An isoperimetric problem with a competing nonlocal singular term

TL;DR

This paper studies a variational problem balancing perimeter and a nonlocal singular term, revealing symmetry properties of minimizers and their behavior as volume varies, with implications for related inequalities.

Contribution

It introduces a new isoperimetric problem involving a nonlocal singular term and analyzes the symmetry and structure of its minimizers.

Findings

Minimizers exist and are radially symmetric for small volume.

Minimizers lose symmetry for large volume, either splitting or developing fingers.

Connections are made to a fractional Gagliardo-Nirenberg inequality.

Abstract

In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or develop fingers of a prescribed width. Finally, we connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality (a Gagliardo-Nirenberg type inequality for fractional perimeter).