Convergence of discrete schemes for the Perona-Malik equation

We prove the convergence, up to a subsequence, of the spatial semidiscrete scheme for the one-dimensional Perona-Malik equation $u_t = (\phi'(u_x))_x$, $\phi(p) := \log(1+p^2)/2$, when the initial datum $\bar u$ is 1-Lipschitz out of a finite number of jump points, and we characterize the problem satisfied by the limit solution. In the more difficult case when $\bar u$ has a whole interval where $\phi''(\bar u_x)$ is negative, we construct a solution by a careful inspection of the behaviour of the approximating solutions in a space-time neighbourhood of the jump points. The limit solution u we obtain is the same as the one obtained by replacing $\phi(\cdot)$ with the truncated function $\min(\phi(\cdot),1)$, and it turns out that $u$ solves a free boundary problem. The free boundary consists of the points dividing the region where $|u_x| > 1$ from the region where $|u_x| \leq 1$. Finally, we consider the full space-time discretization (implicit in time) of the Perona-Malik equation, and we show that, if the time step is small with respect to the spatial grid h, then the limit is the same as the one obtained with the spatial semidiscrete scheme. On the other hand, if the time step is large with respect to h, then the limit solution equals $\bar u$, i.e., the standing solution of the convexified problem.