The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spaces

Abstract

We study the obstacle problem for unbounded sets in a proper metric measure space supporting a (p,p)-Poincare inequality. We prove that there exists a unique solution. We also prove that if the measure is doubling and the obstacle is continuous, then the solution is continuous, and moreover p-harmonic in the set where it does not touch the obstacle. This includes, as a special case, the solution of the Dirichlet problem for p-harmonic functions with Sobolev type boundary data.

18 pages

References in corpus (1)

• Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology

Cited by in corpus (7)

• Local and semilocal Poincaré inequalities on metric spaces
• The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces
• Classification of metric measure spaces and their ends using $p$-harmonic functions
• Resolutivity and invariance for the Perron method for degenerate equations of divergence type
• Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces
• Condenser capacities and capacitary potentials for unbounded sets, and global $p$-harmonic Green functions on metric spaces
• Semiregular and strongly irregular boundary points for $p$-harmonic functions on unbounded sets in metric spaces