The Perona-Malik problem: singular perturbation and semi-discrete approximation

We consider the Perona-Malik equation, which is a forward-backward parabolic equation that can be seen as the formal gradient-flow of a non-convex functional, and we regularize the problem in two different ways: by adding an higher order term or by space discretization.Concerning the higher order regularization, we present some recent results concerning the asymptotic behavior of minimizers of the regularized functional when a fidelity term is added. We show that these minimizers exhibit a multi-scale structure by characterizing the possible blow-up at different scales. The main results are obtained in the one-dimensional case, but some partial generalizations to any dimension are also provided.In the final chapter, we consider the discrete approximation of the dynamic problem in the one-dimensional case and we study the evolution of the maximum and of the total variation of all limits of discrete evolutions. We provide an example in which these quantities do not pass to the limit from the discrete to the continuous setting. Nevertheless, we show that these quantities inherit the same monotonicity properties that hold at the discrete level. These monotonicity results actually hold for a general class of one-dimensional evolution curves that we call uvw-evolutions.