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. Nonlinearity 18, 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. Nonlinearity 20, 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 Analysis 226, 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'.