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 ΩΩ of RnRn 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 ΩΩ), while in the case of the normalized operator they are new. In the normalized case, we also show that stadium-like domains are precisely the unique convex sets in RnRn where the solution to a Dirichlet problem is of class C1,1(Ω)C1,1(Ω).