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.

受最近关于约束相互作用能的最小化和梯度流的研究的启发，我们证明了这些能量是作为众所周知的聚集-扩散能的缓慢扩散极限而产生的。我们表明，聚集扩散能的极小值收敛到约束相互作用能的极小值，并且梯度流收敛到梯度流。我们的结果适用于各种相互作用势，包括奇异的吸引力和排斥力的幂律势。在获得慢扩散极限的过程中，我们还扩展了聚集扩散方程和Wasserstein梯度流的适定性理论，以接受广泛的非凸相互作用势。通过应用我们的结果得出一种受约束的相互作用能的数值方法，