Characterization of stadium-like domains via boundary value problems for the infinity Laplacian

We give a complete characterization, as “stadium-like domains”, of convex subsets $\mathrm{\Omega }$ of ${\mathbb{R}}^n$ where a solution exists to Serrin-type overdetermined boundary value problems in which the operator is either the infinity Laplacian or its normalized version. In case of the not-normalized operator, our results extend those obtained in a previous work, where the problem was solved under some geometrical restrictions on $\mathrm{\Omega }$. In case of the normalized operator, we also show that stadium-like domains are precisely the unique convex sets in ${\mathbb{R}}^n$ where the solution to a Dirichlet problem is of class $C^{1,1}(\mathrm{\Omega })$.