Comparisons of relative BV-capacities and Sobolev capacity in metric spaces

Abstract

We study relations between the variational Sobolev 1-capacity and versions of variational BV-capacity in a complete metric space equipped with a doubling measure and supporting a weak $(1,1)$-Poincaré inequality. We prove the equality of 1-modulus and 1-capacity, extending the known results for $1<p<\infty$ to also cover the more geometric case $p=1$. Then we give alternative definitions for variational BV-capacities and obtain equivalence results between them. Finally we study relations between total 1-capacity and versions of BV-capacity.

30 pages