Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces

TL;DR

This paper investigates the asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces, revealing that tangent limits are Lipschitz functions of least gradient and asymptotic boundary sets are quasiminimal with regular perimeter measures.

Contribution

It establishes the asymptotic limits of BV functions as Lipschitz functions of least gradient and characterizes the tangent sets of finite perimeter sets as quasiminimal with regular perimeter measures.

Findings

Asymptotic limits of BV functions are Lipschitz functions of least gradient.

Almost every boundary point of a finite perimeter set has an asymptotic limit set that is quasiminimal.

Perimeter measure of asymptotic limit sets is Ahlfors co-dimension 1 regular.

Abstract

In this paper, we study the asymptotic behavior of BV functions in complete metric measure spaces equipped with a doubling measure supporting a $1$-Poincar\'e 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 $E$ of finite perimeter, there is an asymptotic limit set $(E)_\infty$ corresponding to the asymptotic expansion of $E$ and that every such asymptotic limit $(E)_\infty$ is a quasiminimal set of finite perimeter. We also show that the perimeter measure of $(E)_\infty$ is Ahlfors co-dimension $1$ regular.