1. Introduction

The main work of this paper is study the properties of the solutions to the nonhomogeneous -harmonic equation for differential forms

(1.1)

When is a 0-form, that is, is a function, (1.1) is equivalent to

(1.2)

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].

As prior estimates, the Caccioppoli estimate, the weak reverse Hölder inequality, and the Harnack inequality play important roles in PDEs. In this paper, we will prove some Caccioppoli estimates for the solution to (1.1), where the operators and satisfy the following conditions on a bounded convex domain :

(1.3)

for almost every , all -differential forms and -differential forms . Where is a positive constant and through are measurable functions on satisfying:

(1.4)

with some , and is the Poincaré constant.

Now we introduce some notations and operations about exterior forms. Let denote the standard orthogonal basis of . For , we denote the linear space of all -vectors by , spanned by the exterior product , corresponding to all ordered -tuples , . The Grassmann algebra is a graded algebra with respect to the exterior products. For and , then its inner product is obtained by

(1.5)

with the summation over all and all integers . The Hodge star operator : is defined by the rule

(1.6)

for all . Hence the norm of can be given by

(1.7)

Throughout this paper, is an open subset. For any constant , denotes a cube such that , where denotes the cube which center is as same as , and . We say is a differential -form on , if every coefficient of is Schwartz distribution on . We denote the space spanned by differential -form on by . We write for the -form on with for all ordered -tuple . Thus is a Banach space with the norm

(1.8)

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

Definition 1.1 ([3]).

We denote the exterior derivative by

(1.9)

and its formal adjoint (the Hodge co-differential) is the operator

(1.10)

The operators and are given by the formulas

(1.11)

By [3, Lemma  2.3], we know that a solution to (1.1) is an element of the Sobolev space such that

(1.12)

for all with compact support.

Remark 1.2.

In fact, the usual -harmonic equation is the particular form of the equation (1.1) when and satisfies

(1.13)

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]).

Let be a positive exponent, and let , , , be two sets of real numbers such that and . Suppose that is a positive number satisfying the inequality

(2.1)

then

(2.2)

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]).

Let be a bounded convex domain in , then for any differential form , one has

(2.3)

Lemma 2.3 ([5]).

If and for any nonnegative , one has

(2.4)

then for any , one has

(2.5)

Theorem 2.4.

Suppose that is a bounded convex domain in , and is a solution to (1.1) which satisfies (1.3), and , then for any , there exist constants and , such that

(2.6)

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

Proof.

We assume that . For any nonnegative , we let , then we have . By using in the equation (1.12), we can obtain

(2.7)

that is,

(2.8)

By the elementary inequality

(2.9)

(2.8) becomes

(2.10)

Using the inequality

(2.11)

then (2.10) becomes

(2.12)

Since ,so we can deduce

(2.13)

Now we let , then . We use in (1.12), then we can obtain

(2.14)

So we have

(2.15)

By (1.3), (2.13), (2.15) and Lemma 2.2, we have

(2.16)

where

We suppose that , and let then we have and

(2.17)

Combining (2.16) and (2.17), we have

(2.18)

where , and By simple computations, we get and

The terms on the right-hand side of the preceding inequality can be estimated by using the Hölder inequality, Minkowski inequality, Poincaré inequality and Lemma 2.2. Thus

(2.19)
(2.20)

By the similar computation, we can obtain

(2.21)

We insert the three previous estimates (2.19), (2.20) and (2.21) into the right-hand side of (2.15), and set

(2.22)

the result can be written

(2.23)

Applying Lemma 2.1 and simplifying the result, we obtain

(2.24)

or in terms of the original quantities

(2.25)

Combining (2.17) and (2.25), we can obtain

(2.26)

If in Theorem 2.4, we can obtain the following.

Corollary 2.5.

Suppose that is a bounded convex domain in , and is a solution to (1.1) which satisfies (1.3), and , then for any , there exist constants and , such that

(2.27)

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)

where ; ; with for some Then for any and any cubes or balls such that , one has

(2.28)

where and are constants depending only on the above conditions and is the diameter of . One can write them

(2.29)

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

Corollary 2.7.

Suppose that is a bounded convex domain in , and is a solution to (1.1) which satisfies (1.3), and , then for any and any cubes or balls such that , there exist constants and , such that

(2.30)

where , , and is the Poincaré constant.

3. Some Examples

Example 3.1.

The Sobolev inequality cannot be deduced to differential forms. For any we only let

(3.1)

then and

(3.2)

So we cannot obtain

(3.3)

Example 3.2.

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

Lemma 3.3 ([5]).

Let , and , then is in and

(3.4)

for any ball or cube , where for and for .