We develop an "improved convergence theorem" for a case study variational problem with singularities, namely, the isoperimetric problem on planar bubble clusters. We exploit this theorem in the description of isoperimetric clusters, possibly perturbed by a potential. Our methods are not specific to bubble clusters, and should provide a starting point to address similar issues in other variational problems where minimizers are known to possibly develop singularities. Further applications and extensions are discussed in companion papers.
Improved convergence theorems for bubble clusters. I. The planar case / Marco, Cicalese; Leonardi, Gian Paolo; Francesco, Maggi. - In: INDIANA UNIVERSITY MATHEMATICS JOURNAL. - ISSN 0022-2518. - ELETTRONICO. - 65:6(2016), pp. 1979-2050. [10.1512/iumj.2016.65.5932]
Improved convergence theorems for bubble clusters. I. The planar case
LEONARDI, Gian Paolo;
2016
Abstract
We develop an "improved convergence theorem" for a case study variational problem with singularities, namely, the isoperimetric problem on planar bubble clusters. We exploit this theorem in the description of isoperimetric clusters, possibly perturbed by a potential. Our methods are not specific to bubble clusters, and should provide a starting point to address similar issues in other variational problems where minimizers are known to possibly develop singularities. Further applications and extensions are discussed in companion papers.
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