Rigidity results for variational infinity ground states

We prove two rigidity results for a variational infinity ground state $u$ of an open bounded convex domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. They state that $u$ coincides with a multiple of the distance from the boundary of $\mathrm{\Omega }$ if either $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }$ is constant on $\mathrm{\partial }\mathrm{\Omega }$, or $u$ is of class $C ^ {1,1}$outsidethehighridgeof$\Omega$\mathrm{.}Consequently,inbothcases$\Omega$ can be geometrically characterized as a “stadium-like domain”.