Cheeger集接触曲面的维数下界
Dimensional lower bounds for contact surfaces of Cheeger sets
摘要：
对Cheeger集$E$与其周围空间$Ω$的边界的接触面的大小进行了分析。我们发现，这个大小是密切相关的$\partial Ω$的正则性提供界的Hausdorff维数的$\partial E\cap \partialΩ$。特别是，我们表明，如果$\partial Ω$有$C^{1,α}$的规律性，那么$\mathcal{H}^{d-2+α}(\partial E\cap \partialΩ)>0$。这表明，确保$\mathcal{H}^{d-1}(\partial E\cap \partial Ω)>0$的充分条件是$\partial Ω$具有$C^{1,1}$正则性。由于Hausdorff界也可以根据$\partial E$的正则性来推断，我们得到了$Ω$凸，从而产生$\partial E\in C^{1,1}$，也是一个充分条件。最后，我们构造的例子表明，这样的界限是最佳的尺寸$d=2$。
英文摘要：
We carry on an analysis of the size of the contact surface of a Cheeger set $E$ with the boundary of its ambient space $Ω$. We show that this size is strongly related to the regularity of $\partial Ω$ by providing bounds on the Hausdorff dimension of $\partial E\cap \partialΩ$. In particular we show that, if $\partial Ω$ has $C^{1,α}$ regularity then $\mathcal{H}^{d-2+α}(\partial E\cap \partialΩ)>0$. This shows that a sufficient condition to ensure that $\mathcal{H}^{d-1}(\partial E\cap \partial Ω)>0$ is that $\partial Ω$ has $C^{1,1}$ regularity. Since the Hausdorff bounds can be inferred in dependence of the regularity of $\partial E$ as well, we obtain that $Ω$ convex, which yields $\partial E\in C^{1,1}$, is also a sufficient condition. Finally, we construct examples showing that such bounds are optimal in dimension $d=2$.
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