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Archive for Rational Mechanics and Analysis

, Volume 217, Issue 3, pp 1197–1217 | Cite as

Existence of Compactly Supported Global Minimisers for the Interaction Energy

  • José A. Cañizo
  • José A. Carrillo
  • Francesco S. Patacchini
Open Access
Article

Abstract

The existence of compactly supported global minimisers for continuum models of particles interacting through a potential is shown under almost optimal hypotheses. The main assumption on the potential is that it is catastrophic, or not H-stable, which is the complementary assumption to that in classical results on thermodynamic limits in statistical mechanics. The proof is based on a uniform control on the local mass around each point of the support of a global minimiser, together with an estimate on the size of the "gaps" it may have. The class of potentials for which we prove the existence of global minimisers includes power-law potentials and, for some range of parameters, Morse potentials, widely used in applications. We also show that the support of local minimisers is compact under suitable assumptions.

Keywords

Probability Measure Global Minimiser Obstacle Problem Morse Potential Porous Medium Equation 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© The Author(s) 2015

Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.

Authors and Affiliations

  • José A. Cañizo
    • 1
  • José A. Carrillo
    • 2
  • Francesco S. Patacchini
    • 2
  1. 1.School of Mathematics, Watson BuildingUniversity of BirminghamBirminghamUK
  2. 2.Department of MathematicsImperial College LondonLondonUK

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