Serrin-type overdetermined problem for the $p$-Laplacian with Robin boundary conditions

Abstract

Let $Ω$ be an open bounded connected subset of $\mathbb{R}^N$, $N \geq 2$, of class $C^{2,α}$, for $α\in (0,1)$. Let $p \geq 2$ and $β>0$. We prove the symmetry of the solution of the $p$-torsion problem with Robin boundary condition subject to the natural overdetermined condition coming from a shape derivative argument and to an extra condition on $β$ and on the minimum of the principal curvatures of $\partialΩ$. The proof is based on some new integral identities, involving the linearized operator of the $p$-Laplacian applied to the standard $P$-function. In passing we prove some other rigidity results in the spirit of Serrin's and Alexandrov's Theorems.