Local Regularity and Local Boundedness Results for Very Weak Solutions of Obstacle Problems
© Gao Hongya et al. 2010
Received: 25 September 2009
Accepted: 18 March 2010
Published: 30 March 2010
1. Introduction and Statement of Results
Definition 1.1 (see ).
This is the classical definition for -obstacle problem; see  for some details of solutions of -obstacle problem.
This paper deals with local regularity and local boundedness for very weak solutions of obstacle problems. Local regularity and local boundedness properties are important among the regularity theories of nonlinear elliptic systems; see the recent monograph  by Bensoussan and Frehse. Meyers and Elcrat  first considered the higher integrability for weak solutions of (1.1) in 1975; see also . Iwaniec and Sbordone  obtained the regularity result for very weak solutions of the -harmonic (1.1) by using the celebrated Gehring's Lemma. The local and global higher integrability of the derivatives in obstacle problem was first considered by Li and Martio  in 1994 by using the so-called reverse Hölder inequality. Gao et al.  gave the definition for very weak solutions of obstacle problem of -harmonic (1.1) and obtained the local and global higher integrability results. The local regularity results for minima of functionals and solutions of elliptic equations have been obtained in . For some new results related to -harmonic equation, we refer the reader to [9–11]. Gao and Tian  gave the local regularity result for weak solutions of obstacle problem with the obstacle function . Li and Gao  generalized the result of  by obtaining the local integrability result for very weak solutions of obstacle problem. The main result of  is the following proposition.
We omit the proof of this corollary. This corollary shows that the condition in the main result of  is not necessary.
The second part of this paper considers local boundedness for very weak solutions of -obstacle problem. The local boundedness for solutions of obstacle problems plays a central role in many aspects. Based on the local boundedness, we can further study the regularity of the solutions. For the local boundedness results of weak solutions of nonlinear elliptic equations, we refer the reader to . In this paper we consider very weak solutions and show that if the obstacle function is , then a very weak solution to the -obstacle problem is locally bounded.
As far as we are aware, Theorem 1.6 is the first result concerning local boundedness for very weak solutions of obstacle problems.
We recall two lammas which will be used in the proof of Theorem 1.4.
Lemma 1.8 (see ).
Lemma 1.9 (see ).
We need the following definition.
Definition 1.10 (see ).
We recall a lemma from  which will be used in the proof of Theorem 1.6.
Lemma 1.11 (see ).
2. Local Regularity
Proof of Theorem 1.4.
3. Local Boundedness
Proof of Theorem 1.6.
The authors would like to thank the referee of this paper for helpful comments upon which this paper was revised. The first author is supported by NSFC (10971224) and NSF of Hebei Province (07M003). The third author is supported by NSF of Zhejiang province (Y607128) and NSFC (10771195).
- Iwaniec T, Sbordone C: Weak minima of variational integrals. Journal für die reine und angewandte Mathematik 1994, 454: 143–161.MATHMathSciNetGoogle Scholar
- Gao HY, Wang M, Zhao HL: Very weak solutions for obstacle problems of the -harmonic equation. Journal of Mathematical Research and Exposition 2004, 24(1):159–167.MATHMathSciNetGoogle Scholar
- Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs. The Clarendon Press, Oxford, UK; 1993:vi+363.Google Scholar
- Bensoussan A, Frehse J: Regularity Results for Nonlinear Elliptic Systems and Applications, Applied Mathematical Sciences. Volume 151. Springer, Berlin, Germany; 2002:xii+441.View ArticleGoogle Scholar
- Meyers NG, Elcrat A: Some results on regularity for solutions of non-linear elliptic systems and quasi-regular functions. Duke Mathematical Journal 1975, 42: 121–136. 10.1215/S0012-7094-75-04211-8MATHMathSciNetView ArticleGoogle Scholar
- Strdulinsky EW: Higher integrability from reverse Hölder inequalities. Indiana University Mathematics Journal 1980, 29: 408–413.Google Scholar
- Li GB, Martio O: Local and global integrability of gradients in obstacle problems. Annales Academiae Scientiarum Fennicae. Series A 1994, 19(1):25–34.MATHMathSciNetGoogle Scholar
- Giachetti D, Porzio MM: Local regularity results for minima of functionals of the calculus of variation. Nonlinear Analysis: Theory, Methods & Applications 2000, 39(4):463–482. 10.1016/S0362-546X(98)00215-6MATHMathSciNetView ArticleGoogle Scholar
- Xing Y, Ding S: Inequalities for Green's operator with Lipschitz and BMO norms. Computers & Mathematics with Applications 2009, 58(2):273–280. 10.1016/j.camwa.2009.03.096MATHMathSciNetView ArticleGoogle Scholar
- Ding S: Lipschitz and BMO norm inequalities for operators. Nonlinear Analysis: Theory, Methods & Applications 2009, 71(12):e2350-e2357. 10.1016/j.na.2009.05.032MATHView ArticleGoogle Scholar
- Gao H, Qiao J, Wang Y, Chu Y: Local regularity results for minima of anisotropic functionals and solutions of anisotropic equations. Journal of Inequalities and Applications 2008, 2008:-11.MathSciNetView ArticleGoogle Scholar
- Gao H, Tian H: Local regularity result for solutions of obstacle problems. Acta Mathematica Scientia B 2004, 24(1):71–74.MATHMathSciNetGoogle Scholar
- Li J, Gao H: Local regularity result for very weak solutions of obstacle problems. Radovi Matematički 2003, 12(1):19–26.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
- Hong MC: Some remarks on the minimizers of variational integrals with nonstandard growth conditions. Bollettino dell'Unione Matematica Italiana 1992, 6(1):91–101.MATHGoogle Scholar
- Iwaniec T, Migliaccio L, Nania L, Sbordone C: Integrability and removability results for quasiregular mappings in high dimensions. Mathematica Scandinavica 1994, 75(2):263–279.MATHMathSciNetGoogle Scholar
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