Nonlocal anisotropic interactions of Coulomb type

Abstract: In this paper we review some recent results on nonlocal interaction problems. The focus is on interaction kernels that are anisotropic variants of the classical Coulomb kernel. In other words, while preserving the same singularity at zero of the Coulomb kernel, they present preferred directions of interaction. For kernels of this kind and general confinement we will prove existence and uniqueness of minimisers of the corresponding energy. In the case of a quadratic confinement we will review a recent result by Carrillo & Shu about the explicit characterisation of minimisers, and present a new proof, which has the advantage of being extendable to higher dimension. In light of this result, we will re-examine some previous works motivated by applications to dislocation theory in materials science. Finally, we will discuss some related results and open questions.