Spectral energies with repulsion

In the talk I will introduce some variational models where an aggregating term, like the perimeter or a Dirichlet-type energy, is in competition with a repulsive one. Examples of such models arise naturally in different fields of physics. It is the case of the Gamow [liquid drop] model and the Hartree energies in quantum mechanics, or the Rayleigh liquid charged drop model in electrowetting theories. I will give an overview of the recent strategies to get well-or-ill posedness of these energies. Then I will focus on a particular case -the reduced Hartree energy of the atom of Helium in a confined potential field- and show a strategy to characterize minimizers for such an energy. The talk is mostly motivated by an ongoing project with Dario Mazzoleni (Pavia) and Cyrill B. Muratov (Pisa).