A Rigidity Result for the Robin Torsion Problem

Let $\mathrm{\Omega }\subset {\mathbb{R}}^2$ be an open, bounded and Lipschitz set. We consider the torsion problem for the Laplace operator associated to $\mathrm{\Omega }$ with Robin boundary conditions. In this setting, we study the equality case in the Talenti-type comparison, proved in arXiv:1909.11950. We prove that the equality is achieved only if $\mathrm{\Omega }$ is a disk and the torsion function $u$ is radial.