Sharp Stability for the Interaction Energy

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.