A Nonlocal Isoperimetric Problem with Dipolar Repulsion

Document Type

Article

Publication Date

12-1-2019

Abstract

We study a geometric variational problem for sets in the plane in which the perimeter and a regularized dipolar interaction compete under a mass constraint. In contrast to previously studied nonlocal isoperimetric problems, here the nonlocal term asymptotically localizes and contributes to the perimeter term to leading order. We establish existence of generalized minimizers for all values of the dipolar strength, mass and regularization cutoff and give conditions for existence of classical minimizers. For subcritical dipolar strengths we prove that the limiting functional is a renormalized perimeter and that for small cutoff lengths all mass-constrained minimizers are disks. For critical dipolar strength, we identify the next-order Γ -limit when sending the cutoff length to zero and prove that with a slight modification of the dipolar kernel there exist masses for which classical minimizers are not disks.

Identifier

85065260295 (Scopus)

Publication Title

Communications in Mathematical Physics

External Full Text Location

https://doi.org/10.1007/s00220-019-03455-y

e-ISSN

14320916

ISSN

00103616

First Page

1059

Last Page

1115

Issue

3

Volume

372

Grant

1614948

Fund Ref

National Science Foundation

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