Variational solutions to the total variation flow on metric measure spaces

TL;DR

This paper develops a variational framework for analyzing the total variation flow on metric measure spaces, establishing conditions for solution continuity using advanced mathematical tools like De Giorgi classes and upper gradients.

Contribution

It introduces a novel variational approach to total variation flow on metric measure spaces, linking solution continuity to specific mathematical conditions.

Findings

Established a necessary and sufficient condition for solution continuity.

Applied De Giorgi classes and upper gradients in the analysis.

Extended total variation flow theory to metric measure spaces.

Abstract

We discuss a purely variational approach to the total variation flow on metric measure spaces with a doubling measure and a Poincar\'e inequality. We apply the concept of parabolic De Giorgi classes together with upper gradients, Newtonian spaces and functions of bounded variation to prove a necessary and sufficient condition for a variational solution to be continuous at a given point.