A nonlocal isoperimetric problem with dipolar repulsion

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.

arXiv comment54 pages, 6 figures

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Cited by in corpus (4)

• On the Ternary Ohta-Kawasaki Free Energy and Its One-dimensional Global Minimizers
• Large mass rigidity for a liquid drop model in 2D with kernels of finite moments
• Second order expansion for the nonlocal perimeter functional
• Existence and nonexistence of minimizers for classical capillarity problems in presence of nonlocal repulsion and gravity