Improved convergence theorems for bubble clusters. I. The planar case

created by maggi on 23 Sep 2014
modified on 21 Oct 2016

[BibTeX]

Accepted Paper

Inserted: 23 sep 2014
Last Updated: 21 oct 2016

Journal: Indiana U Math Journal
Year: 2015

Abstract:

We describe a quantitative construction of almost-normal diffeomorphisms between embedded orientable manifolds with boundary to be used in the study of geometric variational problems with stratified singular sets. We then apply this construction to isoperimetric problems for planar bubble clusters. In this setting we develop an improved convergence theorem, showing that a sequence of almost-minimizing planar clusters converging in $L^1$ to a limit cluster has actually to converge in a strong $C^{1,\alpha}$-sense. Applications of this improved convergence result to the classification of isoperimetric clusters and the qualitative description of perturbed isoperimetric clusters are also discussed. Analogous results for three-dimensional clusters are presented in part two, while further applications are discussed in some companion papers.