平面$N$泡的粘盘极限
Sticky-disk limit of planar $N$-bubbles
摘要：
我们研究平面$N$ -集群，最大限度地减少，面积约束下，加权周长$P_\varepsilon$取决于一个小参数$\varepsilon>0$。具体而言，我们将内部腔室之间的边界加权为$2-\varepsilon$，将内部腔室和外部腔室之间的边界加权为$1$。我们证明，作为$\varepsilon\to 0$极小的$P_\varepsilon$收敛到配置的不相交的磁盘，最大限度地提高相切的数量，每个加权的调和平均的半径的两个相切的磁盘。我们还获得了一些信息的结构上的极小$\varepsilon$。
英文摘要：
We study planar $N$-clusters that minimize, under an area constraint, a weighted perimeter $P_\varepsilon$ depending on a small parameter $\varepsilon>0$. Specifically we weight $2-\varepsilon$ the boundary between the interior chambers and $1$ the boundary between an interior chamber and the exterior one. We prove that as $\varepsilon\to 0$ minimizers of $P_\varepsilon$ converge to configurations of disjoint disks that maximize the number of tangencies, each weighted by the harmonic mean of the radii of the two tangent disks. We also obtain some information on the structure of minimizers for small $\varepsilon$.
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