Some properties of functions of bounded variation in Carnot-Carathéodory spaces

Abstract:

Functions of bounded variations (BV for short) arise naturally in many problems in Calculus of Variations and their notion in general metric spaces has been received considerable attention, so far. One of the most fruitful environments in which a theory of BV functions has been partially extended is represented by Carnot-Carathéodory spaces (CC for short). 

CC spaces can be thought as RnRn endowed with a length metric for which admissible paths have a direction that is linear combination of mm fixed, bracket generating and linearly independent vector fields  (m≤nm≤n).

The first part of the seminar will be devoted to the definition of a CC space and the notions of approximate continuity, approximate differentiability and approximate jump in such a setting. Then, as in the classical Euclidean case, we will show that BV functions are approximate differentiable almost everywhere and that their jump set is rectifiable whenever the sets of finite perimeter of the CC space have rectifiable (reduced) boundary. Assuming a further  geometric hypothesis on the space we also show the validity of a decomposition formula for the distributional derivative of BV functions. Eventually, the validity of a Rank-One property for BV functions in a class of Carnot groups is shown. 

These are joint works with Annalisa Massaccesi and Davide Vittone.