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

TL;DR

This paper introduces a duality-based framework for analyzing gradient flows of linear growth functionals, enabling existence and uniqueness results under minimal regularity assumptions.

Contribution

It develops a novel duality approach to study gradient flows with linear growth, relaxing traditional differentiability and structural requirements.

Findings

Established existence and uniqueness of solutions.

Reduced regularity assumptions on the Lagrangian.

Provided a general solution framework for linear growth functionals.

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.