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

then

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

Lemma 2.3 ([5]).

If and for any nonnegative , one has

then for any , one has

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

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

that is,

By the elementary inequality

(2.8) becomes

Using the inequality

then (2.10) becomes

Since ,so we can deduce

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

So we have

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

where

We suppose that , and let then we have and

Combining (2.16) and (2.17), we have

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

By the similar computation, we can obtain

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

the result can be written

Applying Lemma 2.1 and simplifying the result, we obtain

or in terms of the original quantities

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

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

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

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

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

where , , and is the Poincaré constant.