- Research Article
- Open Access
Some Caccioppoli Estimates for Differential Forms
© Zhenhua Cao et al. 2009
- Received: 31 March 2009
- Accepted: 26 June 2009
- Published: 14 July 2009
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.
- Differential Form
- Sobolev Inequality
- Harnack Inequality
- Weighted Sobolev Space
- Minkowski Inequality
In , 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 . In 1993, Iwaniec and Lutoborskidiscussed the Poincaré inequality for differential forms when in , and the Poincaré inequality for differential forms was generalized to in . In 1999, Nolder gave the reverse Hölder inequality for the solution to the -harmonic equation in , 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 , where the operator satisfies . In 2004, D'Onofrio and Iwaniec introduced the -harmonic type system in , 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].
with some , and is the Poincaré constant.
Similarly, denotes those -forms on which all coefficients belong to . The following definition can be found in [3, page 596].
Definition 1.1 ().
for all with compact support.
We notice that the nonhomogeneous -harmonic equation and the -harmonic type equation are special forms of (1.1).
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 ().
where depends only on and where
By the inequalities (2.13) and (3.28) in , One has the following lemma.
Lemma 2.2 ().
Lemma 2.3 ().
where , , , and is the Poincaré constant. (i.e., when , and when ).
where , and By simple computations, we get and
If in Theorem 2.4, we can obtain the following.
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 , we can obtain the following result which is the improving result of [1, Theorem 2].
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
If we let and is a bump function, then we have the following.
where , , and is the Poincaré constant.
The Poincaré inequality can be deduced to differential forms. We can see the following lemma.
Lemma 3.3 ().
for any ball or cube , where for and for .
This work is supported by the NSF of China (no.10771044 and no.10671046).
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