Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

Kyungkeun Kang; Hwa Kil Kim; Tongseok Lim; Geuntaek Seo

doi:10.1007/s00526-020-01882-7

Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

Kyungkeun Kang, Hwa Kil Kim, Tongseok Lim, Geuntaek Seo

Department of Mathematics

Research output: Contribution to journal › Article › peer-review

1 Citation (Scopus)

Abstract

In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive–attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the λ-Wasserstein metric topology with 1 ≤ λ< ∞, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of ∞-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.

Original language	English
Article number	15
Journal	Calculus of Variations and Partial Differential Equations
Volume	60
Issue number	1
DOIs	https://doi.org/10.1007/s00526-020-01882-7
Publication status	Published - 2021 Feb

Bibliographical note

Funding Information:
Kyungkeun Kang’s work is supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. Hwa Kil Kim’s work is supported by NRF-2018R1D1A1B07049357. Tongseok Lim gratefully acknowledges support from ShanghaiTech University, and in addition, TL is grateful for the support of the University of Toronto and its Fields Institute for the Mathematical Sciences, where parts of this work were performed. Geuntaek Seo’s work is supported by NRF-2017R1A2B4006484.

Publisher Copyright:
© 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

All Science Journal Classification (ASJC) codes

Analysis
Applied Mathematics

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10.1007/s00526-020-01882-7

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abstract = "In this paper, we are concerned with local minimizers of an interaction energy governed by repulsive–attractive potentials of power-law type in one dimension. We prove that sum of two Dirac masses is the unique local minimizer under the λ-Wasserstein metric topology with 1 ≤ λ< ∞, provided masses and distance of Dirac deltas are equally half and one, respectively. In addition, in case of ∞-Wasserstein metric, we characterize stability of steady-state solutions depending on powers of interaction potentials.",

author = "Kyungkeun Kang and Kim, {Hwa Kil} and Tongseok Lim and Geuntaek Seo",

note = "Funding Information: Kyungkeun Kang{\textquoteright}s work is supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. Hwa Kil Kim{\textquoteright}s work is supported by NRF-2018R1D1A1B07049357. Tongseok Lim gratefully acknowledges support from ShanghaiTech University, and in addition, TL is grateful for the support of the University of Toronto and its Fields Institute for the Mathematical Sciences, where parts of this work were performed. Geuntaek Seo{\textquoteright}s work is supported by NRF-2017R1A2B4006484. Publisher Copyright: {\textcopyright} 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.",

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N1 - Funding Information: Kyungkeun Kang’s work is supported by NRF-2019R1A2C1084685 and NRF-2015R1A5A1009350. Hwa Kil Kim’s work is supported by NRF-2018R1D1A1B07049357. Tongseok Lim gratefully acknowledges support from ShanghaiTech University, and in addition, TL is grateful for the support of the University of Toronto and its Fields Institute for the Mathematical Sciences, where parts of this work were performed. Geuntaek Seo’s work is supported by NRF-2017R1A2B4006484. Publisher Copyright: © 2021, The Author(s), under exclusive licence to Springer-Verlag GmbH, DE part of Springer Nature.

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Uniqueness and characterization of local minimizers for the interaction energy with mildly repulsive potentials

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