# Boundary regularity for $p$-harmonic functions and solutions of obstacle problems on unbounded sets in metric spaces

## TL;DR

This paper extends boundary regularity theory for p-harmonic functions and obstacle problems to unbounded sets in metric spaces, establishing local regularity criteria and barrier classifications.

## Contribution

It introduces boundary regularity results and barrier characterizations for p-harmonic functions on unbounded metric spaces, a novel extension of classical theory.

## Key findings

- Barrier classification of regular boundary points established
- Regularity is shown to be a local boundary property
- Boundary regularity results for obstacle problem solutions obtained

## Abstract

The theory of boundary regularity for $p$-harmonic functions is extended to unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. The barrier classification of regular boundary points is established, and it is shown that regularity is a local property of the boundary. We also obtain boundary regularity results for solutions of the obstacle problem on open sets, and characterize regularity further in several other ways.