Static, Quasistatic and Dynamic Analysis for Scaled Perona-Malik Functionals

We present an asymptotic description of local minimization problems, and of quasistatic and dynamic evolutions of discrete one-dimensional scaled Perona-Malik functionals. The scaling is chosen in such a way that these energies Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\varGamma $\end{document}-converge to the Mumford-Shah functional by a result by Morini and Negri. This continuum approximation still provides a good description of quasistatic and gradient-flow type evolutions, while it must be suitably corrected to maintain the pattern of local minima and to account for long-time evolution.