Speaker: Monica Musso, University of Bath

Abstract for the mini course: Introduction to the Liquid Drop Model: Minimizers, Stability and Nonexistence

The liquid drop model, introduced by Gamow to describe atomic nuclei, is a classical problem in the calculus of variations involving the competition between surface tension, which favours compact shapes, and Coulomb repulsion, which favours spreading and fragmentation.

In this introductory lecture, I will present the mathematical formulation of the model and review some fundamental results concerning the existence and geometry of energy minimizers. Particular attention will be given to the role of balls, their minimality and stability properties, and the nonexistence of minimizers for sufficiently large masses. I will discuss the main mathematical ideas behind these results, as well as some open questions.

Abstract for the talk: Beyond Minimizers: Constructing Non-Spherical Equilibria

While balls are stationary configurations for the liquid drop energy at every volume, the existence of other critical points is a much more delicate question.

In this talk, I will discuss recent results on the existence and construction of smooth, compact, non-spherical equilibria of the liquid drop model, in both the small- and large-mass regimes. For small masses, I will describe the construction of configurations consisting of two nearly spherical regions connected by a very thin catenoidal neck. For large masses, I will present equilibria resembling necklaces of pearls, whose geometry is related to Delaunay unduloids. I will also discuss bifurcation phenomena from spherical equilibria, which give rise to further families of non-spherical stationary configurations.

The emphasis will be on the geometric ideas and analytical techniques behind these results, particularly singular perturbation methods and Lyapunov–Schmidt reduction. Together, these constructions reveal the richness of the critical point structure of the liquid drop energy, far beyond the theory of minimizers.

This work is in collaboration with M. del Pino, F. De Regibus, R. Frank, M. Grossi and A. Zuñiga.