A nonlocal isoperimetric problem with dipolar repulsion

TL;DR

This paper investigates a geometric variational problem involving perimeter and dipolar interactions, revealing conditions for minimizers and the effects of dipolar strength and regularization on their shape.

Contribution

It introduces a new nonlocal isoperimetric problem where the dipolar term localizes and influences the perimeter, providing existence results and characterizing minimizers.

Findings

Existence of generalized minimizers for all dipolar strengths, masses, and regularizations.

For small dipolar strength, minimizers are disks, and the limiting functional is a renormalized perimeter.

At critical dipolar strength, non-disk minimizers can occur with a modified dipolar kernel.

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.