- Research Article
- Open Access
Weighted Decomposition Estimates for Differential Forms
© Y. Wang and G. Li. 2010
- Received: 28 October 2009
- Accepted: 24 March 2010
- Published: 4 May 2010
After introducing the definition of -weights, we establish the -weighted decomposition estimates and -weighted Caccioppoli-type estimates for -harmonic tensors. Furthermore, by Whitney covering lemma, we obtain the global results in domain . These results can be used to study the integrability of differential forms and to estimate the integrals for differential forms.
- Differential Form
- Weight Estimate
- Nonlinear Elliptic Equation
- Star Operator
- Local Weight
where are two constants, and is a fixed exponent dependent on (1.2).
The solutions to (1.5) are called -harmonic tensors. See  for the recent research on -harmonic equation.
Caccioppoli-type estimates have been widely studied and frequently used in analysis and related fields, including partial differential equations and the theory of elasticity. These inequalities provide upper bounds for the -norm of if is a function or if is a form with the -norm of the differential form . Different versions of the Caccioppoli-type inequality have been established in [2, 3].
We first introduce the following definition of -weights (or the two-weight), and then establish the local -weighted Caccioppoli-type inequality for solutions to the homogeneous -harmonic equation.
where is a constant.
In , Nolder obtains the following local Caccioppoli-type estimate.
for all balls or cubes with and all closed forms . Here .
The next lemma is the generalized Hölder inequality which will be widely used in this paper.
for any measurable set .
The following weak reverse Hölder inequality plays an important role in founding the integral estimate of the nonhomogeneous and homogeneous -harmonic tensor; see .
for all balls or cubes with .
The following Whitney covering lemma appears in .
for all and some , where is the characteristic function for a set . Moreover, if , then there exists a cube (this cube does not need to be a member of ) in such that .
Now we are ready to prove the local weighted Caccioppoli-type inequality for homogeneous -harmonic tensors.
for all balls or cubes with and all closed forms .
Here is any constant with .
Since is a closed form and is a solution to (1.5), is still a solution to (1.5).
The proof of Theorem 2.6 is completed.
Since there are fore real parameters, , in Theorem 2.6, we can obtain some desired versions of weighted Caccioppoli-type estimates by different choices of them. Let in Theorem 2.6, we obtain the following corollaries.
for all closed forms .
Furthermore, choosing in Corollary 2.7, we get the following result.
for all closed forms .
The Hodge decomposition theorem has been playing an important part in partial differential equation, the operator theory, and so on. In the recent years, there are some interesting conclusions on the Hodge decomposition of differential forms; see [5, 6]. In , there is the following decomposition theorem for differential form .
here is a positive constant.
The next lemma states the reverse Hölder inequality for -weight; see .
where is a real number.
In this section, we will extend (3.2) to the weighted form.
here , , , , and is a constant.
for any .
So inequality (3.4) is true for .
Based on the local weighted estimate for the decomposition and the Whitney covering lemma, we get the global weighted estimate on domain .
where and is a constant.
By Lemma 2.5, has a modified Whitney cover of cubes . Let .
The proof of Theorem 4.1 is completed.
In Theorems 3.4 and 4.1, there is a real parameter , which makes the results more flexible. By choosing different value of the parameter , we get the estimates in different forms. For example, if we take , we have the following corollary.
where is a constant.
This work was supported by the National Natural Science Foundation of China (Grant no. 10771044) and the Natural Science Foundation of Heilongjiang Province (Grant no. 200605).
- Wang Y, Wu C: Global Poincaré inequalities for Green's operator applied to the solutions of the nonhomogeneous -harmonic equation. Computers & Mathematics with Applications 2004, 47(10–11):1545–1554. 10.1016/j.camwa.2004.06.006MATHMathSciNetView ArticleGoogle Scholar
- Ding S: Weighted Caccioppoli-type estimates and weak reverse Hölder inequalities for -harmonic tensors. Proceedings of the American Mathematical Society 1999, 127(9):2657–2664. 10.1090/S0002-9939-99-05285-5MATHMathSciNetView ArticleGoogle Scholar
- Perić I, Žubrinić D: Caccioppoli's inequality for quasilinear elliptic operators. Mathematical Inequalities and Applications 1999, 2(2):251–261.MATHMathSciNetGoogle Scholar
- Nolder CA: Hardy-Littlewood theorems for -harmonic tensors. Illinois Journal of Mathematics 1999, 43(4):613–631.MATHMathSciNetGoogle Scholar
- Scott C: theory of differential forms on manifolds. Transactions of the American Mathematical Society 1995, 347(6):2075–2096. 10.2307/2154923MATHMathSciNetGoogle Scholar
- Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Grundlehren der Mathematischen Wissenschaften. Volume 224. 2nd edition. Springer, Berlin, Germany; 1983:xiii+513.View ArticleGoogle Scholar
- Iwaniec T, Lutoborski A: Integral estimates for null Lagrangians. Archive for Rational Mechanics and Analysis 1993, 125(1):25–79. 10.1007/BF00411477MATHMathSciNetView ArticleGoogle Scholar
- Garnett JB: Bounded Analytic Functions. Academic Press, New York, NY, USA; 1970.Google Scholar
This article is published under license to BioMed Central Ltd. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.