| Abstract: |
| We deal with the {\em liquid drop model}, introduced by Gamow (1930) and Bohr-Wheeler (1939) in nuclear physics to describe the structure of atomic nuclei. The problem consists of finding a surface $\Sigma =\partial \Omega$ in $\mathbb{R}^3$ that is critical for the following energy of regions $\Omega \subset \mathbb{R}^3$:
$$
{\mathcal E}(\Omega) = \hbox{Per }(\Omega ) + \frac 12 \int_{\Omega\times \Omega } \frac {dxdy}{|x-y|}
$$
under the volume constraint $|\Omega| = m$.
The associated Euler-Lagrange equation is
$$
H_\Sigma (x) + \int_{\Omega } \frac {dy}{|x-y|} = \lambda \quad \forall x\in \Sigma, \quad |\Omega| = m,
$$
where $\lambda$ is a constant Lagrange multiplier. 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 solutions with large volumes, consisting of ``pearl collars with an axis located on a large circle, with a shape close to a Delaunay`s unduloid surface with constant mean curvature. This is joint work with Monica Musso and Andr\`es Z\`u\~niga. |
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