Proof of spherical flocking based on quantitative rearrangement inequalities

TL;DR

This paper proves that in large mass regimes, the optimal density profile for spherical flocking is a perfect sphere, using a strict rearrangement inequality with quantitative error estimates derived from recent mathematical advances.

Contribution

It establishes that the ground state density profile in spherical flocking problems is a perfect sphere, employing new quantitative rearrangement inequalities.

Findings

The ground state density profile is a round ball.

Quantitative rearrangement inequalities are effective in this context.

The proof relies on recent deep results by M. Christ.

Abstract

Our recent work on the Burchard-Choksi-Topaloglu flocking problem showed that in the large mass regime the ground state density profile is the characteristic function of some set. Here we show that this set is, in fact, a round ball. The essential mathematical structure needed in our proof is a strict rearrangement inequality with a quantitative error estimate, which we deduce from recent deep results of M. Christ.