Calculus of Variations and Geometric Measure Theory

D. Carazzato - A. Pratelli - I. Topaloglu

On the existence of minimizing sets for a weakly-repulsive non-local energy

created by carazzato on 06 Jul 2023
modified by topaloglu1 on 27 Jan 2025

[BibTeX]

Published Paper

Inserted: 6 jul 2023
Last Updated: 27 jan 2025

Journal: Pure and Applied Analysis
Volume: 6
Number: 4
Pages: 995–1015
Year: 2024
Doi: https://doi.org/10.2140/paa.2024.6.995

ArXiv: 2307.01830 PDF

Abstract:

We consider a non-local interaction energy over bounded densities of fixed mass $m$. We prove that under certain regularity assumptions on the interaction kernel these energies admit minimizers given by characteristic functions of sets when $m$ is sufficiently small (or even for every $m$, in particular cases). We show that these assumptions are satisfied by particular interaction kernels in power-law form, and give a certain characterization of minimizing sets. Finally, following a recent result of Davies, Lim and McCann, we give sufficient conditions on the interaction kernel so that the minimizer of the energy over probability measures is given by Dirac masses concentrated on the vertices of a regular $(N+1)$-gon of side length 1 in $\mathbb{R}^N$.


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