The variational capacity with respect to nonopen sets in metric spaces

TL;DR

This paper systematically studies the variational capacity in metric spaces, especially for nonopen sets, establishing its properties, equivalences, and behavior under metric changes, with applications in potential theory.

Contribution

It introduces a comprehensive treatment of variational capacity for nonopen sets in metric spaces, including new definitions, properties, and examples, extending classical theory.

Findings

Variational capacity is a Choquet capacity under standard assumptions.

On open sets in weighted R^n, it matches classical variational capacity.

A related capacity is introduced that retains Choquet capacity properties on nonopen sets.

Abstract

We pursue a systematic treatment of the variational capacity on metric spaces and give full proofs of its basic properties. A novelty is that we study it with respect to nonopen sets, which is important for Dirichlet and obstacle problems on nonopen sets, with applications in fine potential theory. Under standard assumptions on the underlying metric space, we show that the variational capacity is a Choquet capacity and we provide several equivalent definitions for it. On open sets in weighted R^n it is shown to coincide with the usual variational capacity considered in the literature. Since some desirable properties fail on general nonopen sets, we introduce a related capacity which turns out to be a Choquet capacity in general metric spaces and for many sets coincides with the variational capacity. We provide examples demonstrating various properties of both capacities and counterexamples for when they fail. Finally, we discuss how a change of the underlying metric space influences the variational capacity and its minimizing functions.