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Communications in Mathematical Physics

, Volume 372, Issue 3, pp 1059–1115 | Cite as

A Nonlocal Isoperimetric Problem with Dipolar Repulsion

  • Cyrill B. MuratovEmail author
  • Thilo M. Simon
Article

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 \(\Gamma \)-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.

Notes

Acknowledgements

This work was supported by NSF via grant DMS-1614948. The authors wish to acknowledge valuable discussions with A. Bernoff, V. Julin, and M. Novaga.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Department of Mathematical SciencesNew Jersey Institute of TechnologyNewarkUSA

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