标题： 一个紧性引理及其在液体滴模型极小化问题存在性中的应用 显示英文标题

标题： A compactness lemma and its application to the existence of minimizers for the liquid drop model

摘要： 古代伽莫夫核能液滴模型在变分法中成为一个有趣的问题而焕发新生：寻找一个体积为A的集合$\Omega \subset \mathbb R^3$，使其表面积和库仑自能之和最小。一个球体最小化前者并最大化后者，但猜想是球体总是最小值——当存在最小值时。对于这个有趣的几何问题，最小值的存在性尚未在一般情况下得到证明。我们证明了能量除以$A$（结合能每粒子）的绝对最小值（在所有$A$中）的存在性。我们工作的第二个结果是展示几何最小化问题中最优集存在性的通用方法，我们称之为“缺失质量法”。第三个要点是将拉回紧性引理从$W^{1,p}$扩展到$BV$。 显示英文摘要

摘要： The ancient Gamow liquid drop model of nuclear energies has had a renewed life as an interesting problem in the calculus of variations: Find a set $\Omega \subset \mathbb R^3$ with given volume A that minimizes the sum of its surface area and its Coulomb self energy. A ball minimizes the former and maximizes the latter, but the conjecture is that a ball is always a minimizer -- when there is a minimizer. Even the existence of minimizers for this interesting geometric problem has not been shown in general. We prove the existence of the absolute minimizer (over all $A$) of the energy divided by $A$ (the binding energy per particle). A second result of our work is a general method for showing the existence of optimal sets in geometric minimization problems, which we call the `method of the missing mass'. A third point is the extension of the pulling back compactness lemma from $W^{1,p}$ to $BV$.