Semiregular and strongly irregular boundary points for $p$-harmonic functions on unbounded sets in metric spaces

TL;DR

This paper classifies boundary points for p-harmonic functions in unbounded metric spaces, distinguishing regular, semiregular, and strongly irregular points, and provides characterizations of these types.

Contribution

It introduces a trichotomy for boundary points in unbounded sets within metric spaces and characterizes semiregular and strongly irregular points using various analytical tools.

Findings

Established local properties of boundary point types.

Provided characterizations of semiregular points via capacity and measures.

Extended the classification to unbounded sets in metric spaces.

Abstract

The trichotomy between regular, semiregular, and strongly irregular boundary points for $p$-harmonic functions is obtained for unbounded open sets in complete metric spaces with a doubling measure supporting a $p$-Poincar\'e inequality, $1<p<\infty$. We show that these are local properties. We also deduce several characterizations of semiregular points and strongly irregular points. In particular, semiregular points are characterized by means of capacity, $p$-harmonic measures, removability, and semibarriers.