Rough traces of BV functions in metric measure spaces

Abstract

Following a Maz'ya-type approach, we adapt the theory of rough traces of functions of bounded variation (BV ) to the context of doubling metric measure spaces supporting a Poincaré inequality. This eventually allows for an integration by parts formula involving the rough trace of such functions. We then compare our analysis with the study done in a recent work by Lahti and Shanmugalingam, where traces of BV functions are studied by means of the more classical Lebesgue-point characterization, and we determine the conditions under which the two notions coincide.

On $BV$ functions and essentially bounded divergence-measure fields in metric spaces

“…We refer to [23] for a more detailed exposition of the history of the divergence-measure fields and their applications in R n , and to [21,25,26,27,38] for some recent developments. It seems natural to investigate the possibility to extend the theory of divergence-measure fields and Gauss-Green formulas to non-Euclidean settings, and indeed there have been researches in this direction: in [15,16,40] the authors considered doubling metric measure spaces supporting a Poincaré inequality, while in [22] the class of horizontal divergencemeasure fields in stratified groups is studied. In particular, in [40] the authors employed the Cheeger differential structure (see [17]) to prove a Gauss-Green formula on the socalled regular balls, and later, in [15,16], a Maz'ya-type approach based on [39, Section 9.5], allowed to write a similar formula in terms of the rough trace of a BV function.…”

On BV functions and essentially bounded divergence-measure fields in metric spaces

“…It seems natural to investigate the possibility to extend the theory of divergencemeasure fields and Gauss-Green formulas to non-Euclidean settings, and indeed there has been some research in this direction: in [18,19,42], the authors considered doubling metric measure spaces supporting a Poincaré inequality, while in [25] the class of horizontal divergence-measure fields in stratified groups is studied; lastly, in the more recent paper [17], a Gauss-Green formula for sets of finite perimeter was proved in the context of an RCD.K; N / space by means of Sobolev vector fields in the sense of [36]. In particular, in [42] the authors employed the Cheeger differential structure (see [20]) to prove a Gauss-Green formula on the so-called regular balls, and later, in [18,19], a Maz'yatype approach based on [44], Section 9.5, allowed to write a similar formula in terms of the rough trace of a BV function. On the other hand, in [25] the authors exploited the approach developed in [26] and proved that it is possible to extend the method to stratified groups.…”