Overdetermined Boundary Value Problems for the ∞-Laplacian

We consider overdetermined boundary value problems for the ∞-Laplacian in a domain Ω of ℝⁿ and discuss what kind of implications on the geometry of Ω the existence of a solution may have. The classical ∞-Laplacian, the normalized or game-theoretic ∞-Laplacian, and the limit of the p-Laplacian as p →∞are considered and provide different answers, even if we restrict our domains to those that have only web-functions as solutions.