Sharp stability for the interaction energy

TL;DR

This paper establishes stability estimates for interaction energies with radially decreasing potentials, linking energy deviations to asymmetry and Wasserstein distance, and confirms a conjecture for Coulomb energy stability.

Contribution

It extends stability results to general densities, relating energy stability to asymmetry and Wasserstein distance, and addresses a conjecture for Coulomb energy stability.

Findings

Proved stability estimates using $L^1$ asymmetry.

Derived stability bounds in terms of 2-Wasserstein distance.

Confirmed Guo's conjecture for Coulomb energy stability.

Abstract

This paper is devoted to stability estimates for the interaction energy with strictly radially decreasing interaction potentials, 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, Frank-Lieb and Fusco-Pratelli 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.