Rigidity of critical points for a nonlocal Ohta-Kawasaki energy

TL;DR

This paper studies the shape of critical points in a nonlocal energy model, proving conditions under which they are spherical and providing examples where symmetry does not hold without diameter constraints.

Contribution

It extends known results on global minimizers to all critical points under volume constraints and highlights the necessity of diameter constraints for symmetry.

Findings

Small volume critical points are spherical under certain conditions.

Existence of non-spherical critical points with small volume in one dimension.

Diameter constraints are essential for ensuring radial symmetry.

Abstract

We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers. We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.