Almost classical solutions to the total variation flow

TL;DR

This paper introduces the concept of 'almost classical' solutions for the one-dimensional total variation flow, enabling the analysis of facet evolution and finite-time convergence to equilibrium.

Contribution

It presents a novel approach using 'almost classical' solutions to analyze facet dynamics and steady states in total variation flow with Dirichlet conditions.

Findings

Solutions reach equilibrium in finite time.

The approach handles irregular functions like x^2.

Numerical simulations validate the method.

Abstract

The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key element of our approach is the natural regularity determined by nonlinear elliptic operator, for which $x^2$ is an irregular function. Such a point of view allows us to construct solutions. We apply this idea to implement our approach to numerical simulations for typical initial data. Due to the nature of Dirichlet data any monotone function is an equilibrium. We prove that each solution reaches such steady state in a finite time.