An infinite double bubble theorem

  • Lia Bronsard

    McMaster University, Hamilton, Canada
  • Michael Novack

    Louisiana State University, Baton Rouge, USA
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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