Back
 AM  Vol.4 No.1 , January 2013
An Extension of the Poincar’e Lemma of Differential Forms
Abstract: This paper is to extend the Poincar’e 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