On the stability of the annulus for the torsion of multiply connected domains

TL;DR

This paper proves that domains with nearly optimal torsional rigidity are geometrically close to an annulus, providing a quantitative stability result for the isoperimetric inequality in multiply connected domains.

Contribution

It introduces a quantitative stability estimate for the isoperimetric inequality related to torsion in multiply connected domains, characterizing how near-optimal torsion implies proximity to an annulus.

Findings

Domains with near-optimal torsion are close to an annulus in shape.

The result quantifies the stability of the isoperimetric inequality for torsion.

Optimal shape for torsion among multiply connected domains is an annulus.

Abstract

We establish a quantitative version of the isoperimetric inequality for the torsion of multiply connected domains, among sets with given area and with given joint area of the holes. Since the optimal shape is the annulus, we investigate how a given domain approaches an annular configuration when its torsion is close to the optimal value. Our result shows that when the torsional rigidity is nearly optimal, the domain $\Omega$ must be close to an annulus.