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

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 $BV$.