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.