A spherical rearrangement proof of the stability of a Riesz-type inequality and an application to an isoperimetric type problem

TL;DR

This paper proves the sharp stability of the ball as a minimizer for certain shape functionals using mass transportation, and applies this to an isoperimetric problem involving attraction, repulsion, and perimeter penalties.

Contribution

It introduces a spherical rearrangement proof of stability for Riesz-type inequalities and applies it to complex shape optimization problems with attraction and repulsion.

Findings

Stability exponent of 1/2 is sharp.

Ball minimizes a shape functional with attraction and repulsion.

Application to a fractional isoperimetric inequality.

Abstract

We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is $1/2$ and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.