A note to paper: Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential

TL;DR

This paper discusses the proof of a lemma related to stationary solutions of the Doi-Onsager equation on the sphere with Maier-Saupe potential, offering additional details and a simpler proof approach.

Contribution

It provides clarification and an alternative, simplified proof for a key lemma in the analysis of stationary solutions of the Doi-Onsager equation.

Findings

Additional details for Lemma 3.6 proof

Simpler proof based on existing arguments

Enhanced understanding of stationary solutions

Abstract

This note serves to provide additional details for the proof of Lemma 3.6 in our paper [Liu, Zhang and Zhang, Comm. Math. Sci., 3(2005), pp.201-218]. Moreover, we will also present an alternative, yet simpler, proof based on arguments in [Wang, Zhang and Zhang, CPAM, 68(2015), no. 8, 1326-1398].