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

  • Panu Lahti EMAIL logo

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 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.

MSC 2010: 30L99; 31E05; 26B30

Communicated by Frank Duzaar


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

Acknowledgements

Most of the research for this paper was conducted during the author’s visit to the University of Cincinnati, whose hospitality the author wishes to acknowledge. The author also wishes to thank Anders and Jana Björn for discussions on variational p-capacities, as well as the anonymous referee for suggestions on how to improve the paper.

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Received: 2018-05-01
Revised: 2018-09-12
Accepted: 2018-09-25
Published Online: 2018-10-21
Published in Print: 2021-04-01

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