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The ancient Gamow liquid drop model of nuclear energies has had a renewedlife 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 itssurface area and its Coulomb self energy. A ball minimizes the former andmaximizes the latter, but the conjecture is that a ball is always a minimizer-- when there is a minimizer. Even the existence of minimizers for thisinteresting geometric problem has not been shown in general. We prove theexistence 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 generalmethod for showing the existence of optimal sets in geometric minimizationproblems, which we call the `method of the missing mass'. A third point is theextension of the pulling back compactness lemma from $W^{1,p}$ to $BV$.

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