Calculus of Variations and Geometric Measure Theory

M. G. Mora

Nonlocal anisotropic interactions of Coulomb type

created by mora on 22 May 2023


Submitted Paper

Inserted: 22 may 2023
Last Updated: 22 may 2023

Year: 2023

Part of the material of this paper is based on the content of the course "Nonlocal interaction problems in dislocation theory", that the author taught at the Summer School on Analysis and Applied Mathematics in Münster in September 2022. The author would like to express her gratitude to the organisers and to the University of Münster for the support and the hospitality.


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.

Keywords: Potential theory, Nonlocal energy, anisotropic interaction, Coulomb potential