An overdetermined problem for the infinity-Laplacian around a set of positive reach

We consider an overdetermined problem associated to an inhomogeneous infinity-Laplace equation. More precisely, the domain of the problem is required to contain a given compact set K of positive reach, and the boundary of the domain must lie within the reach of K. We look for a solution vanishing at the boundary and such that the outer derivative depends only on the distance from K. We prove that if the boundary gradient grows fast enough with respect to such distance (faster than the distance raised to 31 ), then the problem is solvable if and only if the domain is a tubular neighborhood of K, thus extending a previous result valid in the case when K is made up of a single point.