Speaker
Description
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 one
predicted 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.