Inserted: 27 nov 2023
We continue the asymptotic analysis of minimizers for the singularly perturbed Perona-Malik functional in dimension one that we started in a previous paper, in which we had shown that the blow-up at a suitable scale of these minimizers converge to a staircase-looking piecewise constant function. We develop our analysis by considering blow-up at finer scales in both the horizontal and vertical regions of these staircases. In the vertical regime, we show that the transition between consecutive steps resembles a cubic polynomial. In the horizontal regime we show that minimizers are actually flat by providing a quantitative uniform estimate on their derivatives.