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Asymptotic behavior of $BV$ functions and sets of finite perimeter in metric measure spaces
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by Sylvester Eriksson-Bique, James T. Gill, Panu Lahti and Nageswari Shanmugalingam PDF

Trans. Amer. Math. Soc. 374 (2021), 8201-8247 Request permission

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 $E$ 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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