On the definition of solution to the total variation flow

TL;DR

This paper establishes the equivalence of different weak solution concepts for the total variation flow under certain boundary conditions, using a duality approach based on approximations of the total variation functional.

Contribution

It demonstrates the coincidence of weak solution notions for total variation flow through a duality argument and approximation techniques, clarifying theoretical foundations.

Findings

Weak solutions based on Anzellotti pairing and variational inequality coincide under boundary restrictions.

A duality result for total variation functional is established.

Approximation of total variation by area-type functionals is key to the proof.

Abstract

We show that the notions of weak solution to the total variation flow based on the Anzellotti pairing and the variational inequality coincide under some restrictions on the boundary data. The key ingredient in the argument is a duality result for the total variation functional, which is based on an approximation of the total variation by area-type functionals.