H. Poincaré, <A HREF="http://gallica.bnf.fr/ark:/12148/bpt6k30607/f621.item.r=Poincare">Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide,</A> Comptes Rendus de L’Academie des Sciences **104**, 622–625 (1887).

As discussed by <A HREF="http://www.ams.org/journals/bull/1941-47-10/S0002-9904-1941-07541-5/S0002-9904-1941-07541-5.pdf">G.C. Evans,</A>
Poincaré assumes tacitly that there do exist one or more bodies of a given volume, with smooth boundaries, which provide relative minima for the electrostatic capacity $C$ with respect to neighboring forms; and his treatment amounts to a proof that among these the sphere provides the absolute minimum. (Note that the Coulomb energy $U=Q^2/2C$, so minimal $C$ corresponds to maximal $U$ for given total charge $Q$.)

A complete proof, without this assumption, was given by G. Szegö, <A HREF="https://eudml.org/doc/168197">Über einige Extremalaufgaben der Potentialtheorie</A> (1930).