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

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.