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

TL;DR

This paper extends the Perron method for solving the Dirichlet problem for p-harmonic functions to unbounded sets within complete metric spaces, establishing resolutivity and invariance properties.

Contribution

It generalizes the Perron method to unbounded open sets in metric spaces with doubling measures and Poincaré inequalities, including results on resolutivity and invariance.

Findings

Continuous and quasicontinuous functions are resolutive.

Perron solutions coincide with p-harmonic extensions.

Perron solutions are invariant under perturbations on capacity-zero sets.

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 $p$-Poincar\'e 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.