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A duality-based approach to gradient flows of linear growth functionals

Wojciech Górny, José M. Mazón

TL;DR

The paper addresses gradient flows of linear-growth functionals on BV spaces, formulated via $F(u)=\int_\Omega f(x,Du)$. It introduces a duality-based framework that replaces multivalued subdifferentials with a field $\mathbf{z}\in X_2(\Omega)$ and employs Fenchel–Rockafellar duality to characterize the subdifferentials $\partial\mathcal{F_N}$ and $\partial\mathcal{F}_h$, proving existence and uniqueness of weak solutions for Neumann and Dirichlet problems under weak regularity assumptions on $f$. This yields a unified theory that covers the total variation flow and differentiable linear-growth functionals, while relaxing differentiability and growth requirements through the asymptotic function $f^0$. The results provide a robust variational-duality approach to linear-growth gradient flows, with the nonparametric area functional and TV flow as canonical examples, and they establish a streamlined, general existence theory applicable to geometric PDEs and variational problems with measure-valued gradients.

Abstract

We study gradient flows of general functionals with linear growth with very weak assumptions. Classical results concerning characterisation of solutions require differentiability of the Lagrangian, as for the time-dependent minimal surface equation, or a special form of the Lagrangian as in the total variation flow. We propose to study this problem using duality techniques, give a general definition of solutions and prove their existence and uniqueness. This approach also allows us to reduce the regularity and structure assumptions on the Lagrangian.

A duality-based approach to gradient flows of linear growth functionals

TL;DR

The paper addresses gradient flows of linear-growth functionals on BV spaces, formulated via . It introduces a duality-based framework that replaces multivalued subdifferentials with a field and employs Fenchel–Rockafellar duality to characterize the subdifferentials and , proving existence and uniqueness of weak solutions for Neumann and Dirichlet problems under weak regularity assumptions on . This yields a unified theory that covers the total variation flow and differentiable linear-growth functionals, while relaxing differentiability and growth requirements through the asymptotic function . The results provide a robust variational-duality approach to linear-growth gradient flows, with the nonparametric area functional and TV flow as canonical examples, and they establish a streamlined, general existence theory applicable to geometric PDEs and variational problems with measure-valued gradients.

Abstract

We study gradient flows of general functionals with linear growth with very weak assumptions. Classical results concerning characterisation of solutions require differentiability of the Lagrangian, as for the time-dependent minimal surface equation, or a special form of the Lagrangian as in the total variation flow. We propose to study this problem using duality techniques, give a general definition of solutions and prove their existence and uniqueness. This approach also allows us to reduce the regularity and structure assumptions on the Lagrangian.
Paper Structure (7 sections, 13 theorems, 212 equations)

This paper contains 7 sections, 13 theorems, 212 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. BV functions and Anzellotti pairings
  4. Function of a measure
  5. Estimates in terms of $f^0$
  6. The Neumann problem
  7. The Dirichlet Problem

Key Result

Proposition 2.3

The distribution $({\bf z},Du)$ is a Radon measure in $\Omega$. Moreover, for any Borel set $B \subseteq \Omega$. In particular, $({\bf z},Du)$ is absolutely continuous with respect to $\vert Du \vert$. Furthermore, so $({\bf z},Du)$ agrees on Sobolev functions with the dot product of ${\bf z}$ and $\nabla u$.

Theorems & Definitions (30)

  • Definition 2.1
  • Definition 2.2
  • Proposition 2.3
  • Theorem 2.4
  • Theorem 2.5
  • Theorem 2.6
  • Proposition 2.7
  • proof
  • Remark 2.8
  • Definition 3.1
  • ...and 20 more