Some results on an isoperimetric model for charged liquid drops

We consider an isoperimetric model, originally proposed by Lord Rayleigh, aimed to describe the (lack of) equilibria of a liquid conducting drop in presence of a charge on its surface. The resulting functional contains an attracting term, usually modeled by the perimeter of the drop, and a repulsive term depending on the amount of charge considered and the electric capacity of the drop. We show that, quite surprisingly, the resulting variational problem is ill posed. We then consider several modification of it and we investigate existence, uniqueness and stability issues about those problem. The talk is based on works in collaborations with M. Goldman, C. Muratov and M. Novaga.