Research Article | Open | Published:
Some Priori Estimates about Solutions to Nonhomogeneous A-Harmonic Equations
Journal of Inequalities and Applicationsvolume 2010, Article number: 520240 (2010)
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.
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
for , where is a constant independent of and . The nonlinear mapping is formulated by
for The problem of weak solutions defined as follows, concerned with , which was considered in  to give the priori estimate for weak solutions.
Definition 1.1 (see ).
Let , and . A differential form is said to be a weak solution of equation
if (1) and (2) for each test form .
Theorem 1.2 (see ).
For each A-harmonic equation (1.3) there exist and a constant such that every weak solution , with and , satisfies
It is easy to see that the mapping given by (1.2) satisfies the following conditions
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)
with the conditions (1.5). A series of norm comparison theorems for a pair of solutions to (1.6) were established in . 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 ).
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
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 . 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
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
with all coefficients . Thus , , is a Banach space with norm
The space is the subspace of with the condition
The Sobolev space of -forms is
We denote the exterior derivative by for , which means
Its formal adjoint operator is defined by
which is called the Hodge codifferential.
Theorem 2.1 (Hodge decomposition ).
For each , there exist differential forms and such that
The forms and are unique and satisfy the uniform estimate
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
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 ).
Suppose that . Then
for each , where
3. Priori Estimates for Solutions
For convenience of estimates we reformulate the condition (1.5). Let the mapping satisfy the following conditions
for almost every and all .
Let , and If is a weak solution to the equation
with the conditions (3.1), then there exist and such that
For the and in Theorem 3.1 there exists a constant K, independent of and , such that
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 . For the sake of completeness we give its proof which displays how to use the Hodge decomposition and the interpolation theorem.
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
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 ).
If and , then
for any positive numbers .
Lemma 3.4 (Hölder's inequality ).
Let , and . If and are measure functions on , then
Proof of Theorem 3.1.
In view of (3.1) we have
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
Using Hölder's inequality and Lemma 3.2 yields
Applying Young's inequality to the first term of the right-hand side in (3.10), we have
Choosing proper and leads to
whenever , where . Thus, by the elementary inequality and (3.12), we can conclude that
where is independent of and , which finishes the proof.
Another kind of restrictive conditions about was given in , 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
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  were based on the conclusion of Theorem 3.1, we can obtain responding results on a larger scale. Taking Lemma in  for example, we can establish the following theorem.
Let be the same as in Theorem 3.1. Suppose that is a weak solution for some to the homogeneous A-harmonic equation
with the assumptions (H1)–(H3). Then for any concentric cubes one has
where and stands for the integral mean over .
Now we consider the nonhomogeneous conjugate A-harmonic equation
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.
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
It is enough to check both (3.17) and (3.18). First, from (3.1) and (3.16), we have
Applying the elementary inequality to the above inequality leads to
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
Integrating (3.21) over and then using Hölder's inequality, we have
Using Young's inequality to (3.22), we get
Taking small enough and using the elementary inequality , we obtain
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).
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
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.
A weight is called weight, where , and we write if
where the supremum is over all balls and is the Lebesgue measure of .
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
where and .
Let , and , that is, . Using Hölder's inequality, we have
Next, we estimate by Hölder's inequality with and Noticing and , we have
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
In view of the elementary inequality for and , from (3.3), we have
Using (4.5) and (4.6) to plug (4.7) and then applying to (4.4), we have
Since , we have from (4.1)
Thus applying this to (4.8), we can finish the proof.
Let satisfy the conditions of Corollary 3.7 in a ball and for some . Then for , there exists a constant , independent of , such that
Putting , that is, into (3.25), we have
Taking a notice to another form of (3.3), we have
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
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:
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The research of the first author was supported by the NSF of China (no. 10971125) and was carried out during his visit to Seattle University, who would like to thank Professor Shusen Ding there for his warm encouragement and careful guidance.