Some Priori Estimates about Solutions to Nonhomogeneous A-Harmonic Equations

Abstract

We deal with the nonhomogeneous A-harmonic equation and the related conjugate A-harmonic equation . Some priori estimates about solutions to these equations are obtained, which generalize some existing results. Particularly, we obtain the same estimate given by Theorem 1 of Iwaniec (1992) for the weak solution to the first equation under weaker conditions by a simpler method.

1. Introduction

The A-harmonic equation and the related conjugate A-harmonic equation for differential forms originated from the Laplace equation and Cauchy-Riemann equation for functions and in the plane , which are the characteristics of analytic functions in the two-dimensionalplane. Their general forms are -harmonic equations and A-harmonic equations that have been playing a significant role in the development of the theory of quasiconformal and quasiregular mappings, being generalized from analytic functions. Many classic partial differential equations concerned with physical problems may be formulated compactly as A-harmonic equations for differential forms. So the exploration of these kinds of equations has unique interests and meanings, which are referred to [1–9].

Let be a bounded measure function on with values in symmetric linear transformations of , the linear space of -covectors in for . Assume that

(1.1)

for , where is a constant independent of and . The nonlinear mapping is formulated by

(1.2)

for The problem of weak solutions defined as follows, concerned with , which was considered in [10] to give the priori estimate for weak solutions.

Definition 1.1 (see [10]).

Let , and . A differential form is said to be a weak solution of equation

(1.3)

if (1) and (2) for each test form .

Theorem 1.2 (see [10]).

For each A-harmonic equation (1.3) there exist and a constant such that every weak solution , with and , satisfies

(1.4)

It is easy to see that the mapping given by (1.2) satisfies the following conditions

(1.5)

where is the bound of in , that is, for all . In this paper we obtain the same result of Theorem 1.2 under the weaker hypotheses (1.5).

On the other hand, it is interesting to investigate the conjugate A-harmonic equation related to (1.3)

(1.6)

with the conditions (1.5). A series of norm comparison theorems for a pair of solutions to (1.6) were established in [5]. The following is the fundamental conclusion there, which will be extended in this paper to the situation that the conjugateness of and is not required.

Theorem 1.3 (see [5]).

Let and be a pair of solutions to (1.3) in a domain . If and , then if and only if . Moreover, there exist constants , independent of and , such that

(1.7)

for all balls with . Here .

As the extension of some results mentioned above, we give their weighted forms by the weight function in the final section.

2. Some Preliminaries about Differential Forms

The majority of notations and preliminaries used throughout this paper can be found in [1]. For the sake of convenience we list them briefly in this section.

Let denote the standard orthogonal basis of . Suppose that is the linear space of -covectors, generated by the exterior products , corresponding to all ordered -tuples The Grassmann algebra is a graded algebra with respect to the exterior products. For and , the inner product in is given by with summation over all -tuples and all integers . We define the Hodge star operator by

(2.1)

where is a -form, is a permutation of , and is the signature of the permutation. The norm of is given by the formula .

Now and later on the notation, stands for a ball or cube in , even though we do not always need this strong restriction on it. A differential -form is a Schwartz distribution on with values in . We use to denote the space of all deferential -forms, and to denote the -forms

(2.2)

with all coefficients . Thus , , is a Banach space with norm

(2.3)

The space is the subspace of with the condition

(2.4)

The Sobolev space of -forms is

We denote the exterior derivative by for , which means

(2.5)

Its formal adjoint operator is defined by

(2.6)

which is called the Hodge codifferential.

Theorem 2.1 (Hodge decomposition [10]).

For each , there exist differential forms and such that

(2.7)

The forms and are unique and satisfy the uniform estimate

(2.8)

for some constant independent of

It is noticeable that the Hodge decomposition (2.7) corresponds to two bounded linear operators and from to , defined by

(2.9)

for , .

To consider priori estimates for the nonhomogeneous A-harmonic equation we need the bounds of and in the sense of the -norm for some special differential forms . The following interpolation theorem plays a key role in dealing with this problem. Let be a measure space and let be a complex Hilbert space. The notation denotes the norm of bounded linear operators for all , where

Theorem 2.2 (see [9]).

Suppose that . Then

(2.10)

for each , where

(2.11)

3. Priori Estimates for Solutions

For convenience of estimates we reformulate the condition (1.5). Let the mapping satisfy the following conditions

(3.1)

for almost every and all .

Theorem 3.1.

Let , and If is a weak solution to the equation

(3.2)

with the conditions (3.1), then there exist and such that

(3.3)

for

Lemma 3.2.

For the and in Theorem 3.1 there exists a constant K, independent of and , such that

(3.4)

where and are given by the Hodge decomposition

This lemma can be directly deduced from Theorem 2.2 (so-called interpolation theorem) as shown in [9]. For the sake of completeness we give its proof which displays how to use the Hodge decomposition and the interpolation theorem.

Proof.

For and its Hodge decomposition we can define a bounded linear operator from to by . In view of the restriction for and we have So taking such that and choosing , and in Theorem 2.2 yield

(3.5)

where does not depend on and . The second inequality is an immediate result from the first one. The proof is complete.

