7 problems

Degond–Frouvelle–Liu conjecture. The only stable anisotropic equilibria occur when $k=1$ and $eta>eta_1^*$; consequently, every stable anisotropic equilibrium is axisymmetric. For…

Let $\rho_0$ be the isotropic state of the Onsager potential, and consider its bifurcations as the interaction parameter $\mu$ varies. A critical point is non-axisymmetric if it is…

Consider the Onsager potential and its oblate branch of axisymmetric critical points as the parameter $\mu$ increases. A secondary bifurcation is a bifurcation from this primary ob…

Consider the Onsager potential with interaction parameter $\mu>0$. For a positive integer $n$, let $D^{2n\mathrm{h}}$ denote the relevant horizontal dihedral symmetry group, and ca…

Let $\mu$ be the Onsager interaction parameter, and consider the local and global minimizers of the Onsager free-energy functional. A state is axisymmetric if its density is invari…

Fix $\sigma_0$ and suppose that the coupled dipolar/Maier–Saupe potential has a non-axisymmetric critical point at parameter $(\sigma_0,\kappa_0)$. Monotonicity conjecture. If a no…

Let $(\sigma_0,\kappa_0)$ be a parameter pair for the coupled dipolar/Maier–Saupe potential, and identify critical points that differ only by a rotation. A critical point is non-ax…