On the Dirichlet and Serrin problems for the inhomogeneous infinity Laplacian in convex domains: Regularity and geometric results

Given an open bounded subset $\mathrm{\Omega }$ of ${\mathbb{R}}^n$, which is convex and satisfies an interior sphere condition, we consider the pde $-\Delta_{\infty} u = 1$in$\Omega$,subjecttothehomogeneousboundarycondition$u = 0$on$\partial \Omega$. We prove that the unique solution to this Dirichlet problem is power-concave (precisely, 3/4 concave) and it is of class $C^1(\mathrm{\Omega })$. We then investigate the overdetermined Serrin-type problem obtained by adding the extra boundary condition $\mathrm{\mid }\mathrm{\nabla }u\mathrm{\mid }=a$ on $\mathrm{\partial }\mathrm{\Omega }$; by using a suitable $P$-function we prove that, if $\mathrm{\Omega }$ satisfies the same assumptions as above and in addition contains a ball with touches $\mathrm{\partial }\mathrm{\Omega }$ at two diametral points, then the existence of a solution to this Serrin-type problem implies that necessarily the cut locus and the high ridge of $\mathrm{\Omega }$ coincide. In turn, in dimension $n=2$, this entails that $\mathrm{\Omega }$ must be a stadium-like domain, and in particular it must be a ball in case its boundary is of class $C^2$.