Global energy minimizers for a diffusion-aggregation model on sphere

We investigate the ground states of a free energy functional on sphere. The energy consists of an entropy and a nonlocal interaction term that are in competition with each other, as they favour spreading and aggregation, respectively. Specifically, the entropy corresponds to slow nonlinear diffusion and the interaction term is modeled by a quadratic interaction potential. We investigate the transitions that occur in the equilibria and the global minimizers of the energy, in terms of the strength of the nonlocal attractive interactions. We consider separately various ranges of the diffusion exponent, which give qualitatively different behaviours of equilibria and ground states. In terms of applications, we note that the energy we consider here is a generalization to nonlinear diffusion of the Onsager free energy with dipolar potential, used to study phase transitions in polymer orientation.