无穷双气泡定理
An Infinite Double Bubble Theorem
AI总结：
本文证明了双气泡定理的一个变体，其中两个腔室具有无穷体积，证明了加权透镜簇是局部最小化的，并在特定对称性和增长假设下确立了其唯一性，同时得到了局部最小化 (N,2)-cluster 的闭包定理。
摘要：
经典的双气泡定理刻画了将 $\mathbb{R}^n$ 划分为三个腔室的最小化分区，其中两个腔室具有规定的有限体积。在本文中，我们证明了双气泡定理的一个变体，其中两个腔室具有无穷体积。这种构型是 (1,2)-cluster 的一个例子，即将 $\mathbb{R}^n$ 划分为三个腔室的分区，其中两个腔室具有无穷体积，仅有一个腔室具有有限体积。如果对于任意 $B_r(0)$，在所有支撑紧包含于 $B_r(0)$ 且保持 $\mathscr{X}(1)$ 体积的变分中，它能最小化界面能 $\sum_{j<k} c_{jk} \mathscr{H}^n(\partial \mathscr{X}(j) \cap \partial\mathscr{X}(k) \cap B_r(0))$，则该 $(1,2)$-cluster 相对于一族权重 $\{c_{jk}\}$ 是局部最小化的。对于 $(1,2)$-cluster，加权双气泡的类似物是加权透镜簇，我们证明了它是局部最小化的。此外，在包含等权重情形的关于 $\{c_{jk}\}$ 的对称性假设下，当 $n\leq 7$ 时，加权透镜簇是 $\mathbb{R}^n$ 中唯一的局部最小化子；在自然增长假设下，当 $n\geq 8$ 时在 $\mathbb{R}^n$ 中也具有相同的唯一性。我们还得到了局部最小化 $(N,2)$-cluster 的闭包定理。
英文摘要：
The classical double bubble theorem characterizes the minimizing partitions of $\mathbb{R}^n$ into three chambers, two of which have prescribed finite volume. In this paper we prove a variant of the double bubble theorem in which two of the chambers have infinite volume. Such a configuration is an example of a (1,2)-cluster, or a partition of $\mathbb{R}^n$ into three chambers, two of which have infinite volume and only one of which has finite volume. A $(1,2)$-cluster is locally minimizing with respect to a family of weights $\{c_{jk}\}$ if for any $B_r(0)$, it minimizes the interfacial energy $\sum_{j<k} c_{jk} \mathscr{H}^n(\partial \mathscr{X}(j) \cap \partial\mathscr{X}(k) \cap B_r(0))$ among all variations with compact support in $B_r(0)$ which preserve the volume of $\mathscr{X}(1)$. For $(1,2)$ clusters, the analogue of the weighted double bubble is the weighted lens cluster, and we show that it is locally minimizing. Furthermore, under a symmetry assumption on $\{c_{jk}\}$ that includes the case of equal weights, the weighted lens cluster is the unique local minimizer in $\mathbb{R}^n$ for $n\leq 7$, with the same uniqueness holding in $\mathbb{R}^n$ for $n\geq 8$ under a natural growth assumption. We also obtain a closure theorem for locally minimizing $(N,2)$-clusters.
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