For $\Omega\subset \mathbb{R}^3$ a region with $|\Omega| = |B\_1|$, let
$$
C(\Omega) = \int\_\Omega\int\_\Omega \frac{dxdy}{|x-y|}
$$
denote the Coulomb (or gravitational, etc) energy. 


Poincaré is credited with an incomplete proof that (\*) $C(\Omega) \leq C(B\_1)$. I know the (now standard) proof using symmetrization, but my impression is that this is not Poincaré's approach. My question is:



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Poincare's argument for maximizing the Coulomb energy
H. Poincaré, [Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide,](http://gallica.bnf.fr/ark:/12148/bpt6k30607/f621.item.r=Poincare) Comptes Rendus de L’Academie des Sciences **104**, 622–625 (1887).


As discussed by [G.C. Evans,](http://www.ams.org/journals/bull/1941-47-10/S0002-9904-1941-07541-5/S0002-9904-1941-07541-5.pdf)
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, [Über einige Extremalaufgaben der Potentialtheorie](https://eudml.org/doc/168197) (1930).


