Overdetermined problems for the normalized $p$-Laplacian

Abstract

We extend the symmetry result of Serrin and Weinberger from the Laplacian operator to the highly degenerate game-theoretic $p$-Laplacian operator and show that viscosity solutions of $-{\Delta }_p^{N}u=1$ in $\Omega$, $u=0$ and $\frac{\partial u}{\partial \nu }=-c\ne 0$ on $\partial \Omega$ can only exist on a bounded domain $\Omega$ if $\Omega$ is a ball.

revised version ( a few typos corrected)