The Double Obstacle Problem on Metric Spaces

• Main Author: Farnana, Zohra
• Format: Doctoral Thesis
• Language: English
• Published: Linköpings universitet, Tillämpad matematik 2009
• Subjects: MATHEMATICS; MATEMATIK
• Online Access: http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-51588; http://nbn-resolving.de/urn:isbn:978-91-85831-00-5

In this thesis we investigate the double obstacle problem for p-harmonic functions on metric spaces. We minimize the p-energy integral among all functions which have prescribed boundary values and lie between two given obstacles. This is a generalization of the Dirichlet problem for p-harmonic functions, in which case the obstacles are —∞ and ∞. We show the existence and uniqueness of solutions, and their continuity when the obstacles are continuous. Moreover we show that the continuous solution is p-harmonic in the open set where it does not touch the continuous obstacles. If the obstacles are not continuous, but satisfy a Wiener type regularity condition, we prove that the solution is still continuous. The Hölder continuity for solutions is shown, when the obstacles are Hölder continuous. Boundary regularity of the solutions is also studied. Furthermore we study two kinds of convergence problems for the solutions. First we let the obstacles and the boundary values vary and show the convergence of the solutions. We also consider generalized solutions for insoluble obstacle problems, using the convergence results. Moreover we show that for soluble obstacle problems the generalized solution coincides, locally, with the standard solution. Second we consider an increasing sequence of open sets, with union Ω, and fix the obstacles and the boundary values. We show that the solutions of the obstacle problems in these sets converge to the solution of the corresponding problem in Ω.