Remarks on Manifolds with Two-Sided Curvature Bounds

We discuss folklore statements about distance functions in manifolds with two-sided bounded curvature. The topics include regularity, subsets of positive reach and the cut locus.

Bernoulli’s Problem for the Infinity-Laplacian Near a Set with Positive Reach

We consider the exterior as well as the interior free-boundary Bernoulli problem associated with the infinity-Laplacian under a non-autonomous boundary condition. Recall that the Bernoulli problem involves two domains: one is given, the other is unknown. Concerning the exterior problem we assume that the given domain has a positive reach, and prove an existence and uniqueness result together with an explicit representation of the solution. Concerning the interior problem, we obtain a similar result under the assumption that the complement of the given domain has a positive reach. In particular, for the interior problem we show that uniqueness holds in contrast to the usual problem associated to the Laplace operator.

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 {\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.