Abstract
We study a measure-theoretic notion of connectedness for sets of finite perimeter in the setting of doubling metric measure spaces supporting a weak (1,1)-Poincaré inequality. The two main results we obtain are a decomposition theorem into indecomposable sets and a characterisation of extreme points in the space of BV functions. In both cases, the proof we propose requires an additional assumption on the space, which is called isotropicity and concerns the Hausdorff-type representation of the perimeter measure.
Highlights
The classical Euclidean theory of functions of bounded variation and sets of finite perimeter—whose cornerstones are represented, for instance, by [6,15,17,22,29,36] – has been successfully generalised in different directions, to several classes of metric structures
The basic theory of BV functions can be developed on abstract metric measure spaces, it is in the framework of doubling spaces supporting a weak (1, 1)-Poincaré inequality that quite a few fine properties are satisfied
The aim of the present paper is to study the notion of indecomposable set of finite perimeter on doubling spaces supporting a weak (1, 1)-Poincaré inequality
Summary
The classical Euclidean theory of functions of bounded variation and sets of finite perimeter—whose cornerstones are represented, for instance, by [6,15,17,22,29,36] – has been successfully generalised in different directions, to several classes of metric structures. In the remaining part of the Introduction, we will briefly describe our two main results: the decomposition theorem for sets of finite perimeter and the characterisation of extreme points in the space of BV functions. In both cases, the natural setting to work in is that of PI spaces satisfying an additional condition—called isotropicity—which we are going to describe in the following paragraph. As we will discuss in Example 1.31, the class of isotropic PI spaces includes weighted Euclidean spaces, Carnot groups of step 2 and non-collapsed RC D spaces Another key feature of the theory of sets of finite perimeter in PI spaces is given by the relative isoperimetric inequality (see Theorem 1.17 below), which has been obtained by M. We refer to the discussion at the beginning of Sect. 3.1
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