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

2014/09/23 by Cicalese, Marco, Leonardi, Gian Paolo, Maggi, Francesco
#49Q20 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.1409.6652

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 L¹ to a limit cluster has actually to converge in a strong C^1,α-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.

Related

• Improved convergence theorems for bubble clusters. II. The\n three-dimensional case
• Sticky-disk limit of planar N-bubbles
• Quantitative estimate on singularities in isoperimetric clusters
• On the connectedness of a minimizing cluster's boundary
• A sharp quantitative version of Hales' isoperimetric honeycomb theorem