Abstract

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a 1-Poincaré inequality. We show that at almost every point x outside the Cantor and jump parts of a BV function, the asymptotic limit of the function is a Lipschitz continuous function of least gradient on a tangent space to the metric space based at x. We also show that, at co-dimension 1 Hausdorff measure almost every measure-theoretic boundary point of a set (Ε) of finite perimeter, there is an asymptotic limit set Ε_∞ corresponding to the asymptotic expansion of Ε and that every such asymptotic limit (Ε)_∞ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of Ε_∞ is Ahlfors co-dimension 1 regular.