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arXiv · 1810.05310

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

Sylvester Eriksson-Bique·James T. Gill·Panu Lahti·Nageswari Shanmugalingam
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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.

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Sylvester Eriksson-Bique, James T. Gill, Panu Lahti, Nageswari Shanmugalingam. 2018-10-12. Asymptotic behavior of BV functions and sets of finite perimeter in metric measure spaces. https://arxiv.org/abs/1810.05310

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