The double sphere solution in the liquid drop model

Abstract

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 solve this problem. We construct an axially symmetric, non-minimizing solution that, for a sufficiently small $V>0$, resembles the union of two balls with volume $V/2$ connected by a tiny, approximately catenoidal neck with a width of the order $V^{\frac 43}$.