AI 中文总结
该研究探讨带有汤川势的液滴模型，证明了不同参数下极小化子的存在性、球的唯一性及稳定性阈值，克服了汤川核非齐次性的技术难题。
AI 中文摘要
我们研究周长泛函的长程扰动，该扰动由通过汤川核定义的非局部排斥项给出，并在体积约束下取极小值。由于该核呈指数衰减，表面张力与排斥力之间的竞争由两个独立量——屏蔽率和体积——决定。具体而言，我们证明：(i) 当屏蔽率高于一个显式临界值时，极小化子在任意体积下均存在，而此前的工作要求体积足够大；(ii) 当体积低于一个显式阈值时，极小化子在任意屏蔽率下均存在；(iii) 在小体积下，且对屏蔽率一致成立，球是唯一的极小化子（平移除外）；(iv) 存在一个依赖于屏蔽率的尖锐体积阈值，使得球为受体积约束的稳定临界点。该阈值对任意屏蔽率均以闭式给出，且当α=0时可约化为已知的无屏蔽值。我们的证明克服了汤川核缺乏齐次性带来的新技术挑战。
英文摘要
We study a long-range perturbation of the perimeter functional by a nonlocal repulsive term defined through the Yukawa kernel, minimized under a volume constraint. Because the kernel decays exponentially, the competition between surface tension and repulsion is governed by two independent quantities, the screening rate and the volume. Specifically we prove that (i) above an explicit critical screening rate, minimizers exist at every volume, whereas earlier work required the volume to be large; (ii) below an explicit volume threshold minimizers exist for every screening rate; (iii) at small volume, and uniformly in the screening rate, the ball is the unique minimizer up to translation; and (iv) there exists a sharp volume threshold, depending on the screening rate, for the ball to be a stable volume-constrained critical point. The threshold is given in closed form for every screening rate and reduces to the known unscreened value when $α=0$. Our proofs overcome new technical changes due to the lack of homogeneity in the Yukawa kernel.
Comments30 pages, comments welcome!
