Submitted Paper

Inserted: 28 nov 2024
Last Updated: 2 dec 2024

Year: 2024

Abstract:

We study a generalization of the manifold-valued Rudin-Osher-Fatemi (ROF) model, which involves an initial datum f mapping from a curved compact surface with smooth boundary to a complete, connected and smooth n-dimensional Riemannian manifold. We prove the existence and uniqueness of minimizers under curvature restrictions on the target and topological ones on the range of f. We obtain a series of regularity results on the associated PDE system of a relaxed functional with Neumann boundary condition. We apply these results to the ROF model to obtain Lipschitz regularity of minimizers without further requirements on the convexity of the boundary. Additionally, we provide a versions of the regularity statement of independent interest: for 1-dimensional domains (related to signal denoising), local Lipschitz regularity (meaningful for image processing) and Lipschitz regularity for a version of the Mosolov problem coming from fluid mechanics.

Keywords: Lipschitz regularity, Total variation functional, manifold-constrained PDE

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• ROF-manifolds-def.pdf