Calculus of Variations and Geometric Measure Theory

S. Eriksson-Bique - J. Gill - P. Lahti - N. Shanmugalingam

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

created by shanmugal on 12 Oct 2018
modified on 30 Jan 2020



Inserted: 12 oct 2018
Last Updated: 30 jan 2020

Year: 2018


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.

Tags: GeMeThNES