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.

中文翻译：

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加倍度量空间上零边界值的变分 1-容量和 BV 函数

在配备加倍测度并支持庞加莱不等式的度量空间的设置下，我们定义并研究了一类边界值为零的 BV 函数。特别是，我们证明了该类是 BV 范数中紧凑支持的 BV 函数的闭包。利用这一理论，我们然后研究了变分 1-容量及其 Lipschitz 和 BV 类似物。我们证明了这些中的每一个都是外部容量，并且对于某些集合，不同的容量是相等的。