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

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 $Ω$ must be close to an annulus.