Overdetermined problems for the normalized 𝑝-Laplacian

Abstract

We extend the symmetry result of Serrin \cite{S} and Weinberger \cite{W} from the Laplacian operator to the highly degenerate game-theoretic p p -Laplacian operator and show that viscosity solutions of − Δ p N u = 1 -\Delta _p^Nu=1 in Ω \Omega , u = 0 u=0 and ∂ u ∂ ν = − c ≠ 0 \tfrac {\partial u}{\partial \nu }=-c\neq 0 on ∂ Ω \partial \Omega can only exist on a bounded domain Ω \Omega if Ω \Omega is a ball.