Cut locus on compact manifolds and uniform semiconcavity estimates for a variational inequality

We study a family of gradient obstacle problems on a compact Riemannian manifold. We prove that the solutions of these free boundary problems are uniformly semiconcave and, as a consequence, we obtain some fine convergence results for the solutions and their free boundaries. More precisely, we show that the elastic and the λ-elastic sets of the solutions Hausdorff converge to the cut locus and the λ-cut locus of the manifold.