Axisymmetry of critical points for the Onsager functional

A simple proof is given of the classical result (Fatkullin I, Slastikov V. 2005 Critical points of the Onsager functional on a sphere. Nonlinearity18, 2565-2580 (doi:10.1088/0951-7715/18/6/008); Liu H et al. 2005 Axial symmetry and classification of stationary solutions of Doi-Onsager equation on the sphere with Maier-Saupe potential. Commun. Math. Sci.3, 201-218 (doi:10.4310/CMS.2005.v3.n2.a7)) 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 H et al. 2007 Characterization of stable kinetic equilibria of rigid, dipolar rod ensembles for coupled dipole-dipole and Maier-Saupe potentials. Nonlinearity20, 277-297 (doi:10.1088/0951-7715/20/2/003)). The proof avoids spherical polar coordinates, instead using an integral identity on the sphere S2. For general interactions with absolutely continuous kernels the smoothness of all critical points is established, generalizing a result in (Vollmer MAC. 2017 Critical points and bifurcations of the three-dimensional Onsager model for liquid crystals. Archive for Rational Mechanics and Analysis226, 851-922 (doi:10.1007/s00205-017-1146-8)) for the Onsager interaction. It is also shown that non-axisymmetric critical points exist for a wide variety of interactions including that of Onsager. This article is part of the theme issue 'Topics in mathematical design of complex materials'.