A symmetry problem for the infinity Laplacian

Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain $\Omega \subset \R ^n$ in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in $\Omega$ with constant source term admits a viscosity solution depending only on the distance from $\partial \Omega$. This problem was previously addressed and studied by Buttazzo and Kawohl in [7]. In the light of some geometrical achievements reached in our recent paper [14], we revisit the results obtained in [7] and we prove strengthened versions of them, where any regularity assumption on the domain and on the solution is removed. Our results require a delicate analysis based on viscosity methods. In particular, we need to build suitable viscosity test functions, whose construction involves a new estimate of the distance function $d_{\partial \Omega}$ near singular points.