Delaunay-Like Compact Equilibria in the Liquid Drop Model

TL;DR

This paper discovers new equilibrium shapes in the liquid drop model that resemble a pearl necklace, challenging previous assumptions about possible solutions.

Contribution

The paper introduces a new class of compact, non-spherical solutions with large volumes in the liquid drop model.

Findings

A new class of compact, embedded solutions resembling a 'pearl necklace' is identified.

These solutions have a geometry similar to Delaunay's unduloid surface of constant mean curvature.

Such equilibria were previously thought to be non-existent due to classical results in constant mean curvature problems.

Abstract

The liquid drop model was introduced by Gamow in 1928 and Bohr–Wheeler in 1938 to model atomic nuclei. The model describes the competition between the surface tension, which keeps the nuclei together, and the Coulomb force, corresponding to repulsion among protons. More precisely, the problem consists of finding a surface \documentclass[12pt]{minimal}
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\begin{document}$$\Sigma =\partial \Omega $$\end{document}Σ=∂Ω in \documentclass[12pt]{minimal}
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\begin{document}$${\mathbb {R}}^3$$\end{document}R3 that is critical for the energy \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} {\mathcal {E}} (\Omega ) = {{{\textrm{Per}}}\,} (\Omega ) + \frac{1}{2} \int _\Omega \int _\Omega \frac{{\text {d}}x{\text {d}}y}{|x-y|} \end{aligned}$$\end{document}E(Ω)=Per(Ω)+12∫Ω∫Ωdxdy|x-y|under the volume constraint \documentclass[12pt]{minimal}
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\begin{document}$$|\Omega | = m$$\end{document}|Ω|=m. The term \documentclass[12pt]{minimal}
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\begin{document}$$\mathrm{Per\,} (\Omega ) $$\end{document}Per(Ω) corresponds to the surface area of \documentclass[12pt]{minimal}
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\begin{document}$$\Sigma $$\end{document}Σ. The associated Euler–Lagrange equation is \documentclass[12pt]{minimal}
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\begin{document}$$\begin{aligned} H_\Sigma (x) + \int _{\Omega } \frac{{\text {d}}y}{|x-y|} = \lambda \quad \hbox { for all } x\in \Sigma , \quad \end{aligned}$$\end{document}HΣ(x)+∫Ωdy|x-y|=λfor allx∈Σ,where \documentclass[12pt]{minimal}
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\begin{document}$$H_\Sigma $$\end{document}HΣ stands for the mean curvature of the surface, and where \documentclass[12pt]{minimal}
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\begin{document}$$\lambda \in {\mathbb {R}}$$\end{document}λ∈R is the Lagrange multiplier associated to the constraint \documentclass[12pt]{minimal}
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\begin{document}$$|\Omega |=m$$\end{document}|Ω|=m. Round spheres enclosing balls of volume m are always solutions; they are minimizers for sufficiently small m. Since the two terms in the energy compete, finding non-minimizing solutions can be challenging. We find a new class of compact, embedded solutions with large volumes, whose geometry resembles a “pearl necklace” with an axis located on a large circle, with a shape close to a Delaunay’s unduloid surface of constant mean curvature. The existence of such equilibria is not at all obvious, since, for the closely related constant mean curvature problem \documentclass[12pt]{minimal}
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\begin{document}$$H_\Sigma = \lambda $$\end{document}HΣ=λ, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.

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