Besides, Young's inequality and Hölder's inequality play a very important role in a variety of estimates throughout this paper and are listed as follows.

Lemma 3.3 (Young's inequality [11]).

If and , then

(3.6)

for any positive numbers .

Lemma 3.4 (Hölder's inequality [1]).

Let , and . If and are measure functions on , then

(3.7)

Proof of Theorem 3.1.

In view of (3.1) we have

(3.8)

where and are given by Lemma 3.2. Taking as the test form in the definition of weak solutions and integrating (3.8) over and using (3.1) again, we have

(3.9)

Using Hölder's inequality and Lemma 3.2 yields

(3.10)

Applying Young's inequality to the first term of the right-hand side in (3.10), we have

(3.11)

Choosing proper and leads to

(3.12)

whenever , where . Thus, by the elementary inequality and (3.12), we can conclude that

(3.13)

where is independent of and , which finishes the proof.

Another kind of restrictive conditions about was given in [9], where the same result as Theorem 3.1 was obtained under the following hypotheses (H1) and (H2). Let be a constant and be a nonlinear operator satisfying

(H1)

(H2)

(H3)

for almost every and

Notice that (3.1) and both (H1) and (H2) are not mutual of inclusion. But all (H1)–(H3) may lead to (3.1) except for constants. Since the main results with (H3) in [9] were based on the conclusion of Theorem 3.1, we can obtain responding results on a larger scale. Taking Lemma in [9] for example, we can establish the following theorem.

Theorem 3.5.

Let be the same as in Theorem 3.1. Suppose that is a weak solution for some to the homogeneous A-harmonic equation

(3.14)

with the assumptions (H1)–(H3). Then for any concentric cubes one has

(3.15)

where and stands for the integral mean over .

Now we consider the nonhomogeneous conjugate A-harmonic equation

(3.16)

and establish the norm comparison theorem for and . It generalizes Theorem 1.3 because here and do not generally satisfy the conjugate condition which is demanded there.

Theorem 3.6.

Let and be a pair of solutions to the nonhomogeneous A-harmonic equation (3.16) with the condition (3.1) in the domain If and , where then if and only if . Moreover, there exist constants and , independent of and , such that

(3.17)

(3.18)

Proof.

It is enough to check both (3.17) and (3.18). First, from (3.1) and (3.16), we have

(3.19)

Applying the elementary inequality to the above inequality leads to

(3.20)

where does not depend on and . Integrating (3.20) over , we get (3.17).

Next, we use the trick used in the proof of Theorem 3.1 to check (3.18). Notice that

(3.21)

Integrating (3.21) over and then using Hölder's inequality, we have

(3.22)

Using Young's inequality to (3.22), we get

(3.23)

Taking small enough and using the elementary inequality , we obtain

(3.24)

where and are constants independent of and . It is not difficult to get (3.18) from (3.24), and so the proof is complete.

Combining Theorems 3.1 and 3.6, we obtain immediately the norm estimate for by means of and , which can be viewed as the symmetrical result to (3.3).

Corollary 3.7.

If and simultaneously satisfy the hypothesis of both Theorem 3.1 and Theorem 3.6, then there is a constant , independent of and , such that

(3.25)

4. Some Weighted Estimates

In this section we give the weighted estimates for some results obtained in the front part. A function is called a weight if a.e. and . Among all weights the function is one of the most important weights and is widely applied to the theory of harmonic analysis, quasiconformal mappings, differential forms, and so on.

Definition 4.1.

A weight is called weight, where , and we write if

(4.1)

where the supremum is over all balls and is the Lebesgue measure of .

The weight and the related Radon measure have many interesting properties; see [12, 13] for details. In order to express weighted integrals briefly, we introduce the notation as follows:

(4.2)

Theorem 4.2.

Let be a weak solution to the nonhomogeneous A-harmonic equation (3.2) in a ball , and let for some . Then, for , there exists a constant , independent of , such that

(4.3)

where and .

Proof.

Let , and , that is, . Using Hölder's inequality, we have

(4.4)

Next, we estimate by Hölder's inequality with and Noticing and , we have

(4.5)

With the same method shown above, we can get the weighted estimate for . But, as a matter of fact, we have a shortcut to obtain the same result. Since , replacing in (4.5) with , we get right away

(4.6)

In view of the elementary inequality for and , from (3.3), we have

(4.7)

Using (4.5) and (4.6) to plug (4.7) and then applying to (4.4), we have

(4.8)

Since , we have from (4.1)

(4.9)

Thus applying this to (4.8), we can finish the proof.

Corollary 4.3.

Let satisfy the conditions of Corollary 3.7 in a ball and for some . Then for , there exists a constant , independent of , such that

(4.10)

where and

Proof.

Putting , that is, into (3.25), we have

(4.11)

Taking a notice to another form of (3.3), we have

(4.12)

which is the source of (4.3). Making a comparison between (4.11) and (4.12), we can obtain the conditions by means of , and that guarantee (4.10) to hold. Specifically, noticing , we have and

(4.13)

Based on this approach it is easy to get the weighted forms of (3.17) and (3.18) if the domain in Theorem 3.6 is replaced by a ball . For example, under the same hypotheses of Theorem 4.2, we have the weighted form of (3.18) as follows:

(4.14)