The variational 1-capacity and BV functions with zero boundary values on metric spaces

TL;DR

This paper develops a theory of BV functions with zero boundary values in metric spaces with doubling measures and Poincaré inequalities, and studies related variational capacities and their properties.

Contribution

It introduces a new class of BV functions with zero boundary values and analyzes their properties and associated capacities in metric spaces.

Findings

BV functions with zero boundary values form the closure of compactly supported BV functions.

Variational 1-capacity and its Lipschitz and BV analogs are outer capacities.

Different capacities coincide for certain sets.

Abstract

In the setting of a metric space that is equipped with a doubling measure and supports a Poincar\'e inequality, we define and study a class of BV functions with zero boundary values. In particular, we show that the class is the closure of compactly supported BV functions in the BV norm. Utilizing this theory, we then study the variational 1-capacity and its Lipschitz and BV analogs. We show that each of these is an outer capacity, and that the different capacities are equal for certain sets.