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A distributional approach to nonlocal curvature flows

Filippo Cagnetti, Massimiliano Morini, Dario Reggiani

TL;DR

This work develops a nonlocal, distributional framework for curvature-driven flows by extending the distance-function approach to both Minkowski-type and $s$-fractional perimeters. It introduces nonlocal Cahn–Hoffman fields and a nonlocal divergence operator to formulate weak and level-set flows, and it proves key results including a comparison principle, the convergence of Almgren–Taylor–Wang minimizing movements, and global well-posedness for the nonlocal flows. The paper provides a unified treatment that covers crystalline-like and fractional-perimeter motions, and it demonstrates convergence to classical mean curvature flow as the nonlocal scale vanishes, with extensions to anisotropic variants. These contributions offer a robust analytic framework for nonlocal geometric evolutions applicable to image processing, materials science, and related fields, and they connect nonlocal models with established viscosity solutions in the fractional setting.

Abstract

In \cite{CMP17} a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the $(N-1)$-dimensional Minkowski pre-content.

A distributional approach to nonlocal curvature flows

TL;DR

This work develops a nonlocal, distributional framework for curvature-driven flows by extending the distance-function approach to both Minkowski-type and -fractional perimeters. It introduces nonlocal Cahn–Hoffman fields and a nonlocal divergence operator to formulate weak and level-set flows, and it proves key results including a comparison principle, the convergence of Almgren–Taylor–Wang minimizing movements, and global well-posedness for the nonlocal flows. The paper provides a unified treatment that covers crystalline-like and fractional-perimeter motions, and it demonstrates convergence to classical mean curvature flow as the nonlocal scale vanishes, with extensions to anisotropic variants. These contributions offer a robust analytic framework for nonlocal geometric evolutions applicable to image processing, materials science, and related fields, and they connect nonlocal models with established viscosity solutions in the fractional setting.

Abstract

In \cite{CMP17} a novel distributional approach has been introduced to provide a well-posed formulation of a class of crystalline mean curvature flows. In this paper, such an approach is extended to the nonlocal setting. Applications include the fractional mean curvature flow and the Minkowski flow; i.e., the geometric flow generated by the -dimensional Minkowski pre-content.

Paper Structure

This paper contains 18 sections, 18 theorems, 231 equations.

Table of Contents

  1. Introduction
  2. Minkowski pre-content type nonlocal perimeter
  3. The distance function approach.
  4. The distance function approach in the nonlocal setting.
  5. The $s$-fractional perimeter
  6. Final remarks
  7. Preliminaries
  8. The Minkowski--type mean curvature flow
  9. The generalised ROF problem
  10. Weak Formulation of the flow
  11. Minimizing movements
  12. Comparison with the evolution of balls
  13. Convergence of the scheme
  14. Comparison principle and well-posedness of the level set flow
  15. Regularised Minkowski type flows
  16. ...and 3 more sections

Key Result

Proposition 3.1

Let $h>0$. Then, for every $g\in \mathds{R} + L^2 (\mathds{R}^N)$ there exists a unique solution $u_h^g$ to the problem nlROF2, with $u_h^g \in \mathds{R} + L^2 (\mathds{R}^N)$. Moreover, if $g_1, g_2 \in \mathds{R} + L^2 (\mathds{R}^N)$ and $g_1 \leq g_2$ a.e. in $\mathds{R}^N$, then $u^{g_1}_h \le where $z_w$ stands for the evaluation of $z$ at the point $w$, and $d z_w(x,y)$ for the integration

Theorems & Definitions (52)

  • Proposition 3.1
  • proof
  • Theorem 3.2
  • Remark 3.3
  • Lemma 3.4
  • proof : Proof of Lemma \ref{['structure z']}
  • Remark 3.5
  • Remark 3.6
  • Lemma 3.7
  • proof
  • ...and 42 more