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

philosophical transactions - royal society. mathematical, physical and engineering sciences/philosophical transactions - royal society. mathematical, physical and engineering sciences

Axisymmetry of critical points for the Onsager functional