Remark on a nonlocal isoperimetric problem

TL;DR

This paper investigates a nonlocal isoperimetric problem involving the Newtonian potential, establishing regularity of critical sets and showing that small nonlocal effects favor spherical shapes as stable solutions.

Contribution

It proves the analyticity of regular critical sets and demonstrates the uniqueness of the ball as the stable critical set when nonlocal effects are weak.

Findings

Critical sets are analytic.

The ball is the unique stable critical set for small nonlocal strength.

Regularity results extend to the Ohta-Kawasaki functional.

Abstract

We consider isoperimetric problem with a nonlocal repulsive term given by the Newtonian potential. We prove that regular critical sets of the functional are analytic. This optimal regularity holds also for critical sets of the Ohta-Kawasaki functional. We also prove that when the strength of the nonlocal part is small the ball is the only possible stable critical set.