tailieunhanh - báo cáo hóa học:" Research Article The Obstacle Problem for the A-Harmonic Equation"

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 The Obstacle Problem for the A-Harmonic Equation | Hindawi Publishing Corporation Journal of Inequalities and Applications Volume 2010 Article ID 767150 11 pages doi 2010 767150 Research Article The Obstacle Problem for the A-Harmonic Equation Zhenhua Cao 1 2 Gejun Bao 2 and Haijing Zhu3 1 Department of Mathematics Jiangxi Normal University Nanchang 330022 China 2 Department of Mathematics Harbin Institute of Technology Harbin 150001 China 3 College of Mathematics and Physics Shandong Institute of Light Industry Jinan 250353 China Correspondence should be addressed to Gejun Bao baogj@ Received 9 December 2009 Revised 26 March 2010 Accepted 31 March 2010 Academic Editor Shusen Ding Copyright 2010 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. Firstly we define an order for differential forms. Secondly we also define the supersolution and subsolution of the A-harmonic equation and the obstacle problems for differential forms which satisfy the A-harmonic equation and we obtain the relations between the solutions to A-harmonic equation and the solution to the obstacle problem of the A-harmonic equation. Finally as an application of the obstacle problem we prove the existence and uniqueness of the solution to the A-harmonic equation on a bounded domain Q with a smooth boundary dQ where the A-harmonic equation satisfies d A x du 0 x e Q u p x e 9Q where p is any given differential form which belongs to W1p Q A1-1 . 1. Introduction Recently a large amount of work about the A-harmonic equation for the differential forms has been done. In 1999 Nolder gave some properties for the solution to the A-harmonic equation in 1 and different versions of these properties had been established in 2-4 . The properties of the nonhomogeneous A-harmonic equation have been discussed in 5-10 . In the above papers we can think that the boundary