The Neumann and Dirichlet problems for the total variation flow in metric measure spaces

TL;DR

This paper investigates the existence, uniqueness, and asymptotic behavior of solutions to Neumann and Dirichlet problems for total variation flow in metric measure spaces, extending solution concepts to $L^1$ initial data.

Contribution

It introduces a new notion of solutions for the Neumann problem applicable to $L^1$ initial data and proves their existence and uniqueness.

Findings

Existence and uniqueness of weak solutions for both problems

Asymptotic behavior analysis of solutions

A new solution concept for $L^1$ initial data in Neumann problems

Abstract

We study the Neumann and Dirichlet problems for the total variation flow in metric measure spaces. We prove existence and uniqueness of weak solutions and study their asymptotic behaviour. Furthermore, in the Neumann problem we provide a notion of solutions which is valid for $L^1$ initial data, as well as prove their existence and uniqueness. Our main tools are the first-order linear differential structure due to Gigli and a version of the Gauss-Green formula.