论文标题
具有无限多个区域
Isoperimetric planar clusters with infinitely many regions
论文作者
论文摘要
$ \ mathbb r^d $中的无限群集$ \ mathbf e $是一系列不连接的可测量集，$ e_k \ subset \ subbb r^d $，$ k \ in \ mathbb n $，称为群集的区域。鉴于$ a_k \ ge 0 $ $ e_k $的$ a_k \ ge 0 $，一个自然的问题是存在一个有限且最小的周长$ p（\ mathbf e）$的群集$ \ mathbf e $中的所有群集中，在所有群集中，具有这样的区域。我们证明，在平面案例中存在这样的群集$ d = 2 $，对于任何选择的区域$ a_k $带有$ \ sum \ sqrt a_k <\ sqrt a_k <\ infty $。我们还显示了有限制的最小化器，其中属性$ p（\ mathbf e）= \ mathcal h^1（\ partial \ mathbf e）$，其中$ \ partial mathbf e $表示群集的理论边界。我们还提供了各向异性和分数周围的无限等级簇的几个例子。
An infinite cluster $\mathbf E$ in $\mathbb R^d$ is a sequence of disjoint measurable sets $E_k\subset \mathbb R^d$, $k\in \mathbb N$, called regions of the cluster. Given the volumes $a_k\ge 0$ of the regions $E_k$, a natural question is the existence of a cluster $\mathbf E$ which has finite and minimal perimeter $P(\mathbf E)$ among all clusters with regions having such volumes. We prove that such a cluster exists in the planar case $d=2$, for any choice of the areas $a_k$ with $\sum \sqrt a_k < \infty$. We also show the existence of a bounded minimizer with the property $P(\mathbf E)=\mathcal H^1(\partial \mathbf E)$, where $\partial mathbf E$ denotes the measure theoretic boundary of the cluster. We also provide several examples of infinite isoperimetric clusters for anisotropic and fractional perimeters.
