论文标题
Cheeger组接触表面的尺寸下限
Dimensional lower bounds for contact surfaces of Cheeger sets
论文作者
论文摘要
我们对其环境空间$ω$的边界进行了对Cheeger Set $ e $的接触表面大小的分析。我们表明，通过在$ \ partial e \ cap \partialΩ$的hausdorff尺寸上提供界限，这种尺寸与$ \ partialω$的规律性密切相关。特别是，我们表明，如果$ \partialΩ$具有$ c^{1，α} $规则性，则$ \ mathcal {h}^{d-2+α}（\ partial e \ cap \partialΩ）> 0 $。这表明有足够的条件确保$ \ MATHCAL {h}^{d-1}（\ partial e \ cap \ partialω）> 0 $是$ \ partialω$具有$ c^{1,1} $ juropartial。由于可以根据$ \ partial e $的规律性来推断豪斯多夫的界限，因此我们获得了$ω$ cONVEX，它在c^{1,1} $中产生$ \ partial e \，也是一个足够的条件。最后，我们构建了示例，表明这种界限在尺寸中是最佳的$ d = 2 $。
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 $d=2$.
