Minimizers for a nonlocal anisotropic energy

In this work, we study the equilibrium configurations of a system of interacting particles. We focus on nonlocal interactions of Coulomb type modified with the addition of a generic anisotropic term and consider general confinements, in a bidimensional setting. Using a $\Gamma$-convergence argument, we prove that, in the many-particle limit, the equilibrium configurations are given by the minimizers of an energy functional on the space of probability measures. Then, we discuss the existence and uniqueness of the minimizer for such energy, and review a recent result on the explicit characterisation of the minimizer in the case of quadratic confinement. To address the case of general confinements, where the analytic solution is not known, we introduce a novel numerical method for the approximation of the minimizer. Eventually, in the light of the numerical results, we make a conjecture on the shape of the minimizer for quartic confinement.