In this work, we give a cardinality estimation of the support of any global minimizer for the interaction energy with mildly repulsive potentials. For any global minimizer μ, the fact that supp(μ) consists of finitely many points was proved in Carrillo et al. (2017). Moreover, the authors found a cardinality estimation of the support of μ on R1. In order to advance some improvement of this result, we give an upper estimate of card(supp(μ)) in arbitrary finite dimension.
Bibliographical noteFunding Information:
Kyungkeun Kang’s work is supported by NRF, Republic of Korea - 2017R1A2B4006484 and NRF, Republic of Korea - 2015R1A5A1009350 . Hwa Kil Kim’s work was supported by NRF, Republic of Korea - 2018R1D1A1B07049357 . Geuntaek Seo’s work was partially supported by the Graduate School of YONSEI University, Republic of Korea Research Scholarship Grants in 2018. The authors are thankful to the anonymous referees for valuable suggestions.
© 2019 Elsevier B.V.
All Science Journal Classification (ASJC) codes
- Statistical and Nonlinear Physics
- Mathematical Physics
- Condensed Matter Physics
- Applied Mathematics