A symmetry problem for the infinity Laplacian

Aim of this paper is to prove necessary and sufficient conditions on the geometry of a domain Ω ⊂ Rn in order that the homogeneous Dirichlet problem for the infinity-Laplace equation in Ω with constant source term admits a viscosity solution depending only on the distance from ∂Ω. 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∂Ω near singular points.