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

TL;DR

This paper investigates gradient obstacle problems on compact Riemannian manifolds, establishing uniform semiconcavity of solutions and analyzing the convergence of free boundaries to the cut locus and related sets.

Contribution

It introduces uniform semiconcavity estimates for solutions of obstacle problems on manifolds and links free boundary convergence to the geometric concept of the cut locus.

Findings

Solutions are uniformly semiconcave.

Free boundaries Hausdorff converge to the cut locus.

Elastic and λ-elastic sets converge to cut and λ-cut loci.

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. 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.