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
When is a 0-form, that is, 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 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 :
for almost every , all -differential forms and -differential forms . Where is a positive constant and through are measurable functions on satisfying:
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
with the summation over all and all integers . The Hodge star operator : is defined by the rule
for all . Hence the norm of can be given by
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
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
and its formal adjoint (the Hodge co-differential) is the operator
The operators and 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 such that
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
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
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