Journal of Inequalities and Applications
Volume 2010 (2010), Article ID 767150, 11 pages
doi:10.1155/2010/767150
Research Article

The Obstacle Problem for the A-Harmonic Equation

Zhenhua Cao,1,2 Gejun Bao,2 and Haijing Zhu3

1Department of Mathematics, Jiangxi Normal University, Nanchang 330022, China
2Department of Mathematics, Harbin Institute of Technology, Harbin 150001, China
3College of Mathematics and Physics, Shandong Institute of Light Industry, Jinan 250353, China

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.

Abstract

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 Ω with a smooth boundary ∂Ω, where the A-harmonic equation satisfies d⋆A(x,du)=0,x∈Ω; u=ρ,x∈∂Ω, where ρ is any given differential form which belongs to W1,p(Ω,Λl-1).