Cut Locus on Compact Manifolds and Uniform Semiconcavity Estimates for a Variational Inequality

Abstract

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 lambda -elastic sets of the solutions Hausdorff converge to the cut locus and the lambda -cut locus of the manifold.