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

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

2011-04-05

arxiv.org/abs/1104.0797v1

Mathematics

Classical Analysis and ODEs

30 pages

Scientific paper

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 $(1,1)$-Poincar\'e 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.

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