Axisymmetry of critical points for the Onsager functional

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.

14 pages, to be published in Phil. Trans. Royal Society A

References in corpus (1)

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

Cited by in corpus (3)

• Solution landscape of the Onsager model identifies non-axisymmetric critical points
• Generalization of the Maier-Saupe theory to the ferroelectric nematic phase
• Modeling and Computation of Liquid Crystals