Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Lipschitz regularity for manifold-constrained ROF elliptic systems

Esther Cabezas-Rivas, Salvador Moll, Vicent Pallardó-Julià

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 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.

Lipschitz regularity for manifold-constrained ROF elliptic systems

Abstract

We study a generalization of the manifold-valued Rudin-Osher-Fatemi (ROF) model, which involves an initial datum mapping from a curved compact surface with smooth boundary to a complete, connected and smooth -dimensional Riemannian manifold. We prove the existence and uniqueness of minimizers under curvature restrictions on the target and topological ones on the range of . 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.

Paper Structure

This paper contains 32 sections, 29 theorems, 200 equations, 3 figures.

Table of Contents

  1. Introduction and main results
  2. A regularity problem within the framework of image denoising
  3. Main results in the perspective of the literature
  4. A geometric viewpoint to ground the naturality of the hypotheses
  5. About the meaningfulness of the generalization to manifolds
  6. Steady Mosolov problem
  7. Strategy for the proof of Lipschitz regularity
  8. Structure of the paper
  9. Notation, conventions and background material
  10. $p$-tension or rough $p$-Laplacian (intrinsic versus extrinsic viewpoint)
  11. $p$-energy on surfaces
  12. Extrinsic versus intrinsic Sobolev spaces
  13. Further notation and conventions
  14. Total variation functional and technical setup for the Euler-Lagrange equations
  15. Manifold constrained functionals with linear growth
  16. ...and 17 more sections

Key Result

Theorem 1.1

Suppose that $f(\overline\Sigma) \subset B_h(p, R)$ for some $R < R_\kappa$. Then there exists a minimizer $u$ of the functional $\mathcal{E}$, which also satisfies $u(\overline\Sigma) \subset B_h(p, R)$. If we further assume that $\kappa \leq 0$, then

Figures (3)

  • Figure 1: Schematic view of the idea of flattening the ends to construct a global coordinate system outside a geodesic ball.
  • Figure 2: Setup for the problem after locally flattening the boundary. The blue area denotes the neighborhood $\mathcal{U}$ in the left figure and $B_1^+$ in the right figure. The striped area represents $\Omega_r(\bar{x})$, the red segment indicates $\Gamma_1\cap \partial\Omega_r$, and the green segment marks the region where $u=v$.
  • Figure 3: Setting to apply Jacobi field comparison.

Theorems & Definitions (53)

  • Theorem 1.1
  • Corollary 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Lemma 2.1
  • Theorem 3.1
  • proof
  • Remark 3.2
  • Definition 3.3
  • Lemma 4.1
  • ...and 43 more