AM  Vol.4 No.1 , January 2013
An Extension of the Poincar’e Lemma of Differential Forms
ABSTRACT

This paper is to extend the Poincare Lemma for differential forms in a bounded, convex domain [1] in Rn to a more general domain that, we call, is deformable to every point in itself. Then we extend the homotopy operator T in [1] to the domain defromed to every point of itself.


Cite this paper
Z. Tang, J. Zhu, J. Huang and J. Li, "An Extension of the Poincar’e Lemma of Differential Forms," Applied Mathematics, Vol. 4 No. 1, 2013, pp. 16-18. doi: 10.4236/am.2013.41004.
References
[1]   T. Iwaniec and A. Lutoborski, “Integral Estimates for Null Lagrangians,” Archive for Rational Mechanics and Analysis, Vol. 125, No. 1, 1993, pp. 25-79. doi:10 .1007/BF00411477

[2]   H. Flanders, “Differential Forms with Applications to the Physical Sciences,” Dover Publications, Mineola, New York, 1963.

[3]   P. R. Agarwal, S. Ding and C. A. Nolder, “Inequalities for Differential Forms,” Springer, New Mexico, 2009. doi:10.1007/978-0-387-68417-8

[4]   M. Spivak, “Calculus on Manifolds,” Perseus Books Publishing, New York, 1965.

[5]   J. Zhu and J. Li, “Some Priori Estimates about Solutions to Nonhomogeneous A-Harmonic Equations,” Journal of Inequalities and Applications, Vol. 2010, No. 520240, 2010, Article ID: 520240.

[6]   S. S. Ding and J. M. Zhu, “Poincar-Type Inequalities for the Homotopy Operator with Lφ-Norms,” Nonlinear Analysis: Theory, Methods and Applications, Vol. 74, No. 11, 2011, pp. 3728-3735.

[7]   J. Zhu, S. Ding and Z. Tang, “The Reverse Holder and Caccioppoli Type Inequalities for Generalized A-Harmonic Equations,” Under Review.

[8]   S. Ding, “Two-Weight Caccioppoli Inequalities for Solutions of Nonhomogeneous A-Harmonic Equations on Riemannian Manifolds,” Proceedings of the American Mathematical Society, Vol. 132, 2004, pp. 2367-2375. doi:10.1090/S0002-9939-04-07347-2

[9]   S. Ding, “Local and Global Norm Comparison Theorems for Solutions to the Nonhomogeneous A-Harmonic Equation,” Journal of Mathematical Analysis and Applications, Vol. 335, No. 2, 2007, pp. 1274-1293. doi:10.1016/j.jmaa.2007.02.048

[10]   M. Giaquinta and J. Soucek, “Caccioppoli’s Inequality and Legendre-Hadamard Condition,” Mathematische Annalen, Vol. 270, No. 1, 1985, pp. 105-107. doi:10.1007/BF01455535

[11]   T. Iwaniec and G. Sbordone, “Weak Minima of Variational Integrals,” Journal of Reine Angew Math, Vol. 454, 1994, pp. 143-161.

[12]   C. A. Nolder, “Hardy-Littlewood Theorems for A-Harmonic Tensors,” Illinois Journal of Mathematics, Vol. 43, 1999, pp. 613-631.

[13]   C. A. Nolder, “Global Integrability Theorems for A-Harmonic Tensors,” Journal of Mathematical Analysis and Applications, Vol. 247, No. 1, 2000, pp. 236-247. doi:10.1006/jmaa.2000.6850

[14]   C. A. Nolder, “Conjugate Harmonic Functions and Clifford Algebras,” Journal of Mathematical Analysis and Applications, Vol. 302, No. 1, 2005, pp. 137-142. doi:10.1016/j.jmaa.2004.08.008

[15]   B. Stroffolini, “On Weakly A-Harmonic Tensors,” Studia Mathematica, Vol. 114, 1995, pp. 289-301.

 
 
Top