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 $L^{\infty }$ and $L^p$ norms, for $p\ge 1$, of the torsion function, seen as a functional on a bounded simply connected open set $\Omega \subset {\mathbb{R}}^n$, and prove that the balls are critical shapes for these functionals, when the volume of $\Omega$ is preserved.