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 ⊂ 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.
© 2015 Rupert L. Frank and Elliott H. Lieb. Received by the editors March 2, 2015; accepted for publication (in revised form) September 17, 2015; published electronically November 19, 2015. The work of this author was partially supported by U.S. National Science Foundation PHY-1347399 and DMS-1363432. The work of this author was partially supported by U.S. National Science Foundation PHY-1265118. Note added in proof. After our paper was submitted, a preprint by Knüpfer, Muratov, and Novaga appeared [KMN] with similar results but with a different methodology. Some of those results had been mentioned by Muratov at the Fields Center meeting in 2014. We are grateful to C. Muratov for helpful remarks on a previous version of this paper.
Submitted - 1503.00192v1.pdf
Published - 15m1010658.pdf