Discrete minimisers are close to continuum minimisers for the interaction energy

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.

37 pages, 1 figure

References in corpus (4)

• Next Order Asymptotics and Renormalized Energy for Riesz Interactions
• Existence of Ground States of Nonlocal-Interaction Energies
• On global minimizers of repulsive-attractive power-law interaction energies
• Some Free Boundary Problems involving Nonlocal Diffusion and Aggregation

Cited by in corpus (3)

• Many-particle limits and non-convergence of dislocation wall pile-ups
• Adhesion and volume constraints via nonlocal interactions lead to cell sorting
• Particle approximation of nonlocal interaction energies