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Some Caccioppoli Estimates for Differential Forms
Journal of Inequalities and Applications volume 2009, Article number: 734528 (2009)
Abstract
We prove the global Caccioppoli estimate for the solution to the nonhomogeneous -harmonic equation
, which is the generalization of the quasilinear equation
. We will also give some examples to see that not all properties of functions may be deduced to differential forms.
1. Introduction
The main work of this paper is study the properties of the solutions to the nonhomogeneous -harmonic equation for differential forms
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ1_HTML.gif)
When is a 0-form, that is,
is a function, (1.1) is equivalent to
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ2_HTML.gif)
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
:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ3_HTML.gif)
for almost every , all
-differential forms
and
-differential forms
. Where
is a positive constant and
through
are measurable functions on
satisfying:
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ4_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ5_HTML.gif)
with the summation over all and all integers
. The Hodge star operator
:
is defined by the rule
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ6_HTML.gif)
for all . Hence the norm of
can be given by
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ7_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ8_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ9_HTML.gif)
and its formal adjoint (the Hodge co-differential) is the operator
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ10_HTML.gif)
The operators and
are given by the formulas
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ11_HTML.gif)
By [3, Lemma  2.3], we know that a solution to (1.1) is an element of the Sobolev space such that
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ12_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ13_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ14_HTML.gif)
then
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ15_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ16_HTML.gif)
Lemma 2.3 ([5]).
If and for any nonnegative
, one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ17_HTML.gif)
then for any , one has
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ18_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ19_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ20_HTML.gif)
that is,
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ21_HTML.gif)
By the elementary inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ22_HTML.gif)
(2.8) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ23_HTML.gif)
Using the inequality
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ24_HTML.gif)
then (2.10) becomes
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ25_HTML.gif)
Since ,so we can deduce
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ26_HTML.gif)
Now we let , then
. We use
in (1.12), then we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ27_HTML.gif)
So we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ28_HTML.gif)
By (1.3), (2.13), (2.15) and Lemma 2.2, we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ29_HTML.gif)
where
We suppose that ,
and let
then we have
and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ30_HTML.gif)
Combining (2.16) and (2.17), we have
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ31_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ32_HTML.gif)
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ33_HTML.gif)
By the similar computation, we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ34_HTML.gif)
We insert the three previous estimates (2.19), (2.20) and (2.21) into the right-hand side of (2.15), and set
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ35_HTML.gif)
the result can be written
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ36_HTML.gif)
Applying Lemma 2.1 and simplifying the result, we obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ37_HTML.gif)
or in terms of the original quantities
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ38_HTML.gif)
Combining (2.17) and (2.25), we can obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ39_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ40_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ41_HTML.gif)
where and
are constants depending only on the above conditions and
is the diameter of
. One can write them
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ42_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ43_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ44_HTML.gif)
then and
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ45_HTML.gif)
So we cannot obtain
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ46_HTML.gif)
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
![](http://media.springernature.com/full/springer-static/image/art%3A10.1155%2F2009%2F734528/MediaObjects/13660_2009_Article_2001_Equ47_HTML.gif)
for any ball or cube , where
for
and
for
.
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Acknowledgment
This work is supported by the NSF of China (no.10771044 and no.10671046).
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Cao, Z., Bao, G., Xing, Y. et al. Some Caccioppoli Estimates for Differential Forms. J Inequal Appl 2009, 734528 (2009). https://doi.org/10.1155/2009/734528
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DOI: https://doi.org/10.1155/2009/734528