Some Caccioppoli Estimates for Differential Forms
© Zhenhua Cao et al. 2009
Received: 31 March 2009
Accepted: 26 June 2009
Published: 14 July 2009
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].
Similarly, denotes those -forms on which all coefficients belong to . The following definition can be found in [3, page 596].
Definition 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 ().
By the inequalities (2.13) and (3.28) in , One has the following lemma.
Lemma 2.2 ().
Lemma 2.3 ().
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].
3. Some Examples
The Poincaré inequality can be deduced to differential forms. We can see the following lemma.
Lemma 3.3 ().
This work is supported by the NSF of China (no.10771044 and no.10671046).
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