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.
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 ).
Theorem 1.2 (see ).
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 ).
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.
which is called the Hodge codifferential.
Theorem 2.1 (Hodge decomposition ).
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 ).
3. Priori Estimates for Solutions
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.
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 ).
Lemma 3.4 (Hölder's inequality ).
Proof of Theorem 3.1.
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
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.
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.
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.
- Agarwal RP, Ding S, Nolder C: Inequalities for Differential Forms. Springer, New York, NY, USA; 2009:xvi+387.View ArticleMATHGoogle Scholar
- Aronsson G, Lindqvist P: On -harmonic functions in the plane and their stream functions. Journal of Differential Equations 1988, 74(1):157–178. 10.1016/0022-0396(88)90022-8MathSciNetView ArticleMATHGoogle Scholar
- Ball JM: Convexity conditions and existence theorems in nonlinear elasticity. Archive for Rational Mechanics and Analysis 1976/77, 63(4):337–403. 10.1007/BF00279992MathSciNetView ArticleMATHGoogle Scholar
- Ding S: Two-weight Caccioppoli inequalities for solutions of nonhomogeneous -harmonic equations on Riemannian manifolds. Proceedings of the American Mathematical Society 2004, 132(8):2367–2375. 10.1090/S0002-9939-04-07347-2MathSciNetView ArticleMATHGoogle Scholar
- Ding S: Local and global norm comparison theorems for solutions to the nonhomogeneous -harmonic equation. Journal of Mathematical Analysis and Applications 2007, 335(2):1274–1293. 10.1016/j.jmaa.2007.02.048MathSciNetView ArticleMATHGoogle Scholar
- Nolder CA: Hardy-Littlewood theorems for -harmonic tensors. Illinois Journal of Mathematics 1999, 43(4):613–632.MathSciNetMATHGoogle Scholar
- Nolder CA: Global integrability theorems for -harmonic tensors. Journal of Mathematical Analysis and Applications 2000, 247(1):236–245. 10.1006/jmaa.2000.6850MathSciNetView ArticleMATHGoogle Scholar
- Nolder CA: Conjugate harmonic functions and Clifford algebras. Journal of Mathematical Analysis and Applications 2005, 302(1):137–142. 10.1016/j.jmaa.2004.08.008MathSciNetView ArticleMATHGoogle Scholar
- Stroffolini B: On weakly -harmonic tensors. Studia Mathematica 1995, 114(3):289–301.MathSciNetMATHGoogle Scholar
- Iwaniec T: -harmonic tensors and quasiregular mappings. Annals of Mathematics 1992, 136(3):589–624. 10.2307/2946602MathSciNetView ArticleMATHGoogle Scholar
- Gilbarg D, Trudinger NS: Elliptic Partial Differential Equations of Second Order, Classics in Mathematics. Springer, Berlin, Germany; 2001:xiv+517.MATHGoogle Scholar
- Heinonen J, Kilpeläinen T, Martio O: Nonlinear Potential Theory of Degenerate Elliptic Equations. Dover, Mineola, NY, USA; 2006:xii+404.MATHGoogle Scholar
- Stein EM: Harmonic Analysis: Real-Variable Methods, Orthogonality, and Oscillatory Integrals, Princeton Mathematical Series. Volume 43. Princeton University Press, Princeton, NJ, USA; 1993:xiv+695.Google Scholar
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