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

Delta‐convex structure of the singular set of distance functions

“…Incidentally, we also define semi-concavity for later use. This definition is well-posed thanks to independence on the choice of the local chart, see [44,Proposition 2.6] and Lemma 2.1 (ii), (iii) (see also [51] and [30,Section 2.2] for semiconvex functions on curved spaces). Of course, any semi-concave function on a manifold is also delta-convex.…”