The Obstacle Problem for the <inline-formula> <graphic file="1029-242X-2010-767150-i1.gif"/></inline-formula>-Harmonic Equation

• Main Authors: Bao Gejun, Zhu Haijing, Cao Zhenhua
• Format: Article
• Language: English
• Published: SpringerOpen 2010-01-01
• Series: Journal of Inequalities and Applications
• Online Access: http://www.journalofinequalitiesandapplications.com/content/2010/767150
• ISSN: 1025-5834; 1029-242X

<p/> <p>Firstly, we define an order for differential forms. Secondly, we also define the supersolution and subsolution of the <inline-formula> <graphic file="1029-242X-2010-767150-i2.gif"/></inline-formula>-harmonic equation and the obstacle problems for differential forms which satisfy the <inline-formula> <graphic file="1029-242X-2010-767150-i3.gif"/></inline-formula>-harmonic equation, and we obtain the relations between the solutions to <inline-formula> <graphic file="1029-242X-2010-767150-i4.gif"/></inline-formula>-harmonic equation and the solution to the obstacle problem of the <inline-formula> <graphic file="1029-242X-2010-767150-i5.gif"/></inline-formula>-harmonic equation. Finally, as an application of the obstacle problem, we prove the existence and uniqueness of the solution to the <inline-formula> <graphic file="1029-242X-2010-767150-i6.gif"/></inline-formula>-harmonic equation on a bounded domain <inline-formula> <graphic file="1029-242X-2010-767150-i7.gif"/></inline-formula> with a smooth boundary <inline-formula> <graphic file="1029-242X-2010-767150-i8.gif"/></inline-formula>, where the <inline-formula> <graphic file="1029-242X-2010-767150-i9.gif"/></inline-formula>-harmonic equation satisfies <inline-formula> <graphic file="1029-242X-2010-767150-i10.gif"/></inline-formula> where <inline-formula> <graphic file="1029-242X-2010-767150-i11.gif"/></inline-formula> is any given differential form which belongs to <inline-formula> <graphic file="1029-242X-2010-767150-i12.gif"/></inline-formula>.</p>