Optimal design problems with repulsive term

Some new ideas in Calculus of Variation and Geometric Measure Theory allowed, in the last decade, to revisit from a precise mathematical point of view some physical models. Instances of such models are the Lord Rayleigh model of charged liquid drops in electrowetting, the liquid drop model by Gamow to describe nuclear fissions, the Hartree equations in atomic physics. In the seminar I will give a brief overview on such results. Later I will focus on recent results about some of those topics. The topic of the talk is partially based upon works in collaboration with M. Goldman, D. Mazzoleni, C.B. Muratov and M. Novaga.