Axisymmetry of critical points for the Onsager functional

TL;DR

This paper provides a simplified proof that critical points of the Onsager functional are axisymmetric, extends smoothness results for critical points, and demonstrates the existence of non-axisymmetric critical points for various interactions.

Contribution

It offers a new proof avoiding spherical coordinates, generalizes smoothness of critical points, and shows non-axisymmetric solutions exist for diverse interactions.

Findings

Critical points are axisymmetric for Onsager functional.

All critical points are smooth under general conditions.

Non-axisymmetric critical points exist for various interactions.

Abstract

A simple proof is given of the classical result due to Fatkullin and Slastikov (2005), Liu, Zhang and Zhang (2005) that critical points for the Onsager functional with the Maier-Saupe molecular interaction are axisymmetric, including the case of stable critical points with an additional dipole-dipole interaction (Zhou et al 2007). The proof avoids spherical polar coordinates, instead using an integral identity on the sphere $S^2$. For general interactions with absolutely continuous kernels the smoothness of all critical points is established, generalizing a result of Vollmer (2016) for the Onsager interaction. It is also shown that non-axisymmetric critical points exist for a wide variety of interactions including that of Onsager.