BV capacity and Hausdorff content

Abstract

In this thesis, we consider a complete and doubling metric measure space which supports a weak Poincaré inequality. We construct a set function called BV capacity, which is defined in terms of functions of bounded variation. When restricted to compact sets, this capacity is equivalent to a variational capacity which is defined in terms of functions in the first order Sobolev space with integrable gradients. Hence BV capacity is useful to characterise not only the space of bounded variation functions, but also this Sobolev space. We study the basic properties of BV capacity and we recognise its connection to the perimeter outer measure. The main purpose of this work is to understand how, for compact sets, the BV capacity and the Hausdorff content of codimension one are equivalent. The key tools to achieve this are the coarea formula, the relative isoperimetric inequality and a modified version of the boxing inequality for metric spaces.