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arXiv · 2609.00108

The double sphere solution in the liquid drop model

Manuel del Pino·Rupert L. Frank·Monica Musso
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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 $V^{\frac 43}$.

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Manuel del Pino, Rupert L. Frank, Monica Musso. 2026-09-18. The double sphere solution in the liquid drop model. https://arxiv.org/abs/2609.00108

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