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 Ω ⊂ ℝ³ 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 [E. H. Lieb, Invent. Math., 74(1983), pp. 441-448] from W^1,p to BV.