  We investigate the effect of linear diffusion and interactions with the
domain boundary on swarm equilibria by analyzing critical points of the
associated energy functional. Through this process we uncover two properties of
energy minimization that depend explicitly on the spatial domain: (i)
unboundedness from below of the energy due to an imbalance between diffusive
and aggregative forces depends explicitly on a certain volume filling property
of the domain, and (ii) metastable mass translation occurs in domains without
sufficient symmetry. From the first property, we present a sharp condition for
existence (resp. non-existence) of global minimizers in a large class of
domains, analogous to results in free space, and from the second property, we
identify that external forces are necessary to confine the swarm and grant
existence of global minimizers in general domains. We also introduce a
numerical method for computing critical points of the energy and give examples
to motivate further research.
