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Advances in Calculus of Variations

Managing Editor: Duzaar, Frank / Kinnunen, Juha

Editorial Board: Armstrong, Scott N. / Balogh, Zoltán / Cardiliaguet, Pierre / Dacorogna, Bernard / Dal Maso, Gianni / DiBenedetto, Emmanuele / Fonseca, Irene / Gianazza, Ugo / Ishii, Hitoshi / Kristensen, Jan / Manfredi, Juan / Martell, Jose Maria / Mingione, Giuseppe / Nystrom, Kaj / Riviére, Tristan / Schaetzle, Reiner / Shen, Zhongwei / Silvestre, Luis / Tonegawa, Yoshihiro / Touzi, Nizar / Wang, Guofang


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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 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.

Keywords: Metric measure space; bounded variation; zero boundary values; variational capacity; outer capacity; quasi-semicontinuity

MSC 2010: 30L99; 31E05; 26B30

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About the article


Received: 2018-05-01

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, DOI: https://doi.org/10.1515/acv-2018-0024.

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