Sharp estimates for the Laplacian torsional rigidity with negative Robin boundary conditions

Motivated by pioneering works of Bandle and Wagner, given a bounded Lipschitz domain $\mathrm{\Omega }\subset {\mathbb{R}}^d$ with $d\ge 3$, we consider the Robin-Laplacian torsional rigidity ${\tau }_{\alpha }(\mathrm{\Omega })$ with negative boundary parameter $\alpha$ and we show that sharp inequalities for ${\tau }_{\alpha }(\mathrm{\Omega })$ hold if $\mathrm{\mid }\alpha \mathrm{\mid }$ is small enough. In particular, we prove that, if $\mathrm{\mid }\alpha \mathrm{\mid }$ is smaller than the first non-trivial Steklov-Laplacian eigenvalue, then the ball maximises ${\tau }_{\alpha }(\mathrm{\Omega })$ among all convex domains under perimeter or volume constraints.This solves an open problem raised by Bandle and Wagner. We also prove the result in the planar case among simply connected sets and under perimeter constraint.