Some Caccioppoli Estimates for Differential Forms

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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

The main work of this paper is study the properties of the solutions to the nonhomogeneous Open image in new window -harmonic equation for differential forms
When Open image in new window is a 0-form, that is, Open image in new window is a function, (1.1) is equivalent to

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].

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 Open image in new window and Open image in new window satisfy the following conditions on a bounded convex domain Open image in new window :
for almost every Open image in new window , all Open image in new window -differential forms Open image in new window and Open image in new window -differential forms Open image in new window . Where Open image in new window is a positive constant and Open image in new window through Open image in new window are measurable functions on Open image in new window satisfying:

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.

Now we introduce some notations and operations about exterior forms. Let Open image in new window denote the standard orthogonal basis of Open image in new window . For Open image in new window , we denote the linear space of all Open image in new window -vectors by Open image in new window , spanned by the exterior product Open image in new window , corresponding to all ordered Open image in new window -tuples Open image in new window , Open image in new window . The Grassmann algebra Open image in new window is a graded algebra with respect to the exterior products. For Open image in new window and Open image in new window , then its inner product is obtained by
with the summation over all Open image in new window and all integers Open image in new window . The Hodge star operator Open image in new window : Open image in new window is defined by the rule
for all Open image in new window . Hence the norm of Open image in new window can be given by
Throughout this paper, Open image in new window is an open subset. For any constant Open image in new window , Open image in new window denotes a cube such that Open image in new window , where Open image in new window denotes the cube which center is as same as Open image in new window , and Open image in new window . We say Open image in new window is a differential Open image in new window -form on Open image in new window , if every coefficient Open image in new window of Open image in new window is Schwartz distribution on Open image in new window . We denote the space spanned by differential Open image in new window -form on Open image in new window by Open image in new window . We write Open image in new window for the Open image in new window -form Open image in new window on Open image in new window with Open image in new window for all ordered Open image in new window -tuple Open image in new window . Thus Open image in new window is a Banach space with the norm

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]).

We denote the exterior derivative by
and its formal adjoint (the Hodge co-differential) is the operator
The operators Open image in new window and Open image in new window are given by the formulas
By [3, Lemma  2.3], we know that a solution to (1.1) is an element of the Sobolev space Open image in new window such that

for all Open image in new window with compact support.

Remark 1.2.

In fact, the usual Open image in new window -harmonic equation is the particular form of the equation (1.1) when Open image in new window and Open image in new window satisfies

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]).

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

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]).

Let Open image in new window be a bounded convex domain in Open image in new window , then for any differential form Open image in new window , one has

Lemma 2.3 ([5]).

Theorem 2.4.

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

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.

We assume that Open image in new window . For any nonnegative Open image in new window , we let Open image in new window , then we have Open image in new window . By using Open image in new window in the equation (1.12), we can obtain
By the elementary inequality
(2.8) becomes
Using the inequality
then (2.10) becomes
Since Open image in new window ,so we can deduce
Now we let Open image in new window , then Open image in new window . We use Open image in new window in (1.12), then we can obtain
So we have
By (1.3), (2.13), (2.15) and Lemma 2.2, we have

where Open image in new window

Combining (2.16) and (2.17), we have

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

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

Corollary 2.5.

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

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

(ii) Open image in new window

(iii) Open image in new window

where Open image in new window and Open image in new window are constants depending only on the above conditions and Open image in new window is the diameter of Open image in new window . One can write them

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.

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

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.

The Sobolev inequality cannot be deduced to differential forms. For any Open image in new window we only let
So we cannot obtain

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).

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Copyright information

© Zhenhua Cao et al. 2009

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.

Authors and Affiliations

  • Zhenhua Cao
    • 1
  • Gejun Bao
    • 1
  • Yuming Xing
    • 1
  • Ronglu Li
    • 1
  1. 1.Department of MathematicsHarbin Institute of TechnologyHarbinChina

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