The double sphere solution in the liquid drop model

We consider the problem of finding critical domains $Ω\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (Ω) = {\rm Per}\,(Ω) + \frac 12 \iint_{Ω\times Ω} \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|Ω|=V$. We look for smooth, embedded, compact surfaces $\partialΩ$ that