Anisotropic total variation flow of non-divergence type on a higher dimensional torus

TL;DR

This paper extends viscosity solution theory to anisotropic total variation flow of non-divergence type on higher-dimensional tori, providing comparison, stability, and existence results for singular nonlinear parabolic problems.

Contribution

It introduces a novel extension of viscosity solutions to anisotropic total variation flows in higher dimensions, including comparison and existence theorems.

Findings

Established a comparison principle for the flow.

Proved stability under approximation by regularized problems.

Demonstrated existence for continuous initial data.

Abstract

We extend the theory of viscosity solutions to a class of very singular nonlinear parabolic problems of non-divergence form in a periodic domain of an arbitrary dimension with diffusion given by an anisotropic total variation energy. We give a proof of a comparison principle, an outline of a proof of the stability under approximation by regularized parabolic problems, and an existence theorem for general continuous initial data, which extend the results recently obtained by the authors.