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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 to1  3{\frac{1}{3}}   ), 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.

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