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 $-\Delta_p^Nu=1$ in $\Omega$, $u=0$ and $\tfrac{\partial u}{\partial\nu}=-c\neq 0$ on $\partial\Omega$ can only exist on a bounded domain $\Omega$ if $\Omega$ is a ball.