An infinite double bubble theorem

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