A quantitative variational analysis of the staircasing phenomenon for a second order regularization of the Perona-Malik functional

TL;DR

This paper analyzes the staircasing phenomenon in second order regularizations of the Perona-Malik functional in one dimension, revealing microstructure formation through Gamma-convergence and blow-up techniques.

Contribution

It provides a rigorous asymptotic analysis of minimizers showing staircasing, extending understanding of second order regularizations of the Perona-Malik functional.

Findings

Minimizers develop microstructures resembling piecewise constant functions.

Gamma-convergence characterizes the asymptotic behavior of the regularized functional.

The approach can be extended to more general models.

Abstract

We consider the Perona-Malik functional in dimension one, namely an integral functional whose Lagrangian is convex-concave with respect to the derivative, with a convexification that is identically zero. We approximate and regularize the functional by adding a term that depends on second order derivatives multiplied by a small coefficient. We investigate the asymptotic behavior of minima and minimizers as this small parameter vanishes. In particular, we show that minimizers exhibit the so-called staircasing phenomenon, namely they develop a sort of microstructure that looks like a piecewise constant function at a suitable scale. Our analysis relies on Gamma-convergence results for a rescaled functional, blow-up techniques, and a characterization of local minimizers for the limit problem. This approach can be extended to more general models.