In this paper, we investigate the minimization of a functional in which the usual perimeter is competing with a nonlocal singular term comparable (but not necessarily equal to) a fractional perimeter. The motivation for this problem is a cell motility model introduced in some previous work by the first author. We establish several facts about global minimizers with a volume constraint. In particular we prove that minimizers exist and are radially symmetric for small mass, while minimizers cannot be radially symmetric for large mass. For large mass, we prove that the minimizing sequences either split into smaller sets that drift to infinity or must develop intricate non-symmetrical shape. Finally, we connect these two alternatives to a related minimization problem for the optimal constant in a classical interpolation inequality (a Gagliardo–Nirenberg type inequality for fractional perimeter).

在本文中，我们研究了一个功能的最小化，在该功能中，通常的周长与可比拟的（但不一定等于）分数周长的非局部奇异项竞争。这个问题的动机是由第一作者在先前的一些工作中引入的细胞运动模型。我们建立了有关具有体积限制的全局最小化器的几个事实。特别地，我们证明了最小化器存在并且对于小质量是径向对称的，而对于大质量，最小化器不能是径向对称的。对于大质量，我们证明了最小化序列要么分裂成较小的集合，然后漂移到无穷大，要么必须发展出复杂的非对称形状。最后，