On well-posedness of variational models of charged drops

Electrified liquids are well known to be prone to a variety of interfacial instabilities that result in the onset of apparent interfacial singularities and liquid fragmentation. In the case of electrically conducting liquids, one of the basic models describing the equilibrium interfacial configurations and the onset of instability assumes the liquid to be equipotential and interprets those configurations as local minimizers of the energy consisting of the sum of the surface energy and the electrostatic energy. Here we show that, surprisingly, this classical geometric variational model is mathematically ill-posed irrespective of the degree to which the liquid is electrified. Specifically, we demonstrate that an isolated spherical droplet is never a local minimizer, no matter how small is the total charge on the droplet, as the energy can always be lowered by a smooth, arbitrarily small distortion of the droplet's surface. This is in sharp contrast to the experimental observations that a critical amount of charge is needed in order to destabilize a spherical droplet. We discuss several possible regularization mechanisms for the considered free boundary problem and argue that well-posedness can be restored by the inclusion of the entropic effects resulting in finite screening of free charges.