A Compactness Lemma and Its Application to the Existence of Minimizers for the Liquid Drop Model
Rupert L. Frank, Élliott H. Lieb · SIAM Journal on Mathematical Analysis · 2015
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 [E. H. Lieb, Invent. Math., 74 (1983), pp. 441--448] from $W^{1,p}$ to $BV$.