Delaunay-like compact equilibria in the liquid drop model

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 $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the energy $$E(\Omega) = {\rm Per\,} (\Omega ) + \frac 12 \int_\Omega\int_\Omega \frac {dxdy}{|x-y|}$$ under the volume constraint $|\Omega| = m$. The term ${\rm Per\,} (\Omega )$ corresponds to the surface area of $\Sigma$. The associated Euler-Lagrange equation is $$H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \hbox{ for all } x\in \Sigma, \quad$$ where $H_\Sigma$ stands for the mean curvature of the surface, and where $\lambda\in\mathbb{R}$ is the Lagrange multiplier associated to the constraint $|\Omega|=m$. 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 $H_\Sigma = \lambda$, the only compact embedded solutions are spheres, as stated by the classical Alexandrov result.