Some properties of the torsion function with Robin boundary conditions

In this paper we study some properties of the torsion function with Robin boundary conditions. Here we write the shape derivative of the $L^{\infty}$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded simply connected domain $\Omega \in \mathbb{R}^n$, and prove that the balls are a critical shape for these functionals, when the volume is preserved.