Annales Academiæ Scientiarum Fennicæ
Mathematica
Volumen 37, 2012, 35-51

NEIGHBOURHOOD CAPACITIES

Juha Lehrbäck

University of Jyväskylä, Department of Mathematics and Statistics
P.O. Box 35 (MaD), FI-40014 University of Jyväskylä, Finland; juha.lehrback 'at' jyu.fi

Abstract. We study the behaviour of the p-capacity of a compact set E with respect to the t-neighbourhoods of E as t varies. We establish sharp upper and lower bounds for these capacities in terms of Minkowski and Hausdorff type contents of E, respectively, and our results hold in both Euclidean and more general metric spaces. In our lower bounds the porosity of the set E plays an important role, and it is shown by examples that an assumption like this is in general necessary. In addition, we present a self-contained approach to the theory of sets of zero capacity in metric spaces.

2010 Mathematics Subject Classification: Primary 31C15, 28A12.

Key words: Variational capacity, Hausdorff content, Minkowski content, metric space, porosity.

Reference to this article: J. Lehrbäck: Neighbourhood capacities. Ann. Acad. Sci. Fenn. Math. 37 (2012), 35-51.

Full document as PDF file

doi:10.5186/aasfm.2012.3704