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Sharp Stability for the Interaction Energy

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Abstract

This paper is devoted to stability estimates for the interaction energy with radially symmetric interaction potentials that are strictly decreasing in the radial variable, such as the Coulomb and Riesz potentials. For a general density function, we first prove a stability estimate in terms of the \(L^1\) asymmetry of the density, extending some previous results by Burchard–Chambers (Calc Var PDE 54(3):3241–3250, 2015; A stability result for Riesz potentials in higher dimensions. arXiv:2007.11664, 2020). Frank–Lieb (Ann Sc Norm Super Pisa Cl Sci XXII:1241–1263, 2021) and Fusco–Pratelli (ESAIM Control Optim Calcul Var 26:113, 2020) for characteristic functions. We also obtain a stability estimate in terms of the 2-Wasserstein distance between the density and its radial decreasing rearrangement. Finally, we consider the special case of Newtonian potential, and address a conjecture by Guo on the stability for the Coulomb energy.

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Notes

  1. The original statement in [15] focused on \(\rho =1_D\) with \(|D|=1\), but after a simple scaling argument one gets (1.6) for any \(|D|>0\).

  2. For Theorem 1.2–1.3, \(\text {supp }\,\rho \subset B(0,R)\) is already part of the assumption. For Theorem 1.1, although \(\rho \) is not assumed to have compact support (recall that we only assume \(\text {supp }\,\rho ^*\subset B(0,R_*)\)), we will show that the proof can be reduced to the case where \(\text {supp }\,\rho \subset B(0,R)\) for \(R=20R_*\).

  3. We leave it as an open question to the reader to obtain a more refined lower bound of the term \({\mathcal {E}}_{{{\tilde{W}}}}[\rho ^*] - {\mathcal {E}}_{{{\tilde{W}}}}[\rho ]\), instead of the crude estimate that it is nonnegative. Although this crude estimate will be sufficient for us to obtain the sharp power \(\alpha (\rho )^2\) in Theorem 1.1, it could result in a worse constant for certain potentials and densities, since the first parenthesis on the right hand side of (2.4) might dominate the second parenthesis (e.g. if W is singular near the origin, and \(\rho \) is concentrated near the origin).

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Acknowledgements

YY was partially supported by the NSF grants DMS-1715418, DMS-1846745, and Sloan Research Fellowship. The authors would like to thank Yan Guo and Pierre-Emmanuel Jabin for helpful discussions. The authors are grateful to the anonymous referees for their careful reading of the manuscript and valuable suggestions.

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Authors and Affiliations

  1. Department of Mathematics, Oklahoma State University, 401 Mathematical Sciences Building, Stillwater, OK, 74078, USA

    Xukai Yan

  2. Department of Mathematics, National University of Singapore, Block S17, 10 Lower Kent Ridge Road, Singapore, 119076, Singapore

    Yao Yao

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  1. Xukai Yan
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Correspondence to Yao Yao.

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Communicated by A. Figalli.

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Yan, X., Yao, Y. Sharp Stability for the Interaction Energy. Arch Rational Mech Anal 246, 603–629 (2022). https://doi.org/10.1007/s00205-022-01823-y

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