The Perron method for $p$-harmonic functions in unbounded sets in ${\mathbf{R}}^n$ and metric spaces

Abstract

The Perron method for solving the Dirichlet problem for $p$-harmonic functions is extended to unbounded open sets in the setting of a complete metric space with a doubling measure supporting a $p$-Poincaré inequality, $1<p<\infty$. The upper and lower ($p$-harmonic) Perron solutions are studied for $p$-parabolic open sets. It is shown that continuous functions and quasicontinuous Dirichlet functions are resolutive (i.e., that their upper and lower Perron solutions coincide) and that the Perron solution coincides with the $p$-harmonic extension. It is also shown that Perron solutions are invariant under perturbation of the function on a set of capacity zero.

References in corpus (4)

• Obstacle and Dirichlet problems on arbitrary nonopen sets, and fine topology
• The Dirichlet problem for p-harmonic functions with respect to the Mazurkiewicz boundary
• The obstacle and Dirichlet problems associated with p-harmonic functions in unbounded sets in Rn and metric spaces
• The Dirichlet problem for p-harmonic functions on the topologist's comb

Cited by in corpus (8)

• Local and semilocal Poincaré inequalities on metric spaces
• Sphericalization and p-harmonic functions on unbounded domains in Ahlfors regular metric spaces
• 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
• Uniqueness and nonuniqueness of $p$-harmonic Green functions on weighted ${\mathbf{R}}^n$ and 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
• The Perron method associated with finely $p$-harmonic functions on finely open sets