On the minimizers of some energies containing Riesz-like terms

In this thesis, we analyze the Riesz-type functional present in some variational models, i.e. E(μ)=∫∫g(x-y)dμ(x)dμ(y), where μ is a probability measure, and g is a given interaction kernel. Concerning the minimization of the Riesz energy in the space of the probability measures, we consider attractive-repulsive kernels, and we study the regularity of the minimizers in a natural and rather general context. Under suitable assumptions on the kernel, we find some minimizing measures whose density with respect to the Lebesgue measure is uniformly bounded. Restricting the space dimension to 1, we also prove the continuity of such density. In both of the cases, some important conclusions are derived when the kernel is subharmonic, at least in some regions. Our work generalizes other results since we consider quite general kernels, in contrast with the power-like ones considered before.We consider also the same minimization problem set in the space of uniformly bounded densities with an additional mass constraint, and we investigate the saturation phenomena when the kernel is weakly repulsive. In fact, when the kernel is weakly repulsive in the origin, and the mass constraint is small, the height constraint is completely saturated as a result of a concentration effect. This result is obtained through a connection between this problem and the one set in the space of probability measures.Finally, we study a generalized Gamow model, where the Riesz kernel represents the electrostatic energy. In our generalization, the kernel is radial and radially decreasing (thus, it is completely repulsive), and we characterize the balls as the unique minimizers when the measure constraint is small enough. Moreover, we compute precisely the optimal families of balls for this functional, that constitute the ground state in some cases.