On the Serrin's problem with Robin boundary conditions

Abstract

Let $\hat{I}\text{c◯}\subset {\mathbb{R}}^N$, $N\ge 2$, be an open, connected, bounded set with $C^2$ boundary. In this paper we consider the torsion problem with Robin boundary conditions and we study the symmetry of the solutions when suitable extra conditions are imposed on the boundary of $\hat{I}\text{c◯}$. In particular, we prove the Serrin's rigidity result under suitable assumptions on the domain and on the Robin parameter.