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A few topics on total variation flows

Yoshikazu Giga, Hirotoshi Kuroda, Michał Łasica

Abstract

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of calibrability, finite time extinction, and some regularity properties of solutions. We focus on the second-order flow (possibly with weights) and the fourth-order flow. We also discuss the fractional cases.

A few topics on total variation flows

Abstract

Total variation gradient flows are important in several applied fields, including image analysis and materials science. In this paper, we review a few basic topics including definition of a solution, explicit examples and the notion of calibrability, finite time extinction, and some regularity properties of solutions. We focus on the second-order flow (possibly with weights) and the fourth-order flow. We also discuss the fractional cases.
Paper Structure (10 sections, 23 theorems, 255 equations, 2 figures)

This paper contains 10 sections, 23 theorems, 255 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Definition of a solution
  3. Total variation flows
  4. Formulation by the Cahn--Hoffman vector field
  5. Calibrability
  6. Some simple explicit solutions
  7. Upper bounds for the extinction time
  8. Second and fourth-order problems
  9. Fractional case
  10. Regularity

Key Result

Proposition 2.1

Let $H$ be a (real) Hilbert space. Let $\mathcal{E}$ be a lower semicontinuous functional on $H$ with values in $(-\infty,\infty]$ and $\mathcal{E}\not\equiv\infty$. Then for any $u_0\in \overline{D(\mathcal{E})}$, there exists a unique $u\in C\left([0,\infty),H\right)$ with $u_t\in\bigcap_{\delta>0 If $u_0\in D(\mathcal{E})$, then $\delta=0$ is allowed.

Figures (2)

  • Figure 1: Examples of $U$
  • Figure 2: Profile of a solution when $n=2$

Theorems & Definitions (36)

  • Proposition 2.1
  • Definition 2.2
  • Theorem 2.3
  • Theorem 2.4
  • Remark 2.5
  • Lemma 2.6: ACM, Theorem 1.8
  • Theorem 2.7
  • Theorem 2.8
  • Theorem 2.9
  • Definition 2.10
  • ...and 26 more