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# Advances in Calculus of Variations

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

Panu Lahti
Published Online: 2018-10-21 | DOI: https://doi.org/10.1515/acv-2018-0024

## Abstract

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

MSC 2010: 30L99; 31E05; 26B30

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Revised: 2018-09-12

Accepted: 2018-09-25

Published Online: 2018-10-21

The research was funded by a grant from the Finnish Cultural Foundation.

Citation Information: Advances in Calculus of Variations, ISSN (Online) 1864-8266, ISSN (Print) 1864-8258,

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