The double sphere solution in the liquid drop model

Abstract

We consider the problem of finding critical domains $\Omega \subset {\mathbb{R}}^3$ for the energy functional $$\mathcal{E}(\Omega )=\mathrm{Per}(\Omega )+\frac{1}{2}{\iint }_{\Omega \times \Omega }\frac{dxdy}{|x-y|}$$ 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 $V^{\frac{4}{3}}$.