Passing to the limit in maximal slope curves: from a regularized Perona-Malik equation to the total variation flow

TL;DR

This paper demonstrates that solutions of a regularized Perona-Malik equation converge over time to solutions of the total variation flow, establishing a connection between these models through a rigorous limit process.

Contribution

It provides a rigorous proof of convergence from the regularized Perona-Malik equation to the total variation flow using Gamma-convergence and gradient flow theory.

Findings

Global-in-time convergence in any space dimension

Convergence occurs in a slow time scale

The limit of gradient flows equals the gradient flow of the limit

Abstract

We prove that solutions of a mildly regularized Perona-Malik equation converge, in a slow time scale, to solutions of the total variation flow. The convergence result is global-in-time, and holds true in any space dimension. The proof is based on the general principle that "the limit of gradient-flows is the gradient-flow of the limit". To this end, we exploit a general result relating the Gamma-limit of a sequence of functionals to the limit of the corresponding maximal slope curves.