Lipschitz regularity for manifold-constrained ROF elliptic systems

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 variants 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.