Abstract: 

In this talk I will present our treatment of a geometric variational problem arising from modeling the equilibrium shapes of liquid drops whose energy presents a competition of surface tension with the repulsive Coulombic energy of a fixed number of point charges inside the drop. The continuum analog of this problem in which the liquid is treated as a perfect conductor is known to be variationally ill-posed,hence the discrete nature of the charges preserved in our model presents a non-trivial regularization whose properties are far from obvious. In our model, we make a simplification of no dielectric contrast between the liquid and its surroundings, which nevertheless is an appropriate assumption for charged drops of liquid helium that are used in applications to quantum chemistry. For large numbers of charges, we identify a sharp charge threshold as the volume of the drop goes to infinity jointly with the number of charges. This threshold separates the regime of existence of minimizers from that of non-existence and turns out to be considerably lower than the onepredicted by Rayleigh for continuum charge distributions, and below the threshold the minimizer looks like a small perturbation of a ball with charges distributed approximately uniformly over the drop surface. Above the threshold, on the other hand, it is always convenient to evaporate a single charge from the drop and move it to infinity to lower energy. This is joint work with M. Novaga and P. Zaleski.