimal regularity for the Poisson equation. Proceedings of the American Mathematical Society 2009, 137(6):2037–2047. 10.1090/S0002-9939-09-09805-0MathSciNetView ArticleMATHGoogle Scholar
- Weber M: Stochastic processes with value in exponential type Orlicz spaces. The Annals of Probability 1988, 16(3):1365–1371. 10.1214/aop/1176991696MathSciNetView ArticleMATHGoogle Scholar
- Rao MM, Ren ZD: Applications of Orlicz Spaces, Monographs and Textbooks in Pure and Applied Mathematics. Volume 250. Marcel Dekker, New York, NY, USA; 2002:xii+464.Google Scholar
- Acerbi E, Mingione G: Gradient estimates for a class of parabolic systems. Duke Mathematical Journal 2007, 136(2):285–320. 10.1215/S0012-7094-07-13623-8MathSciNetView ArticleMATHGoogle Scholar
- Mingione G: The Calderón-Zygmund theory for elliptic problems with measure data. Annali della Scuola Normale Superiore di Pisa. Classe di Scienze. Serie V 2007, 6(2):195–261.MathSciNetMATHGoogle Scholar
- Lieberman GM: The natural generalization of the natural conditions of Ladyzhenskaya and urall′tseva tseva for elliptic equations. Communications in Partial Differential Equations 1991, 16(2–3):311–361. 10.1080/03605309108820761MathSciNetView ArticleGoogle Scholar
- Chen Y, Wu L: Second Order Elliptic Partial Differential Equations and Elliptic Systems. American Mathematical Society, Providence, RI, USA; 1998.Google Scholar
- Giaquinta M: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Annals of Mathematics Studies. Volume 105. Princeton University Press, Princeton, NJ, USA; 1983:vii+297.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.