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Discrete minimisers are close to continuum minimisers for the interaction energy

  • J. A. Cañizo
  • F. S. Patacchini
Article
  • 147 Downloads

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 iscrete analogue containing all the discrete minimisers.

Mathematics Subject Classification

35A15 35Q70 49M25 82B21 

Notes

Acknowledgements

J. A. Cañizo was supported by the Spanish Ministerio de Economía y Competitividad and the European Regional Development Fund (ERDF/FEDER), project MTM2014-52056-P. F. S. Patacchini thanks Imperial College London for supporting his PhD studies via a Roth studentship. The authors wish to thank the Mittag-Leffler Institute for their support during the programme “Interactions between Partial Differential Equations & Functional Inequalities” from 1 September to 16 December 2016, when progress on this paper was made. The authors are grateful to José Antonio Carrillo for his insightful comments on earlier versions of this paper.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2017

Authors and Affiliations

  1. 1.Departamento de Matemática Aplicada, Facultad de CienciasAvenida de Fuentenueva S/NGranadaSpain
  2. 2.Department of Mathematical SciencesCarnegie Mellon UniversityPittsburghUSA

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