- Research Article
- Open Access
Gradient Estimates for Weak Solutions of -Harmonic Equations
© Fengping Yao. 2010
- Received: 29 December 2009
- Accepted: 23 March 2010
- Published: 30 May 2010
We obtain gradient estimates in Orlicz spaces for weak solutions of -Harmonic Equations under the assumptions that satisfies some proper conditions and the given function satisfies some moderate growth condition. As a corollary we obtain -type regularity for such equations.
- 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.
where is a constant independent from and . Indeed, if with , (1.12) is reduced to the classical estimate.
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.
( ) We remark that the global condition makes the functions grow moderately. For example, for . Examples such as are ruled out by , and those such as are ruled out by .
where and .
() Under condition (1.15), it is easy to check that satisfies and
The Orlicz space is the linear hull of .
Moreover, we give the following lemma.
Assume that and . Then
(1) and is dense in
for any , where .
Now we are set to state the main result.
where and is a constant independent from and .
holds if and only if .
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).
If is a local weak solution of (1.1) and satisfies (1.2)–(1.5), then
(1) satisfies (1.2)–(1.5) with the same constants
for all and .
Thus we complete the proof.
2.2. The Iteration-Covering Procedure
for any and , which implies that (2.15) holds truely.
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 give the following local estimates for problem (1.1).
and then finish the proof by choosing small enough.
where is a fixed point.
We first state the definition of the global weak solutions.
for any .
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.
where and .
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).
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