# Some Caccioppoli Estimates for Differential Forms

- 719 Downloads

## Abstract

We prove the global Caccioppoli estimate for the solution to the nonhomogeneous Open image in new window -harmonic equation Open image in new window , which is the generalization of the quasilinear equation Open image in new window . 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 Open image in new window in the weighted Sobolev spaces, which is a particular form of (1.2). Recently, a large amount of work on the Open image in new window -harmonic equation for differential forms has been done. In 1992, Iwaniec introduced the Open image in new window -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 Open image in new window in [4], and the Poincaré inequality for differential forms was generalized to Open image in new window in [5]. In 1999, Nolder gave the reverse Hölder inequality for the solution to the Open image in new window -harmonic equation in [6], and different versions of the Caccioppoli estimates have been established in [7, 8, 9]. In 2004, Ding proved the Caccioppli estimates for the solution to the nonhomogeneous Open image in new window -harmonic equation Open image in new window in [10], where the operator Open image in new window satisfies Open image in new window . In 2004, D'Onofrio and Iwaniec introduced the Open image in new window -harmonic type system in [11], which is an important extension of the conjugate Open image in new window -harmonic equation. Lots of work on the solution to the Open image in new window -harmonic type system have been done in [5, 12].

with some Open image in new window , Open image in new window Open image in new window and Open image in new window is the Poincaré constant.

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

Definition 1.1 ([3]).

for all Open image in new window with compact support.

Remark 1.2.

We notice that the nonhomogeneous Open image in new window -harmonic equation Open image in new window and the Open image in new window -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 Open image in new window depends only on Open image in new window Open image in new window Open image in new window and where Open image in new window

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 Open image in new window , Open image in new window , Open image in new window , and Open image in new window is the Poincaré constant. (i.e., Open image in new window when Open image in new window , and Open image in new window when Open image in new window ).

Proof.

where Open image in new window

where Open image in new window , Open image in new window and Open image in new window By simple computations, we get Open image in new window and Open image in new window

If Open image in new window in Theorem 2.4, we can obtain the following.

Corollary 2.5.

where Open image in new window and Open image in new window .

When Open image in new window is a Open image in new window -differential form, that is, Open image in new window is a function, we have Open image in new window . Now we use Open image in new window in place of Open image in new window 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 Open image in new window be a solution to the equation Open image in new window in a domain Open image in new window . For any Open image in new window , one denotes Open image in new window . Suppose that the following conditions hold

(i) Open image in new window , where Open image in new window is a constant, Open image in new window such that 2003 Open image in new window

(iii) Open image in new window

If we let Open image in new window and Open image in new window is a bump function, then we have the following.

Corollary 2.7.

where Open image in new window , Open image in new window , and Open image in new window 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 Open image in new window , where Open image in new window for Open image in new window and Open image in new window for Open image in new window .

## Notes

### Acknowledgment

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

### References

- 1.Serrin J:
**Local behavior of solutions of quasi-linear equations.***Acta Mathematica*1964,**111**(1):247–302. 10.1007/BF02391014MATHMathSciNetCrossRefGoogle Scholar - 2.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 - 3.Iwaniec T:
**-harmonic tensors and quasiregular mappings.***Annals of Mathematics*1992,**136**(3):589–624. 10.2307/2946602MATHMathSciNetCrossRefGoogle Scholar - 4.Iwaniec T, Lutoborski A:
**Integral estimates for null Lagrangians.***Archive for Rational Mechanics and Analysis*1993,**125**(1):25–79. 10.1007/BF00411477MATHMathSciNetCrossRefGoogle Scholar - 5.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 - 6.Nolder CA:
**Hardy-Littlewood theorems for -harmonic tensors.***Illinois Journal of Mathematics*1999,**43**(4):613–632.MATHMathSciNetGoogle Scholar - 7.Bao G:
**-weighted integral inequalities for -harmonic tensors.***Journal of Mathematical Analysis and Applications*2000,**247**(2):466–477. 10.1006/jmaa.2000.6851MATHMathSciNetCrossRefGoogle Scholar - 8.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/S0002-9939-99-05285-5MATHMathSciNetCrossRefGoogle Scholar - 9.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)00036-2MATHMathSciNetCrossRefGoogle Scholar - 10.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/S0002-9939-04-07347-2MATHMathSciNetCrossRefGoogle Scholar - 11.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.2004.53.2462MATHMathSciNetCrossRefGoogle Scholar - 12.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

## Copyright information

This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.