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:
What was Poincaré's argument for (*)?
H. Poincaré, Sur une théorème de M. Liapounoff relatif a l’équilibre d’une masse fluide, Comptes Rendus de L’Academie des Sciences 104, 622–625 (1887).
As discussed by G.C. Evans, 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 (1930).
