Inserted: 12 nov 2006
Last Updated: 19 nov 2006
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 $\vert u_x\vert>1$ from the region where $\vert u_x\vert \leq 1$. 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.