Calculus of Variations and Geometric Measure Theory

F. Générau - E. Oudet - B. Velichkov

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

created by velichkov on 15 Jun 2020
modified on 24 Apr 2023


Accepted Paper

Inserted: 15 jun 2020
Last Updated: 24 apr 2023

Journal: Arch. Rat. Mech. Anal.
Year: 2022

ArXiv: 2006.07222 PDF


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.