Discrete minimisers are close to continuum minimisers for the interaction energy

TL;DR

This paper proves that discrete minimisers of interaction energy converge to continuum minimisers as the number of particles increases, using $ ext{Gamma}$-convergence and analyzing support and regularity properties.

Contribution

It establishes the convergence of discrete to continuum minimisers for interaction energies under technical conditions, introducing empirical Morrey measures as a key tool.

Findings

Discrete minimisers converge to continuum minimisers as particles increase.

Discrete interaction energy $ ext{Gamma}$-converges to continuum energy.

Continuum minimisers belong to Morrey spaces.

Abstract

Under suitable technical conditions we show that minimisers of the discrete interaction energy for attractive-repulsive potentials converge to minimisers of the corresponding continuum energy as the number of particles goes to infinity. We prove that the discrete interaction energy $\Gamma$-converges in the narrow topology to the continuum interaction energy. As an important part of the proof we study support and regularity properties of discrete minimisers: we show that continuum minimisers belong to suitable Morrey spaces and we introduce the set of empirical Morrey measures as a natural discrete analogue containing all the discrete minimisers.