- Research Article
- Open Access
Some Priori Estimates about Solutions to Nonhomogeneous A-Harmonic Equations
© J. Zhu and J. Li. 2010
- Received: 31 December 2009
- Accepted: 17 March 2010
- Published: 5 May 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.
- Weak Solution
- Differential Form
- Bounded Linear Operator
- Interpolation Theorem
- Quasiregular Mapping
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].
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 ).
if (1) and (2) for each test form .
Theorem 1.2 (see ).
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).
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 ).
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.
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.
where is a -form, is a permutation of , and is the signature of the permutation. The norm of is given by the formula .
The Sobolev space of -forms is
which is called the Hodge codifferential.
Theorem 2.1 (Hodge decomposition ).
for some constant independent of
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 ).
for almost every and all .
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.
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 ).
for any positive numbers .
Lemma 3.4 (Hölder's inequality ).
Proof of Theorem 3.1.
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.
where and stands for the integral mean over .
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.
where does not depend on and . Integrating (3.20) over , we get (3.17).
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).
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.
where the supremum is over all balls and is the Lebesgue measure of .
where and .
Thus applying this to (4.8), we can finish the proof.
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.
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