  Inspired by recent work on minimizers and gradient flows of constrained
interaction energies, we prove that these energies arise as the slow diffusion
limit of well-known aggregation-diffusion energies. We show that minimizers of
aggregation-diffusion energies converge to a minimizer of the constrained
interaction energy and gradient flows converge to a gradient flow. Our results
apply to a range of interaction potentials, including singular attractive and
repulsive-attractive power-law potentials. In the process of obtaining the slow
diffusion limit, we also extend the well-posedness theory for
aggregation-diffusion equations and Wasserstein gradient flows to admit a wide
range of nonconvex interaction potentials. We conclude by applying our results
to develop a numerical method for constrained interaction energies, which we
use to investigate open questions on set valued minimizers.
