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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中文翻译：

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𝐵𝑉 函数的渐近行为和度量空间中的有限周长集

摘要：在本文中，我们研究了 BV 函数在完全度量测度空间中的渐近行为，该空间配备了支持 $1$-Poincaré 不等式的加倍测度。我们表明，在 BV 函数的 Cantor 和跳跃部分之外的几乎每个点 $x$ 处，函数的渐近极限是在基于 $x$ 的度量空间的切线空间上具有最小梯度的 Lipschitz 连续函数。我们还表明，在共同维度 $1$Hausdorff 测量有限周长的集合 $E$ 的几乎每个测度理论边界点，有一个渐近极限集 $(E)_\infty$ 对应于的渐近展开$E$ 并且每个这样的渐近极限 $(E)_\infty$ 都是有限周长的拟最小集。我们还表明 $(E)_\infty$ 的周长度量是 Ahlfors 共维 $1$ 正则。

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