Functions of bounded variation on good metric spaces

In this paper we give a natural definition of Banach space valued BV functions defined on complete metric spaces endowed with a doubling measure (for the sake of simplicity we will say doubling metric spaces) supporting a Poincare inequality (see Definition 2.5 below). The definition is given starting from Lipschitz functions and taking closure with respect to a suitable convergence; more precisely, we define a total variation functional for every Lipschitz function; then we take the lower semicontinuous envelope with respect to the L-1 topology and define the BV space as the domain of finiteness of the envelope. The main problem of this definition is the proof that the total variation of any BV function is a measure; the techniques used to prove this fact are typical of F-convergence and relaxation. In Section 4 we define the sets of finite perimeter, obtaining a Coarea formula and an Isoperimetric inequality. In the last section of this paper we also compare our definition of B V functions with some definitions already existing in particular classes of doubling metric spaces, such as Weighted spaces, Ahlfors-regular spaces and Carnot-Caratheodory spaces. (C) 2003 Editions scientifiques et medicales Elsevier SAS. All rights reserved.