An infinite double bubble theorem
Lia Bronsard
McMaster University, Hamilton, CanadaMichael Novack
Louisiana State University, Baton Rouge, USA

Abstract
The classical double bubble theorem characterizes the minimizing partitions of 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 into three chambers, two of which have infinite volume and only one of which has finite volume. A -cluster is locally minimizing with respect to a family of weights if for any , it minimizes the interfacial energy among all variations with compact support in which preserve the volume of . For 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 that includes the case of equal weights, the weighted lens cluster is the unique local minimizer in for , with the same uniqueness holding in for under a natural growth assumption. We also obtain a closure theorem for locally minimizing -clusters.
Cite this article
Lia Bronsard, Michael Novack, An infinite double bubble theorem. Ann. Inst. H. Poincaré C Anal. Non Linéaire (2025), published online first
DOI 10.4171/AIHPC/158