Some properties of the torsion function with Robin boundary conditions

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