Geodesic
Geometry of minimizers for the interaction energy with mildly repulsive potentials
Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1299-1308

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We show that the support of any local minimizer of the interaction energy consists of isolated points whenever the interaction potential is of class C2 and mildly repulsive at the origin; moreover, if the minimizer is global, then its support is finite. In addition, for some class of potentials we prove the validity of a uniform upper bound on the cardinal of the support of a global minimizer. Finally, in the one-dimensional case, we give quantitative bounds.

DOI : 10.1016/j.anihpc.2016.10.004
Keywords: Interaction energy, Local minimizers, Mild repulsion 
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     author = {Carrillo, J.A. and Figalli, A. and Patacchini, F.S.},
     title = {Geometry of minimizers for the interaction energy with mildly repulsive potentials},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     pages = {1299--1308},
     publisher = {Elsevier},
     volume = {34},
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     year = {2017},
     doi = {10.1016/j.anihpc.2016.10.004},
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     zbl = {1408.49035},
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Carrillo, J.A.; Figalli, A.; Patacchini, F.S. Geometry of minimizers for the interaction energy with mildly repulsive potentials. Annales de l'I.H.P. Analyse non linéaire, Tome 34 (2017) no. 5, pp. 1299-1308. doi: 10.1016/j.anihpc.2016.10.004

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