Bifurcations in Isoperimetric Problems with Nonlocal Interactions

TL;DR

This paper investigates bifurcations in isoperimetric problems with nonlocal interactions, revealing the existence of non-spherical solutions near certain radii and their absence elsewhere.

Contribution

It demonstrates the occurrence of bifurcations leading to non-spherical solutions in nonlocal isoperimetric problems at specific radii, expanding understanding of solution structures.

Findings

Non-spherical solutions bifurcate from balls at specific radii.

Bifurcating solutions can have arbitrarily large volume.

No bifurcation occurs outside the identified sequence of radii.

Abstract

We study isoperimetric problems modeled on the liquid drop model, with nonlocal interactions under a volume constraint. While balls are natural critical points, we show that, for an unbounded sequence of radii, non-spherical solutions bifurcate from the family of balls. These new solutions lie arbitrarily close to balls and can have arbitrarily large volume. Conversely, at radii outside this sequence, no bifurcation occurs, and nearby solutions are trivial, arising only from rigid motions.