The double sphere solution in the liquid drop model

RESUMO

We consider the problem of finding critical domains $\Omega\subset{\mathbb R}^3$ for the energy functional $$ \mathcal E (\Omega) = {\rm Per}\,(\Omega) + \frac 12 \iint_{\Omega\times \Omega } \frac{dx\,dy}{|x-y|} $$ under the volume constraint $|\Omega|=V$. We look for smooth, embedded, compact surfaces $\partial\Omega$ 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 $V^{\frac 43}$.