一个紧性引理及其在液滴模型极小解存在性中的应用
A compactness lemma and its application to the existence of minimizers for the liquid drop model
摘要：
古老的伽莫夫核能液滴模型作为变分法中的一个有趣问题重新焕发了生命：找到一个具有给定体积A的集合$Ω\subset \mathbb R^3$，使其表面积和库仑自能之和最小。一个球最小化前者，最大化后者，但猜想是，一个球总是一个最小化者-当有一个最小化者。即使是存在的极小这个有趣的几何问题还没有被证明一般。我们证明了能量除以$A$（每个粒子的结合能）的绝对极小值（在所有$A$上）的存在性。我们的工作的第二个结果是一个通用的方法来显示存在的最佳集的几何最小化问题，我们称之为“方法的失踪质量”。第三点是将拉回紧性引理从$W^{1,p}$推广到$BV$。
英文摘要：
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 $Ω\subset \mathbb R^3$ 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 $BV$.
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