- Research Article
- Open Access

# Some Caccioppoli Estimates for Differential Forms

- Zhenhua Cao
^{1}, - Gejun Bao
^{1}Email author, - Yuming Xing
^{1}and - Ronglu Li
^{1}

**2009**:734528

https://doi.org/10.1155/2009/734528

© Zhenhua Cao et al. 2009

**Received:**31 March 2009**Accepted:**26 June 2009**Published:**14 July 2009

## Abstract

We prove the global Caccioppoli estimate for the solution to the nonhomogeneous -harmonic equation , which is the generalization of the quasilinear equation . We will also give some examples to see that not all properties of functions may be deduced to differential forms.

## Keywords

- Differential Form
- Sobolev Inequality
- Harnack Inequality
- Weighted Sobolev Space
- Minkowski Inequality

## 1. Introduction

In [1], Serrin gave some properties of (1.2) when the operator satisfies some conditions. In [2, chapter 3], Heinonen et al. discussed the properties of the quasielliptic equations in the weighted Sobolev spaces, which is a particular form of (1.2). Recently, a large amount of work on the -harmonic equation for differential forms has been done. In 1992, Iwaniec introduced the -harmonic tensors and the relations between quasiregular mappings and the exterior algebra (or differential forms) in [3]. In 1993, Iwaniec and Lutoborskidiscussed the Poincaré inequality for differential forms when in [4], and the Poincaré inequality for differential forms was generalized to in [5]. In 1999, Nolder gave the reverse Hölder inequality for the solution to the -harmonic equation in [6], and different versions of the Caccioppoli estimates have been established in [7–9]. In 2004, Ding proved the Caccioppli estimates for the solution to the nonhomogeneous -harmonic equation in [10], where the operator satisfies . In 2004, D'Onofrio and Iwaniec introduced the -harmonic type system in [11], which is an important extension of the conjugate -harmonic equation. Lots of work on the solution to the -harmonic type system have been done in [5, 12].

with some , and is the Poincaré constant.

Similarly, denotes those -forms on which all coefficients belong to . The following definition can be found in [3, page 596].

Definition 1.1 ([3]).

for all with compact support.

Remark 1.2.

We notice that the nonhomogeneous -harmonic equation and the -harmonic type equation are special forms of (1.1).

## 2. The Caccioppoli Estimate

In this section we will prove the global and the local Caccioppoli estimates for the solution to (1.1) which satisfies (1.3). In the proof of the global Caccioppoli estimate, we need the following three lemmas.

Lemma 2.1 ([1]).

where depends only on and where

By the inequalities (2.13) and (3.28) in [5], One has the following lemma.

Lemma 2.2 ([5]).

Lemma 2.3 ([5]).

Theorem 2.4.

where , , , and is the Poincaré constant. (i.e., when , and when ).

Proof.

where

where , and By simple computations, we get and

If in Theorem 2.4, we can obtain the following.

Corollary 2.5.

where and .

When is a -differential form, that is, is a function, we have . Now we use in place of in (1.3), then (1.1) satisfying (1.3) is equivalent to (5) which satisfies (6) in [1], we can obtain the following result which is the improving result of [1, Theorem 2].

Corollary 2.6.

Let be a solution to the equation in a domain . For any , one denotes . Suppose that the following conditions hold

(i) , where is a constant, such that 2003

(ii)

(iii)

If we let and is a bump function, then we have the following.

Corollary 2.7.

where , , and is the Poincaré constant.

## 3. Some Examples

Example 3.1.

Example 3.2.

The Poincaré inequality can be deduced to differential forms. We can see the following lemma.

Lemma 3.3 ([5]).

for any ball or cube , where for and for .

## Declarations

### Acknowledgment

This work is supported by the NSF of China (no.10771044 and no.10671046).

## Authors’ Affiliations

## References

- Serrin J:
**Local behavior of solutions of quasi-linear equations.***Acta Mathematica*1964,**111**(1):247–302. 10.1007/BF02391014MATHMathSciNetView ArticleGoogle Scholar - Heinonen J, Kilpeläìnen T, Martio O:
*Nonlinear Potential Theory of Degenerate Elliptic Equations, Oxford Mathematical Monographs*. The Clarendon Press, Oxford University Press, New York, NY, USA; 1993:vi+363.MATHGoogle Scholar - Iwaniec T:
**-harmonic tensors and quasiregular mappings.***Annals of Mathematics*1992,**136**(3):589–624. 10.2307/2946602MATHMathSciNetView ArticleGoogle Scholar - Iwaniec T, Lutoborski A:
**Integral estimates for null Lagrangians.***Archive for Rational Mechanics and Analysis*1993,**125**(1):25–79. 10.1007/BF00411477MATHMathSciNetView ArticleGoogle Scholar - Cao Z, Bao G, Li R, Zhu H:
**The reverse Hölder inequality for the solution to -harmonic type system.***Journal of Inequalities and Applications*2008,**2008:**-15.Google Scholar - Nolder CA:
**Hardy-Littlewood theorems for -harmonic tensors.***Illinois Journal of Mathematics*1999,**43**(4):613–632.MATHMathSciNetGoogle Scholar - Bao G:
**-weighted integral inequalities for -harmonic tensors.***Journal of Mathematical Analysis and Applications*2000,**247**(2):466–477. 10.1006/jmaa.2000.6851MATHMathSciNetView ArticleGoogle Scholar - Ding S:
**Weighted Caccioppoli-type estimates and weak reverse Hölder inequalities for -harmonic tensors.***Proceedings of the American Mathematical Society*1999,**127**(9):2657–2664. 10.1090/S0002-9939-99-05285-5MATHMathSciNetView ArticleGoogle Scholar - Yuming X:
**Weighted integral inequalities for solutions of the -harmonic equation.***Journal of Mathematical Analysis and Applications*2003,**279**(1):350–363. 10.1016/S0022-247X(03)00036-2MATHMathSciNetView ArticleGoogle Scholar - Ding S:
**Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -harmonic equations on Riemannian manifolds.***Proceedings of the American Mathematical Society*2004,**132**(8):2367–2375. 10.1090/S0002-9939-04-07347-2MATHMathSciNetView ArticleGoogle Scholar - D'Onofrio L, Iwaniec T:
**The -harmonic transform beyond its natural domain of definition.***Indiana University Mathematics Journal*2004,**53**(3):683–718. 10.1512/iumj.2004.53.2462MATHMathSciNetView ArticleGoogle Scholar - Bao G, Cao Z, Li R: The Caccioppoli estimate for the solution to the -harmonic type system. Proceedings of the 6th International Conference on Differential Equations and Dynaminal Systems (DCDIS '09), 2009 63–67.Google Scholar

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