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Overdetermined boundary value problems for the $\infty$-Laplacian

We consider overdetermined boundary value problems for the $\infty$-Laplacian in a domain $Ω$ of $\R^n$ and discuss what kind of implications on the geometry of $Ω$ the existence of a solution may have. The classical $\infty$-Laplacian, the normalized or game-theoretic $\infty$-Laplacian and the limit of the $p$-Laplacian as $p\to \infty$ are considered and provide different answers.