Accepted Paper

Inserted: 15 jun 2020
Last Updated: 24 apr 2023

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

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.

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• cut_locus6_cvgmt.pdf