# Some Caccioppoli Estimates for Differential Forms

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

**2009**:734528

**DOI: **10.1155/2009/734528

© Zhenhua Cao et al. 2009

**Received: **31 March 2009

**Accepted: **26 June 2009

**Published: **14 July 2009

## Abstract

## 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]).

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 , and By simple computations, we get and

If in Theorem 2.4, we can obtain the following.

Corollary 2.5.

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

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

Corollary 2.7.

## 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]).

## Declarations

### Acknowledgment

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

## Authors’ Affiliations

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