Gradient estimates for the Perona-Malik equation

We consider the Cauchy problem for the Perona–Malik equation in a bounded open set, with Neumann boundary conditions. In the one dimensional case, we prove some a priori estimates. Then we consider the semi-discrete scheme obtained by replacing the space derivatives by finite differences. Extending the previous estimates to the discrete setting we prove a compactness result for this scheme and we characterize the possible limits in some cases. Finally we give examples to show that the corresponding estimates are in general false in the more dimensional case.