- Research Article
- Open Access
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
Local regularity and local boundedness results for very weak solutions of obstacle problems of the -harmonic equation are obtained by using the theory of Hodge decomposition, where .
- Weak Solution
- Lipschitz Domain
- Obstacle Problem
- Local Regularity
- Regular Domain
where is a Carathéodory function satisfying the following conditions:
The function is an obstacle and determines the boundary values.
where is some constant depending only on and .
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.
There exists with , such that any very weak solution to the -obstacle problem belongs to , , provided that , , and .
Notice that in the above proposition we have restricted ourselves to the case , because when , every function in is trivially in for every by the classical Sobolev imbedding theorem.
In the first part of this paper, we continue to consider the local regularity theory for very weak solutions of obstacle problem by showing that the condition in Proposition 1.3 is not necessary.
There exists with , such that any very weak solution to the -obstacle problem belongs to , provided that , , and .
As a corollary of the above theorem, if , that is, if we consider weak solutions of -obstacle problem, then we have the following local regularity result.
Suppose that , . Then a solution to the -obstacle problem belongs to .
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.
There exists with , such that for any with and any , 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 ).
for every and , where is a real positive constant that depends only on and is a real positive constant. Then .
Lemma 1.9 (see ).
We need the following definition.
Definition 1.10 (see ).
for , , where is the -dimensional Lebesgue measure of the set .
We recall a lemma from  which will be used in the proof of Theorem 1.6.
Lemma 1.11 (see ).
in which the constant is determined only by the quantities .
Proof of Theorem 1.4.
where is the constant given by Lemma 1.9. Thus satisfies inequality (1.11) with and . Theorem 1.4 follows from Lemma 1.8.
Proof of Theorem 1.6.
a.e. in .
This result together with the assumptions and yields the desired result.
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).
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