Tuyển tập báo cáo các nghiên cứu khoa học quốc tế ngành hóa học dành cho các bạn yêu hóa học tham khảo đề tài: Research Article Some Caccioppoli Estimates for Differential Forms | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2009 Article ID 734528 11 pages doi 2009 734528 Research Article Some Caccioppoli Estimates for Differential Forms Zhenhua Cao Gejun Bao Yuming Xing and Ronglu Li Department of Mathematics Harbin Institute of Technology Harbin 150001 China Correspondence should be addressed to Gejun Bao baogj@ Received 31 March 2009 Accepted 26 June 2009 Recommended by Shusen Ding We prove the global Caccioppoli estimate for the solution to the nonhomogeneous A-harmonic equation d Afx u du Bfx u du which is the generalization of the quasilinear equation div A x u Vu B x u Vu . We will also give some examples to see that not all properties of functions maybe deduced to differential forms. Copyright 2009 Zhenhua Cao et al. 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. 1. Introduction The main work of this paper is study the properties of the solutions to the nonhomogeneous A-harmonic equation for differential forms d A x u du B x u du . When u is a 0-form that is u is a function is equivalent to div A x u Vu B x u Vu . In 1 Serrin gave some properties of when the operator satisfies some conditions. In 2 chapter 3 Heinonen et al. discussed the properties of the quasielliptic equations - div A x Vu 0 in the weighted Sobolev spaces which is a particular form of . Recently a large amount of work on the A-harmonic equation for differential forms has been done. In 1992 Iwaniec introduced the p-harmonic tensors and the relations between quasiregular mappings and the exterior algebra or differential forms in 3 . In 1993 Iwaniec and Lutoborski discussed the Poincare inequality for differential forms when 1 p n in 4 and the Poincare inequality for differential forms was generalized to p 1 in 5 . In 1999 Nolder gave .
