Geometry of prime end boundary and the Dirichlet problem for bounded domains in metric spaces

TL;DR

This paper investigates the Dirichlet problem in metric spaces using prime end boundaries, establishing resolutivity of certain boundary functions, introducing a new capacity concept, and demonstrating the approach's effectiveness through examples.

Contribution

It introduces a novel prime end boundary framework and a related capacity concept, extending Dirichlet problem solutions in metric spaces and Euclidean domains.

Findings

Functions continuous on prime end boundary are resolutive.

Bounded perturbations on zero-capacity sets do not affect solutions.

Examples show the approach's effectiveness in classical and new contexts.

Abstract

In this note we study the Dirichlet problem associated with a version of prime end boundary of a bounded domain in a complete metric measure space equipped with a doubling measure supporting a Poincare inequality. We show the resolutivity of functions that are continuous on the prime end boundary and are Lipschitz regular when restricted to the subset of all prime ends whose impressions are singleton sets. We also consider a new notion of capacity adapted to the prime end boundary, and show that bounded perturbations of such functions on subsets of the prime end boundary with zero capacity are resolutive and that their Perron solutions coincide with the Perron solution of the original functions. We also describe some examples which demonstrate the efficacy of the prime end boundary approach in obtaining new results even for the classical Dirichlet problem for some Euclidean domains.